Band 13, Heft 1

In this article, we study the following nonlinear Schrödinger system − Δ u 1 + V 1 ( x ) u 1 = α u 1 u 2 + μ u 1 , x ∈ R 4 , − Δ u 2 + V 2 ( x ) u 2 = α 2 u 1 2 + β u 2 2 + μ u 2 , x ∈ R 4 , \left\{\begin{array}{ll}-\Delta {u}_{1}+{V}_{1}\left(x){u}_{1}=\alpha {u}_{1}{u}_{2}+\mu {u}_{1},& x\in {{\mathbb{R}}}^{4},\\ -\Delta {u}_{2}+{V}_{2}\left(x){u}_{2}=\frac{\alpha }{2}{u}_{1}^{2}+\beta {u}_{2}^{2}+\mu {u}_{2},& x\in {{\mathbb{R}}}^{4},\end{array}\right. with the constraint ∫ R 4 ( u 1 2 + u 2 2 ) d x = 1 {\int }_{{{\mathbb{R}}}^{4}}\left({u}_{1}^{2}+{u}_{2}^{2}){\rm{d}}x=1 , where α > 0 \alpha \gt 0 and α > β \alpha \gt \beta , μ ∈ R \mu \in {\mathbb{R}} , V 1 ( x ) {V}_{1}\left(x) , and V 2 ( x ) {V}_{2}\left(x) are bounded functions. Under some mild assumptions on V 1 ( x ) {V}_{1}\left(x) and V 2 ( x ) {V}_{2}\left(x) , we prove the existence of normalized peak solutions by using the finite dimensional reduction method, combined with the local Pohozaev identities. Because of the interspecies interaction between the components, we aim to obtain some new technical estimates.

Consideration herein is the stability issue of peaked solitary wave solution for the modified Camassa-Holm-Novikov equation, which is derived from the shallow water theory. This wave configuration accommodates the ordered trains of the modified Camassa-Holm-Novikov-peaked solitary solution. With the application of conservation laws and the monotonicity property of the localized energy functionals, we prove the orbital stability of this wave profile in the H 1 ( R ) {H}^{1}\left({\mathbb{R}}) energy space according to the modulation argument.

In this study, we investigate the two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity: u t = ∇ ⋅ ( u θ − 1 ∇ u ) − χ ∇ ⋅ u v ∇ v , x ∈ Ω , t > 0 , v t = Δ v − v + u + g ( x , t ) , x ∈ Ω , t > 0 , ( ∗ ) \left\{\begin{array}{ll}{u}_{t}=\nabla \cdot \left({u}^{\theta -1}\nabla u)-\chi \nabla \cdot \left(\frac{u}{v}\nabla v\right),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {v}_{t}=\Delta v-v+u+g\left(x,t),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ \end{array}\right.\hspace{2.0em}\hspace{2.0em}\hspace{2.0em}\left(\ast ) in a bounded domain with smooth boundary. We present the global boundedness of weak solutions to the model ( ∗ \ast ) if θ > 3 2 \theta \gt \frac{3}{2} and (1.10)–(1.11). This result improves our recent work.

In this article, we are concerned about the existence, uniqueness, and nonexistence of the positive solution for: − Δ u − 1 2 ( x ⋅ ∇ u ) = μ h ( x ) u q − 1 + λ u − u p , x ∈ R N , u ( x ) → 0 , as ∣ x ∣ → + ∞ , \left\{\begin{array}{l}-\Delta u-\frac{1}{2}\left(x\cdot \nabla u)=\mu h\left(x){u}^{q-1}+\lambda u-{u}^{p},\hspace{1.0em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\to 0,\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| x| \to +\infty ,\end{array}\right. where N ⩾ 3 N\geqslant 3 , 0 < q < 1 0\lt q\lt 1 , λ > 0 \lambda \gt 0 , p > 1 p\gt 1 , μ > 0 \mu \gt 0 is a parameter and the function h ( x ) h\left(x) satisfies certain conditions. To start with, based on the variational argument and perturbation method, we obtain the existence and uniqueness of the positive solution for the aforementioned singular elliptic differential equation as λ > N 2 \lambda \gt \frac{N}{2} . In addition, there is no solution as λ ⩽ N 2 \lambda \leqslant \frac{N}{2} . Later, from an experimental point of view, we give the numerical solution of the aforementioned singular elliptic differential equation by means of a neural network in some special cases, which enrich the theoretical results. Our conclusions partially extend the results corresponding to the nonsingular case.

In this article, we consider the bounded variation capacity (BV capacity) and characterize the Sobolev-type inequalities related to BV functions in a general framework of strictly local Dirichlet spaces with a doubling measure via the BV capacity. Under a weak Bakry-Émery curvature-type condition, we give the connection between the Hausdorff measure and the Hausdorff capacity, and discover some capacitary inequalities and Maz’ya-Sobolev inequalities for BV functions. The De Giorgi characterization for total variation is also obtained with a quasi-Bakry-Émery curvature condition. It should be noted that the results in this article are proved if the Dirichlet space supports the weak ( 1 , 2 ) \left(1,2) -Poincaré inequality instead of the weak ( 1 , 1 ) \left(1,1) -Poincaré inequality compared with the results in the previous references.

A singular nonlinear differential equation z σ d w d z = a w + z w f ( z , w ) , {z}^{\sigma }\frac{{\rm{d}}w}{{\rm{d}}z}=aw+zwf\left(z,w), where σ > 1 \sigma \gt 1 , is considered in a neighbourhood of the point z = 0 z=0 located either in the complex plane C {\mathbb{C}} if σ \sigma is a natural number, in a Riemann surface of a rational function if σ \sigma is a rational number, or in the Riemann surface of logarithmic function if σ \sigma is an irrational number. It is assumed that w = w ( z ) w=w\left(z) , a ∈ C ⧹ { 0 } a\in {\mathbb{C}}\setminus \left\{0\right\} , and that the function f f is analytic in a neighbourhood of the origin in C × C {\mathbb{C}}\times {\mathbb{C}} . Considering σ \sigma to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions w = w ( z ) w=w\left(z) in a domain that is part of a neighbourhood of the point z = 0 z=0 in C {\mathbb{C}} or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the property lim z → 0 w ( z ) = 0 {\mathrm{lim}}_{z\to 0}w\left(z)=0 is proved and an asymptotic behaviour of w ( z ) w\left(z) is established. Several examples and figures illustrate the results derived. The blow-up phenomenon is discussed as well.

Let Ω ⊂ R n \Omega \subset {{\bf{R}}}^{n} be a smooth bounded domain. In this article, we prove a result of which the following is a by-product: Let q ∈ ] 0 , 1 [ q\in ]0,1{[} , α ∈ L ∞ ( Ω ) \alpha \in {L}^{\infty }\left(\Omega ) , with α > 0 \alpha \gt 0 , and k ∈ N k\in {\bf{N}} . Then, the problem − tan ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x Δ u = α ( x ) u q in Ω u > 0 in Ω u = 0 on ∂ Ω ( k − 1 ) π < ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x < ( k − 1 ) π + π 2 \left\{\begin{array}{ll}-\tan \left(\mathop{\displaystyle \int }\limits_{\Omega }| \nabla u\left(x){| }^{2}{\rm{d}}x\right)\Delta u=\alpha \left(x){u}^{q}\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega \\ u\gt 0\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega \\ u=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega \\ \left(k-1)\pi \lt \mathop{\displaystyle \int }\limits_{\Omega }| \nabla u\left(x){| }^{2}{\rm{d}}x\lt \left(k-1)\pi +\frac{\pi }{2}\hspace{1.0em}\end{array}\right. has a unique weak solution u ˜ \tilde{u} , which is the unique global minimum in H 0 1 ( Ω ) {H}_{0}^{1}\left(\Omega ) of the functional u → 1 2 tan ∫ Ω ∣ ∇ u ˜ ( x ) ∣ 2 d x ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x − 1 q + 1 ∫ Ω α ( x ) ∣ u + ( x ) ∣ q + 1 d x , u\to \frac{1}{2}\tan \left(\mathop{\int }\limits_{\Omega }| \nabla \tilde{u}\left(x){| }^{2}{\rm{d}}x\right)\mathop{\int }\limits_{\Omega }| \nabla u\left(x){| }^{2}{\rm{d}}x-\frac{1}{q+1}\mathop{\int }\limits_{\Omega }\alpha \left(x)| {u}^{+}\left(x){| }^{q+1}{\rm{d}}x, where u + = max { 0 , u } {u}^{+}=\max \left\{0,u\right\} .

In this paper, we are concerned with the following fractional relativistic Schrödinger equation with critical growth: ( − Δ + m 2 ) s u + V ( ε x ) u = f ( u ) + u 2 s * − 1 in R N , u ∈ H s ( R N ) , u > 0 in R N , \left\{\begin{array}{ll}{\left(-\Delta +{m}^{2})}^{s}u+V\left(\varepsilon x)u=f\left(u)+{u}^{{2}_{s}^{* }-1}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{s}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}u\gt 0& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array}\right. where ε > 0 \varepsilon \gt 0 is a small parameter, s ∈ ( 0 , 1 ) s\in \left(0,1) , m > 0 m\gt 0 , N > 2 s N\gt 2s , 2 s * = 2 N N − 2 s {2}_{s}^{* }=\frac{2N}{N-2s} is the fractional critical exponent, ( − Δ + m 2 ) s {\left(-\Delta +{m}^{2})}^{s} is the fractional relativistic Schrödinger operator, V : R N → R V:{{\mathbb{R}}}^{N}\to {\mathbb{R}} is a continuous potential, and f : R → R f:{\mathbb{R}}\to {\mathbb{R}} is a superlinear continuous nonlinearity with subcritical growth at infinity. Under suitable assumptions on the potential V V , we construct a family of positive solutions u ε ∈ H s ( R N ) {u}_{\varepsilon }\in {H}^{s}\left({{\mathbb{R}}}^{N}) , with exponential decay, which concentrates around a local minimum of V V as ε → 0 \varepsilon \to 0 .

We show that the classical strong maximum principle, concerning positive supersolutions of linear elliptic equations vanishing on the boundary of the domain can be extended, under suitable conditions, to the case in which the forcing term is sign-changing. In addition, for the case of solutions, the normal derivative on the boundary may also vanish on the boundary (definition of flat solution). This leads to examples in which the unique continuation property fails. As a first application, we show the existence of positive solutions for a sublinear semilinear elliptic problem of indefinite sign. A second application, concerning the positivity of solutions of the linear heat equation, for some large values of time, with forcing and/or initial datum changing sign, is also given.

In this article, we consider the parabolic equations with nonlocal Monge-Ampère operators ∂ u ∂ t ( x , t ) − D s θ u ( x , t ) = f ( u ( x , t ) ) , ( x , t ) ∈ R + n × R . \frac{\partial u}{\partial t}\left(x,t)-{D}_{s}^{\theta }u\left(x,t)=f\left(u\left(x,t)),\hspace{1.0em}\left(x,t)\in {{\mathbb{R}}}_{+}^{n}\times {\mathbb{R}}. We first prove the narrow region principle and maximal principle for antisymmetric functions, under the condition that u u is uniformly bounded, which weaken the general decay condition u → 0 u\to 0 at infinity. Then, the monotonicity of positive solutions is established using the method of moving planes.

We deal with existence and uniqueness of nonnegative solutions to: − Δ u = f ( x ) , in Ω , ∂ u ∂ ν + λ ( x ) u = g ( x ) u η , on ∂ Ω , \left\{\begin{array}{ll}-\Delta u=f\left(x),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ \frac{\partial u}{\partial \nu }+\lambda \left(x)u=\frac{g\left(x)}{{u}^{\eta }},\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where η ≥ 0 \eta \ge 0 and f , λ f,\lambda , and g g are the nonnegative integrable functions. The set Ω ⊂ R N ( N > 2 ) \Omega \subset {{\mathbb{R}}}^{N}\left(N\gt 2) is open and bounded with smooth boundary, and ν \nu denotes its unit outward normal vector. More generally, we handle equations driven by monotone operators of p p -Laplacian type jointly with nonlinear boundary conditions. We prove the existence of an entropy solution and check that, under natural assumptions, this solution is unique. Among other features, we study the regularizing effect given to the solution by both the absorption and the nonlinear boundary term.

In this article, we study the following quasilinear equation with nonlocal nonlinearity − Δ u − κ u Δ ( u 2 ) + λ u = ( ∣ x ∣ − μ * F ( u ) ) f ( u ) , in R N , -\Delta u-\kappa u\Delta \left({u}^{2})+\lambda u=\left({| x| }^{-\mu }* F\left(u))f\left(u),\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where κ \kappa is a parameter, N ≥ 3 N\ge 3 , μ ∈ ( 0 , N ) \mu \in \left(0,N) , F ( t ) = ∫ 0 t f ( s ) d s F\left(t)={\int }_{0}^{t}f\left(s){\rm{d}}s , and λ \lambda is a positive constant. We are going to analyze two cases: the L 2 {L}_{2} -norm of the solution is not confirmed and the L 2 {L}_{2} -norm of the solution is prescribed. Under the almost optimal assumptions on f f , we obtain the existence of a sequence of radial solutions for two cases.

This article is devoted to a global Calderón-Zygmund estimate in the framework of Lorentz spaces for the m m -order gradients of weak solution to a higher-order elliptic equation with p p -growth. We prove the main result based on a proper power decay estimation of the upper-level set by the principle of layer cake representation for the L γ , q {L}^{\gamma ,q} -estimate of D m u {D}^{m}u , while the coefficient satisfies a small BMO semi-norm and the boundary of underlying domain is flat in the sense of Reifenberg. In particular, a tricky ingredient is to establish the normal component of higher derivatives controlled by the horizontal component of higher derivatives of solutions in the neighborhood at any boundary point, which is achieved by comparing the solution under consideration with that for some reference problems.

The compressible Navier-Stokes-Smoluchowski equations under investigation concern the behavior of the mixture of fluid and particles at a macroscopic scale. We devote to the existence of the global classical solution near the stationary solution based on the energy method under weaker conditions imposed on the external potential compared with Chen et al. (Global existence and time–decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5287–5307). Under further assumptions that the stationary solution ( ρ s ( x ) , 0 , 0 ) T {\left({\rho }_{s}\left(x),0,0)}^{T} is in a small neighborhood of the constant state ( ρ ¯ , 0 , 0 ) T {\left(\bar{\rho },0,0)}^{T} at infinity, we also obtain the time decay rates of the solution by the combination of the energy method and the linear L p {L}^{p} - L q {L}^{q} decay estimates.

This article is devoted to the study of the initial boundary value problem for a mixed pseudo-parabolic r ( x ) r\left(x) -Laplacian-type equation. First, by employing the imbedding theorems, the theory of potential wells, and the Galerkin method, we establish the existence and uniqueness of global solutions with subcritical initial energy, critical initial energy, and supercritical initial energy, respectively. Then, we obtain the decay estimate of global solutions with sub-sharp-critical initial energy, sharp-critical initial energy, and supercritical initial energy, respectively. For supercritical initial energy, we also need to analyze the properties of ω \omega -limits of solutions. Finally, we discuss the finite-time blowup of solutions with sub-sharp-critical initial energy and sharp-critical initial energy, respectively.

In this article, we consider the multiplicity of positive solutions for a static Schrödinger-Poisson-Slater equation of the type − Δ u + u 2 ∗ 1 ∣ 4 π x ∣ u = μ f ( x ) ∣ u ∣ p − 2 u + g ( x ) ∣ u ∣ 4 u in R 3 , -\Delta u+\left({u}^{2}\ast \frac{1}{| 4\pi x| }\right)u=\mu f\left(x){| u| }^{p-2}u+g\left(x){| u| }^{4}u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}, where μ > 0 \mu \gt 0 , 1 < p < 2 1\lt p\lt 2 , f ∈ L 6 6 − p ( R 3 ) f\in {L}^{\tfrac{6}{6-p}}\left({{\mathbb{R}}}^{3}) , and f , g ∈ C ( R 3 , R + ) f,g\in C\left({{\mathbb{R}}}^{3},{{\mathbb{R}}}^{+}) . Using Ekeland’s variational principle and a measure representation concentration-compactness of Lions, when g g has one local maximum point, we obtain two positive solutions for μ > 0 \mu \gt 0 small; while g g has k k strict local maximum points, we prove that the equation has at least k + 1 k+1 distinct positive solutions for μ > 0 \mu \gt 0 small by the Nehari manifold. Moreover, we show that one of the solutions is a ground state solution.

In this article, we consider a weakly coupled p p -Laplacian system of a Bernoulli-type free boundary problem, through minimization of a corresponding functional. We prove various properties of any local minimizer and the corresponding free boundary.

This article is concerned with the following Hamiltonian elliptic system: − ε 2 Δ u + ε b → ⋅ ∇ u + u + V ( x ) v = H v ( u , v ) in R N , − ε 2 Δ v − ε b → ⋅ ∇ v + v + V ( x ) u = H u ( u , v ) in R N , \left\{\begin{array}{l}-{\varepsilon }^{2}\Delta u+\varepsilon \overrightarrow{b}\cdot \nabla u+u+V\left(x)v={H}_{v}\left(u,v)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ -{\varepsilon }^{2}\Delta v-\varepsilon \overrightarrow{b}\cdot \nabla v+v+V\left(x)u={H}_{u}\left(u,v)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array}\right. where ε > 0 \varepsilon \gt 0 is a small parameter, V V is a potential function, and H H is a super-quadratic sub-critical Hamiltonian. Applying suitable variational arguments and refined analysis techniques, we construct a new multiplicity result of semiclassical solutions which depends on the number of global minimum points of V V . This result indicates how the shape of the graph of V V affects the number of semiclassical solutions.

In this article, our main aim is to investigate the existence of radial k k -convex solutions for the following Dirichlet system with k k -Hessian operators: S k ( D 2 u ) = λ 1 ν 1 ( ∣ x ∣ ) ( − u ) p 1 ( − v ) q 1 in ℬ ( R ) , S k ( D 2 v ) = λ 2 ν 2 ( ∣ x ∣ ) ( − u ) p 2 ( − v ) q 2 in ℬ ( R ) , u = v = 0 on ∂ ℬ ( R ) . \left\{\begin{array}{ll}{S}_{k}\left({D}^{2}u)={\lambda }_{1}{\nu }_{1}\left(| x| ){\left(-u)}^{{p}_{1}}{\left(-v)}^{{q}_{1}}& {\rm{in}}\hspace{1em}{\mathcal{ {\mathcal B} }}\left(R),\\ {S}_{k}\left({D}^{2}v)={\lambda }_{2}{\nu }_{2}\left(| x| ){\left(-u)}^{{p}_{2}}{\left(-v)}^{{q}_{2}}& {\rm{in}}\hspace{1em}{\mathcal{ {\mathcal B} }}\left(R),\\ u=v=0& {\rm{on}}\hspace{1em}\partial {\mathcal{ {\mathcal B} }}\left(R).\end{array}\right. Here, u p 1 v q 1 {u}^{{p}_{1}}{v}^{{q}_{1}} is called a Lane-Emden type nonlinearity. The weight functions ν 1 , ν 2 ∈ C ( [ 0 , R ] , [ 0 , ∞ ) ) {\nu }_{1},{\nu }_{2}\in C\left(\left[0,R],\left[0,\infty )) with ν 1 ( r ) > 0 < ν 2 ( r ) {\nu }_{1}\left(r)\gt 0\lt {\nu }_{2}\left(r) for all r ∈ ( 0 , R ] r\in \left(0,R] , p 1 , q 2 {p}_{1},{q}_{2} are nonnegative and q 1 , p 2 {q}_{1},{p}_{2} are positive exponents, ℬ ( R ) = { x ∈ R N : ∣ x ∣ < R } {\mathcal{ {\mathcal B} }}\left(R)=\left\{x\in {{\mathbb{R}}}^{N}:| x| \lt R\right\} , N ≥ 2 N\ge 2 is an integer, N 2 ≤ k ≤ N \frac{N}{2}\le k\le N . In order to achieve our main goal, we first study the existence of radial k k -convex solutions of the above-mentioned systems with general nonlinear terms by using the upper and lower solution method and Leray-Schauder degree. Based on this, by constructing a continuous curve, which divides the first quadrant into two disjoint sets, we obtain the existence and multiplicity of radial k k -convex solutions for the system depending on the parameters λ 1 {\lambda }_{1} , λ 2 {\lambda }_{2} and the continuous curve.

In this article, we consider the following elliptic system of Hamiltonian-type on a bounded domain: − Δ u = K 1 ( ∣ y ∣ ) ∣ v ∣ p − 1 v , in B 1 ( 0 ) , − Δ v = K 2 ( ∣ y ∣ ) ∣ u ∣ q − 1 u , in B 1 ( 0 ) , u = v = 0 on ∂ B 1 ( 0 ) , \left\{\begin{array}{ll}-\Delta u={K}_{1}\left(| y| ){| v| }^{p-1}v,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ -\Delta v={K}_{2}\left(| y| ){| u| }^{q-1}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ u=v=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}_{1}\left(0),\end{array}\right. where K 1 ( r ) {K}_{1}\left(r) and K 2 ( r ) {K}_{2}\left(r) are positive bounded functions defined in [ 0 , 1 ] \left[0,1] , B 1 ( 0 ) {B}_{1}\left(0) is the unit ball in R N {{\mathbb{R}}}^{N} , and ( p , q ) \left(p,q) is a pair of positive numbers lying on the critical hyperbola 1 p + 1 + 1 q + 1 = N − 2 N . \frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}. Under some suitable further assumptions on the functions K 1 ( r ) {K}_{1}\left(r) and K 2 ( r ) {K}_{2}\left(r) , we prove the existence of infinitely many nonradial positive solutions whose energy can be made arbitrarily large. Our proof is based on the reduction method. The most ingredients of the article are using the Green representation and estimating the Green function and its regular part very carefully. For this purpose, some more extra ideas and techniques are needed. We believe that our method and techniques can be applied to other related problems.

This article is devoted to the existence of optimal controls in various control problems associated with a novel nonlinear interface problem on an unbounded domain with non-monotone set-valued transmission conditions. This interface problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models non-monotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which, however, lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting. Boundary integral methods lead to another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Broadening the scope of this article, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results; moreover, we establish a stability result with respect to the extended real-valued function as a parameter. Based on the latter stability result, we prove the existence of optimal controls for four kinds of optimal control problems: distributed control on the bounded domain, boundary control, simultaneous boundary-distributed control governed by the interface problem, and control of the obstacle driven by a related bilateral obstacle interface problem.

This work is devoted to study the convection–reaction–diffusion behavior of contaminant in the recovered fracturing fluid which flows in the wellbore from shale gas reservoir. First, we apply various constitutive laws for generalized non-Newtonian fluids, diffusion principles, and friction relations to formulate the recovered fracturing fluid model. The latter is a partial differential system composed of a nonlinear and nonsmooth stationary incompressible Navier-Stokes equation with a multivalued friction boundary condition, and a nonlinear convection–reaction–diffusion equation with mixed Neumann boundary conditions. Then, we provide the weak formulation of the fluid model which is a hemivariational inequality driven by a nonlinear variational equation. We establish existence of solutions to the recovered fracturing fluid model via a surjectivity theorem for multivalued operators combined with an alternative iterative method and elements of nonsmooth analysis.

Using the Ekeland variational principle and the mountain pass lemma we prove some properties about potential homeomorphisms between a real Banach space and its dual. In particular, we show that a locally monotone homeomorphism is necessarily strictly monotone if it is potential.

In this article, we study the following fractional Schrödinger-Poisson system: ε 2 s ( − Δ ) s u + V ( x ) u + ϕ u = f ( u ) + ∣ u ∣ 2 s * − 2 u , in R 3 , ε 2 t ( − Δ ) t ϕ = u 2 , in R 3 , \left\{\begin{array}{ll}{\varepsilon }^{2s}{\left(-\Delta )}^{s}u+V\left(x)u+\phi u=f\left(u)+{| u| }^{{2}_{s}^{* }-2}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ {\varepsilon }^{2t}{\left(-\Delta )}^{t}\phi ={u}^{2},\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\end{array}\right. where ε > 0 \varepsilon \gt 0 is a small parameter, 0 < s , t < 1 , 2 s + 2 t > 3 0\lt s,t\lt 1,2s+2t\gt 3 , and 2 s * = 6 3 − 2 s {2}_{s}^{* }=\frac{6}{3-2s} is the critical Sobolev exponent in dimension 3. By assuming that V V is weakly differentiable and f ∈ C ( R , R ) f\in {\mathcal{C}}\left({\mathbb{R}},{\mathbb{R}}) satisfies some lower order perturbations, we show that there exists a constant ε 0 > 0 {\varepsilon }_{0}\gt 0 such that for all ε ∈ ( 0 , ε 0 ] \varepsilon \in (0,{\varepsilon }_{0}] , the above system has a semiclassical Nehari-Pohozaev-type ground state solution v ˆ ε {\hat{v}}_{\varepsilon } . Moreover, the decay estimate and asymptotic behavior of { v ˆ ε } \left\{{\hat{v}}_{\varepsilon }\right\} are also investigated as ε → 0 \varepsilon \to 0 . Our results generalize and improve the ones in Liu and Zhang and Ambrosio, and some other relevant literatures.

This article is concerned with the structure of solutions to the elliptic problem for a Hénon-type equation with a forcing term: − Δ u = α ( x ) u p + κ μ , in R N , u > 0 , in R N , ( P κ ) \hspace{11.3em}-\Delta u=\alpha \left(x){u}^{p}+\kappa \mu ,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}u\gt 0,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{13.0em}\left({{\rm{P}}}_{\kappa }) where N ≥ 3 N\ge 3 , p > 1 p\gt 1 , κ > 0 \kappa \gt 0 , and α \alpha is a positive continuous function in R N \ { 0 } {{\mathbb{R}}}^{N}\setminus \left\{0\right\} , and μ \mu is a nonnegative Radon measure in R N {{\mathbb{R}}}^{N} . Under suitable assumptions on the exponent p p , the coefficient α \alpha , and the forcing term μ \mu , we give a complete classification of the existence/nonexistence of solutions to problem ( P κ {{\rm{P}}}_{\kappa } ) with respect to κ \kappa .

We consider the blow-up solutions for the Gross-Pitaveskii equation modeling the attractive Boes-Einstein condensate. First, a new variational characteristic is established by computing the best constant of a generalized Gagliardo-Nirenberg inequality. Then, a lower bound on blow-up rate and a new concentration phenomenon of blow-up solutions are obtained in the L 2 {L}^{2} supercritical case. Finally, in the L 2 {L}^{2} critical case, a delicate limit of blow-up solutions is analyzed.

In this article, we consider the existence of solutions for nonlinear elliptic equations of the form − Δ u + V ( ∣ x ∣ ) u = Q ( ∣ x ∣ ) f ( u ) , x ∈ R 2 , -\Delta u+V\left(| x| )u=Q\left(| x| )f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{2}, where the nonlinear term f ( s ) f\left(s) has critical exponential growth which behaves like e α s 2 {e}^{\alpha {s}^{2}} , the radial potentials V , Q : R + → R V,Q:{{\mathbb{R}}}^{+}\to {\mathbb{R}} are unbounded, singular at the origin or decaying to zero at infinity. By combining the variational methods, Trudinger-Moser inequality, and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a Mountain-pass-type solution for the above problem under some weak assumptions.

We study the following fractional Choquard equation ( − Δ ) s u + u ∣ x ∣ θ = ( I α * F ( u ) ) f ( u ) , x ∈ R N , {\left(-\Delta )}^{s}u+\frac{u}{{| x| }^{\theta }}=({I}_{\alpha }* F\left(u))f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ⩾ 3 N\geqslant 3 , s ∈ 1 2 , 1 s\in \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2},1\right) , α ∈ ( 0 , N ) \alpha \in \left(0,N) , θ ∈ ( 0 , 2 s ) \theta \in \left(0,2s) , and I α {I}_{\alpha } is the Riesz potential. The main purpose of this article is twofold. We first study the regularity of weak solutions for the aforementioned equation with critical nonlinearity, which extends the results of θ = 0 \theta =0 in Moroz-Van Schaftingen [ Existence of groundstates for a class of nonlinear Choquardequations , Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579]. Then, as an application of the regularity results, we establish the existence of ground state solutions for above equation with the nonlinearity involving embedding top and bottom indices, which is related to the Hardy-Littlewood-Sobolev inequality and singular term 1 ∣ x ∣ θ \frac{1}{{| x| }^{\theta }} . It is worth noting that our approach is not involving the concentration-compactness principle.

In this study, we investigate some existence results for double phase anisotropic variational problems involving critical growth. We first establish a Lions-type concentration-compactness principle and its variant at infinity for the solution space, which are our independent interests. Using these results, we obtain a nontrivial nonnegative solution to problems of generalized concave-convex type. We also obtain infinitely many solutions when the nonlinear term is symmetric. Our results are new even for the p ( ⋅ ) p\left(\cdot ) -Laplace equations.

We study the existence and nonexistence of normalized solutions for the nonlinear Schrödinger equation with mixed nonlocal nonlinearities: − Δ u = λ u + μ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u + ( I α ∗ ∣ u ∣ q ) ∣ u ∣ q − 2 u in R N , ∫ R N ∣ u ∣ 2 d x = c , \left\{\begin{array}{ll}-\Delta u=\lambda u+\mu \left({I}_{\alpha }\ast {| u| }^{p}){| u| }^{p-2}u+\left({I}_{\alpha }\ast {| u| }^{q}){| u| }^{q-2}u& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x=c,& \end{array}\right. where N ≥ 1 , N + α N ≤ p < q ≤ N + α + 2 N N\ge 1,\frac{N+\alpha }{N}\le p\lt q\le \frac{N+\alpha +2}{N} , the parameter μ ∈ R \mu \in {\mathbb{R}} and λ \lambda is a Lagrange multiplier. Furthermore, we prove the relationship between minimizers and ground state solutions under the Nehari manifold, which seems to be the first result in the nonlocal context.

We establish an Amann-Zehnder-type result for resonance systems of quasilinear elliptic equations with homogeneous Dirichlet boundary conditions, involving nonlinearities growing asymptotically ( p , q ) \left(p,q) -linear at infinity. The proof relies on a cohomological linking in a product Banach space where the properties of cones of the sublevels are missing, differently from the single quasilinear equation. We also perform critical group computations of the energy functional at the origin, in spite of the lack of its C 2 {C}^{2} regularity, to exclude that the found mini-max solution is trivial. Finally, we furnish a local condition which guarantees that the found solution is not semi-trivial.

We investigate the existence and stability of standing waves with prescribed mass c > 0 c\gt 0 for Choquard equation with noncritical rotation in Bose-Einstein condensation. Then, we consider the mass collapse behavior of standing waves, the ratio of energy to mass and the Lagrange multiplier, as c → 0 + c\to {0}^{+} . Our results extend the existing results.

In this article, we mainly prove low regularity conservation laws for the Fokas-Lenells equation in Besov spaces with small initial data both on the line and on the circle. We develop a new technique in Fourier analysis and complex analysis to obtain the a priori estimates. It is based on the perturbation determinant associated with the Lax pair introduced by Killip, Vişan, and Zhang for completely integrable dispersive partial differential equations. Additionally, we also utilize the perturbation determinant to derive the global a priori estimates for the Schwartz solutions to the Camassa-Holm (CH) equation in H 1 {H}^{1} . Even though the energy conservation law of the CH equation is a fact known to all, the perturbation determinant method indicates that we cannot get any conserved quantities for the CH equation in H k {H}^{k} except k = 1 k=1 .

In this article, we study global multiplicity result for a class of modified quasilinear singular equations involving the critical exponential growth: − Δ u − Δ ( u 2 ) u = λ ( α ( x ) u − q + f ( x , u ) ) in Ω , u > 0 in Ω , u = 0 on ∂ Ω , \left\{\begin{array}{rcl}-\Delta u-\Delta \left({u}^{2})u& =& \lambda (\alpha \left(x){u}^{-q}+f\left(x,u))\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u& \gt & 0\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u& =& 0\hspace{1em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Ω \Omega is a smooth bounded domain in R 2 {{\mathbb{R}}}^{2} , 0 < q < 3 0\lt q\lt 3 , and α : Ω → ( 0 , + ∞ ) \alpha :\Omega \to \left(0,+\infty ) such that α ∈ L ∞ ( Ω ) \alpha \in {L}^{\infty }\left(\Omega ) . The function f : Ω × R → R f:\Omega \times {\mathbb{R}}\to {\mathbb{R}} is continuous and enjoys critical exponential growth of the Trudinger-Moser type. Using a sub-super solution method, we show that there exists some Λ * > 0 {\Lambda }^{* }\gt 0 such that for all λ ∈ ( 0 , Λ * ) \lambda \in \left(0,{\Lambda }^{* }) , the problem has at least two positive solutions, for λ = Λ * \lambda ={\Lambda }^{* } , the problem achieves at least one positive solution for λ > Λ * , \lambda \gt {\Lambda }^{* }, and the problem has no solutions.

This article is devoted to studying the existence of positive solutions to the following fractional Choquard equation: ( − Δ ) s u + u = ∫ Ω ∣ u ( y ) ∣ p ∣ x − y ∣ N − α d y ∣ u ∣ p − 2 u + ε ∫ Ω ∣ u ( y ) ∣ 2 α , s * ∣ x − y ∣ N − α d y ∣ u ∣ 2 α , s * − 2 u , in Ω , u = 0 , on R N \ Ω , \left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+u=\left(\mathop{\displaystyle \int }\limits_{\Omega }\frac{{| u(y)| }^{p}}{{| x-y| }^{N-\alpha }}{\rm{d}}y\right){| u| }^{p-2}u+\varepsilon \left(\mathop{\displaystyle \int }\limits_{\Omega }\frac{{| u(y)| }^{{2}_{\alpha ,s}^{* }}}{{| x-y| }^{N-\alpha }}{\rm{d}}y\right){| u| }^{{2}_{\alpha ,s}^{* }-2}u,& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0,& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\end{array}\right. where Ω \Omega is an exterior domain with smooth boundary ∂ Ω ≠ ∅ \partial \Omega \ne \varnothing such that R N \ Ω {{\mathbb{R}}}^{N}\backslash \Omega is bounded, N > 2 s , 2 < p < 2 α , s * N\gt 2s,2\lt p\lt {2}_{\alpha ,s}^{* } , 2 α , s * ≔ N + α N − 2 s {2}_{\alpha ,s}^{* }:= \frac{N+\alpha }{N-2s} is the fractional Hardy-Littlewood-Sobolev critical exponent, and ε > 0 \varepsilon \gt 0 is a parameter. We establish the limit profiles and uniqueness of positive radial ground-states for the limit equation without the critical exponent as α \alpha sufficiently close to N N . Then, combining variational method, barycentric functions, and Brouwer degree theory, we determine the existence of positive bound-state solutions provided that ε > 0 \varepsilon \gt 0 is sufficiently small.

We give a proof of the distribution property of limit cycles in so-called quadratic systems. We prove that the possible limit cycle distributions are either ( n , 0 ) \left(n,0) or ( n , 1 ) \left(n,1) (where n ∈ { 0 } ∪ N n\in \left\{0\right\}\cup {\mathbb{N}} ). The aim of this article is to simplify and fill gaps in the original proof by Zhang ( On the distribution and number of limit cycles for quadratic systems with two foci , Qual. Theory Dyn. Sys. 3 (2002), 437–463). The sixteenth Hilbert problem asks for an upper bound for the number of limit cycles in polynomial systems H ( n ) H\left(n) , where n n is the maximum degree of the polynomial defining the system. A consequence of the distribution property is that it reduces the study of H ( 2 ) H\left(2) to the study of the maximum number of limit cycles surrounding one singularity.

The aim of this article is to study regularity properties of a local minimizer of a double phase functional of type ℱ ( u ) ≔ ∫ Ω ( ∣ D u ∣ p ( x ) + a ( x ) ∣ D u ∣ p ( x ) log ( e + ∣ D u ∣ ) ) d x , {\mathcal{ {\mathcal F} }}\left(u):= \mathop{\int }\limits_{\Omega }({| Du| }^{p\left(x)}+a\left(x){| Du| }^{p\left(x)}\log \left(e+| Du| )){\rm{d}}x, being p ( x ) , a ( x ) p\left(x),a\left(x) log-continuous functions with p ( x ) > 1 p\left(x)\gt 1 , a ≥ 0 a\ge 0 . Double phase functionals ∫ ( ∣ D u ∣ p + a ( x ) ∣ D u ∣ q ) d x \int ({| Du| }^{p}+a\left(x){| Du| }^{q}){\rm{d}}x , with constant exponents p p and q q ( q ≥ p ≥ 1 ) \left(q\ge p\ge 1) , appeared in the papers by Zhikov, and C 1 , α {C}^{1,\alpha } -regularity of their minimizers was given by Colombo and Mingione. Later, by Baroni, Colombo, and Mingione, the above type functionals with logarithm but with constant exponent, regularity properties were given. They obtained sharp regularity results for minimizers of such functionals. In this article, we treat the case that the exponents are functions of p ( x ) p\left(x) and partly generalize their regularity results.

In this paper, we study coupled elliptic systems with gradient dependent right-hand sides and nonlinear boundary conditions, where the left-hand sides are driven by so-called double phase operators. By applying a surjectivity result for pseudomonotone operators along with an equivalent norm in the function space, we prove that the system has at least one nontrivial solution under very general assumptions on the data. Under slightly stronger conditions, we are also able to show that this solution is unique. As a result of independent interest, we further prove the boundedness of solutions to such elliptic systems by employing Moser’s iteration scheme.

In this article, we investigate the following nonlinear magnetic Schrödinger equations: ( − i ∇ + A ( x ) ) 2 u + V ( x ) u = f 1 ( x , ∣ v ∣ 2 ) v , ( − i ∇ + A ( x ) ) 2 v + V ( x ) v = f 2 ( x , ∣ u ∣ 2 ) u , \left\{\begin{array}{l}{\left(-i\nabla +A\left(x))}^{2}u+V\left(x)u={f}_{1}\left(x,{| v| }^{2})v,\\ {\left(-i\nabla +A\left(x))}^{2}v+V\left(x)v={f}_{2}\left(x,{| u| }^{2})u,\end{array}\right. where V V is the electric potential and A A is the magnetic potential. Assuming that the nonlinear function f i ( i = 1 , 2 ) {f}_{i}\left(i=1,2) satisfies three types of polynomial growth assumptions: super-quadratic, asymptotically quadratic, and local super-quadratic at ∣ x ∣ → ∞ | x| \to \infty , we prove the existence of the Nehari-Pankov type ground state solutions using critical point theory together with the non-Nehari manifold method. The resulting problem engages two major difficulties: the first one is that the associated functional is strongly indefinite, and the second lies in verifying the link geometry and showing the boundedness of Cerami sequences. Our results extend and complement the present ones in the literature.

We consider the functional ℱ ( u ) ≔ ∫ Ω f ( x , D u ( x ) ) d x , {\mathcal{ {\mathcal F} }}\left(u):= \mathop{\int }\limits_{\Omega }f\left(x,Du\left(x)){\rm{d}}x, where f ( x , z ) f\left(x,z) satisfies a ( p , q ) \left(p,q) -growth condition with respect to z z and can be approximated by means of a suitable sequence of functions. We consider B R ⋐ Ω {B}_{R}\hspace{0.33em}\Subset \hspace{0.33em}\Omega and the spaces X = W 1 , p ( B R , R N ) and Y = W 1 , p ( B R , R N ) ∩ W loc 1 , q ( B R , R N ) . X={W}^{1,p}\left({B}_{R},{{\mathbb{R}}}^{N})\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}Y={W}^{1,p}\left({B}_{R},{{\mathbb{R}}}^{N})\cap {W}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{1,q}\left({B}_{R},{{\mathbb{R}}}^{N}). We prove that the lower semicontinuous envelope of ℱ ∣ Y {\mathcal{ {\mathcal F} }}{| }_{Y} coincides with ℱ {\mathcal{ {\mathcal F} }} or, in other words, that the Lavrentiev term is equal to zero for any admissible function u ∈ W 1 , p ( B R , R N ) u\in {W}^{1,p}\left({B}_{R},{{\mathbb{R}}}^{N}) . We perform the approximations by means of functions preserving the values on the boundary of B R {B}_{R} .

This study investigates the properties of solutions about one-dimensional wave equations connected in parallel under the effect of two nonlinear localized frictional damping mechanisms. First, under various growth conditions about the nonlinear dissipative effect, we try to establish the decay rate estimates by imposing minimal amount of support on the damping and provide some examples of exponential decay and polynomial decay. To achieve this, a proper observability inequality has been proposed and constructed based on some refined microlocal analysis. Then, the existence of a global attractor is proved when the damping terms are linearly bounded at infinity, a special weighting function has been used in this part, which eliminates undesirable terms of the higher order while contributing lower-order terms. Finally, we establish that the long-time behavior of solutions of the nonlinear system is completely determined by the dynamics of large finite number of functionals.

In a recent study, we showed that the large ( x , t ) \left(x,t) behavior of a class of physically relevant solutions of Boussinesq’s equation for water waves is described by ten main asymptotic sectors. In the sector adjacent to the positive x x -axis, referred to as Sector I, we stated without proof an exact expression for the leading asymptotic term together with an error estimate. Here, we provide a proof of this asymptotic formula.

We consider the homogeneous Dirichlet problem for the parabolic equation u t − div ( ∣ ∇ u ∣ p ( x , t ) − 2 ∇ u ) = f ( x , t ) + F ( x , t , u , ∇ u ) {u}_{t}-{\rm{div}}({| \nabla u| }^{p\left(x,t)-2}\nabla u)=f\left(x,t)+F\left(x,t,u,\nabla u) in the cylinder Q T ≔ Ω × ( 0 , T ) {Q}_{T}:= \Omega \times \left(0,T) , where Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} , N ≥ 2 N\ge 2 , is a C 2 {C}^{2} -smooth or convex bounded domain. It is assumed that p ∈ C 0 , 1 ( Q ¯ T ) p\in {C}^{0,1}\left({\overline{Q}}_{T}) is a given function and that the nonlinear source F ( x , t , s , ξ ) F\left(x,t,s,\xi ) has a proper power growth with respect to s s and ξ \xi . It is shown that if p ( x , t ) > 2 ( N + 1 ) N + 2 p\left(x,t)\gt \frac{2\left(N+1)}{N+2} , f ∈ L 2 ( Q T ) f\in {L}^{2}\left({Q}_{T}) , ∣ ∇ u 0 ∣ p ( x , 0 ) ∈ L 1 ( Ω ) {| \nabla {u}_{0}| }^{p\left(x,0)}\in {L}^{1}\left(\Omega ) , then the problem has a solution u ∈ C 0 ( [ 0 , T ] ; L 2 ( Ω ) ) u\in {C}^{0}\left(\left[0,T];\hspace{0.33em}{L}^{2}\left(\Omega )) with ∣ ∇ u ∣ p ( x , t ) ∈ L ∞ ( 0 , T ; L 1 ( Ω ) ) {| \nabla u| }^{p\left(x,t)}\in {L}^{\infty }\left(0,T;\hspace{0.33em}{L}^{1}\left(\Omega )) , u t ∈ L 2 ( Q T ) {u}_{t}\in {L}^{2}\left({Q}_{T}) , obtained as the limit of solutions to the regularized problems in the parabolic Hölder space. The solution possesses the following global regularity properties: ∣ ∇ u ∣ 2 ( p ( x , t ) − 1 ) + r ∈ L 1 ( Q T ) , for any  0 < r < 4 N + 2 , ∣ ∇ u ∣ p ( x , t ) − 2 ∇ u ∈ L 2 ( 0 , T ; W 1 , 2 ( Ω ) ) N . {| \nabla u| }^{2\left(p\left(x,t)-1)+r}\in {L}^{1}\left({Q}_{T}),\hspace{1.0em}\hspace{0.1em}\text{for any\hspace{0.5em}}0\lt r\lt \frac{4}{N+2}\text{}\hspace{0.1em},\hspace{1.0em}{| \nabla u| }^{p\left(x,t)-2}\nabla u\in {L}^{2}{\left(0,T;{W}^{1,2}\left(\Omega ))}^{N}.

In this article, we are concerned with the nonlinear Schrödinger equation − Δ u + λ u = μ ∣ u ∣ p − 2 u + f ( u ) , in R 2 , -\Delta u+\lambda u=\mu {| u| }^{p-2}u+f\left(u),\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{2}, having prescribed mass ∫ R 2 ∣ u ∣ 2 d x = a 2 > 0 , \mathop{\int }\limits_{{{\mathbb{R}}}^{2}}{| u| }^{2}{\rm{d}}x={a}^{2}\gt 0, where λ \lambda arises as a Lagrange multiplier, μ > 0 \mu \gt 0 , p ∈ ( 2 , 4 ] p\in \left(2,4] , and the nonlinearity f ∈ C 1 ( R , R ) f\in {C}^{1}\left({\mathbb{R}},{\mathbb{R}}) behaves like e 4 π u 2 {e}^{4\pi {u}^{2}} as ∣ u ∣ → + ∞ | u| \to +\infty . For a L 2 {L}^{2} -critical or L 2 {L}^{2} -subcritical perturbation μ ∣ u ∣ p − 2 u \mu {| u| }^{p-2}u , we investigate the existence of normalized solutions to the aforementioned problem. Moreover, the limiting profiles of solutions have been considered as μ → 0 \mu \to 0 or a → 0 a\to 0 . This result can be considered as a supplement to the work of Soave ( Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case , J. Funct. Anal. 279 (2020), no. 6, 1–43) and Alves et al. ( Normalized solutions for a Schrödinger equation with critical growth in R N {{\mathbb{R}}}^{N} , Calc. Var. Partial Differential Equations 61 (2022), no. 1, 1–24).

In this article, we consider the double-phase problem with nonlocal reaction. For the autonomous case, we introduce the methods of the Pohozaev manifold, Hardy-Littlewood Sobolev subcritical approximation, adding the mass term to prove the existence and nonexistence of normalized solutions to this problem. For the nonautonomous case, we show the existence of normalized solutions to the double-phase problem by using the Pohozaev restrict method and describing the relationship between the energy of this problem and its limit problem. Moreover, we study the existence of normalized solutions to the double-phase problem involving double Hardy-Littlewood-Sobolev critical exponents.

In the present study, we investigate the existence of the normalized solutions to Sobolev critical fractional Schrödinger equation: ( − Δ ) s u + λ u = f ( u ) + ∣ u ∣ 2 s * − 2 u , in R N , ( P m ) ∫ R N ∣ u ∣ 2 d x = m 2 , \hspace{14em}\left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+\lambda u=f\left(u)+{| u| }^{{2}_{s}^{* }-2}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{12em}<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" voffset="-2.9ex">\left({P}_{m})</mml:mpadded>\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={m}^{2},\hspace{1.0em}\end{array}\right. where 0 < s < 1 0\lt s\lt 1 , N ≥ 2 N\ge 2 , m > 0 m\gt 0 , 2 s * ≔ 2 N N − 2 s {2}_{s}^{* }:= \frac{2N}{N-2s} , λ \lambda is an unknown parameter that will appear as a Lagrange multiplier, and f f is a mass supercritical and Sobolev subcritical nonlinearity. Under fairly general assumptions about f f , with the aid of the Pohozaev manifold and concentration-compactness principle, we obtain a couple of the normalized solution to ( P m ) \left({P}_{m}) . We mainly extend the results of Appolloni and Secchi ( Normalized solutions for the fractional NLS with mass supercritical nonlinearity , J. Differential Equations 286 (2021), 248–283) concerning the above problem from Sobolev subcritical setting to Sobolev critical setting, and also extend the results of Jeanjean and Lu ( A mass supercritical problem revisited , Calc. Var. 59 (2020), 174) from classical Schrödinger equation to fractional Schrödinger equation involving Sobolev critical growth. More importantly, our result settles an open problem raised by Soave ( Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case , J. Funct. Anal. 279 (2020), 108610), when s = 1 s=1 .

In this article, we consider the global and local well-posedness of the mild solutions to the Cauchy problem of fractional drift diffusion system with higher-order nonlinearity. The main difficulty comes from the higher-order nonlinearity. Instead of the convention that people always focus on the properties of the solution in critical spaces, here we are interested in non-critical spaces such as supercritical Sobolev spaces and subcritical Lebesgue spaces. For the initial data in these non-critical spaces, using the properties of fractional heat semigroup and the classical Hardy-Littlewood-Sobolev inequality, we obtain the existence and uniqueness of the mild solution, together with the decaying rate estimates in terms of time variable.

This work is concerned with the following Klein-Gordon-Maxwell system: − Δ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = f ( u ) , x ∈ R 2 , Δ ϕ = ( ω + ϕ ) u 2 , x ∈ R 2 , \left\{\begin{array}{ll}-\Delta u+V\left(x)u-\left(2\omega +\phi )\phi u=f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{2},\\ \Delta \phi =\left(\omega +\phi ){u}^{2},\hspace{1.0em}& x\in {{\mathbb{R}}}^{2},\end{array}\right. where ω > 0 \omega \gt 0 is a constant, u , ϕ : R 2 → R u,\phi :{{\mathbb{R}}}^{2}\to {\mathbb{R}} , V ∈ C ( R 2 , R ) V\in {\mathcal{C}}\left({{\mathbb{R}}}^{2},{\mathbb{R}}) , and f ∈ C ( R , R ) f\in {\mathcal{C}}\left({\mathbb{R}},{\mathbb{R}}) obeys exponential critical growth in the sense of the Trudinger-Moser inequality. We give some new sufficient conditions on f f , specifically related to exponential growth, to obtain the existence of nontrivial solutions. Our results improve and extend the previous results. In particular, we give a more precise estimation than the ones in the existing literature about the minimax level.

This article is devoted to study a class of new p ( x ) p\left(x) -Kirchhoff equation. By means of perturbation technique, variational method, and the method invariant sets for the descending flow, the existence and multiplicity of solutions to this problem are obtained.

We establish regularity results for equilibrium configurations of vectorial multidimensional variational problems, involving bulk and surface energies. The bulk energy densities are uniformly strictly quasiconvex functions with p p -growth, p ≥ 2 p\ge 2 , without any further structure conditions. The anisotropic surface energy is defined by means of an elliptic integrand Φ \Phi not necessarily regular. For a minimal configuration ( u , E ) \left(u,E) , we prove partial Hölder continuity of the gradient ∇ u \nabla u of the deformation.

In this article, we aim to establish the optimal time decay rates of strong solutions to a two-phase flow model derived from a type of coupled fluid-kinetic equation. It is proved that the strong solutions converge to the given constant states with algebraic time decay rates under some additional assumptions on the initial data.

We provide bounds for the sequence of eigenvalues { λ i ( Ω ) } i {\left\{{\lambda }_{i}\left(\Omega )\right\}}_{i} of the Dirichlet problem ( I − Δ ) ln u = λ u in Ω , u = 0 in R N \ Ω , {\left(I-\Delta )}^{\mathrm{ln}}u=\lambda u\hspace{1em}{\rm{in}}\hspace{0.33em}\Omega ,\hspace{1.0em}u=0\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N}\setminus \Omega , where ( I − Δ ) ln {\left(I-\Delta )}^{\mathrm{ln}} is the Klein-Gordon operator with Fourier transform symbol ln ( 1 + ∣ ξ ∣ 2 ) \mathrm{ln}\left(1+{| \xi | }^{2}) . The purpose of this study is to obtain the upper and lower bounds for the sum of the first k -eigenvalues by extending the Li-Yau’s method and Kröger’s method, respectively.

In this article, we first introduce a new fractional g g -Laplacian Monge-Ampère operator: F g s v ( x ) ≔ inf P.V. ∫ R n g v ( z ) − v ( x ) ∣ C − 1 ( z − x ) ∣ s d z ∣ C − 1 ( z − x ) ∣ n + s ∣ C ∈ C , {F}_{g}^{s}v\left(x):= \inf \left\{\hspace{0.1em}\text{P.V.}\hspace{0.1em}\mathop{\int }\limits_{{{\mathbb{R}}}^{n}}g\left(\frac{v\left(z)-v\left(x)}{{| {C}^{-1}\left(z-x)| }^{s}}\right)\frac{{\rm{d}}z}{{| {C}^{-1}\left(z-x)| }^{n+s}}| C\in {\mathcal{C}}\right\}, where g g is the derivative of a Young function and the diagonal matrix C {\mathcal{C}} is positive definite, which has a determinant equal to 1. First, we establish some crucial maximum principles for equations involving the fractional g g -Laplacian Monge-Ampère operator. Based on the maximum principles, the direct method of moving planes is applied to study the equation involving the fractional g g -Laplacian Monge-Ampère operator. As a result, the nonexistence of the positive solutions, symmetry, monotonicity, and asymptotic property of solutions are obtained in bounded/unbounded domains.

In this article, we study the following fractional Kirchhoff-type problems with critical and sublinear nonlinearities: a + b ∬ R N × R N ∣ u ( x ) − u ( y ) ∣ 2 ∣ x − y ∣ N + 2 s d x d y ( − Δ ) s u = λ u q − 1 + u 2 s * − 1 , u > 0 , in Ω , u = 0 , in R N \ Ω , ∫ R N u 2 d x = c 2 , \left\{\begin{array}{l}\left(a+b\mathop{\iint }\limits_{{{\mathbb{R}}}^{N}\times {{\mathbb{R}}}^{N}}\frac{{| u\left(x)-u(y)| }^{2}}{{| x-y| }^{N+2s}}{\rm{d}}x{\rm{d}}y\right){\left(-\Delta )}^{s}u=\lambda {u}^{q-1}+{u}^{{2}_{s}^{* }-1},\hspace{1em}u\gt 0,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0\left,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={c}^{2},\end{array}\right. where ( − Δ ) s {\left(-\Delta )}^{s} is the fractional Laplacian, Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with Lipschitz boundary, 0 < s < 1 , 2 s < N < 4 s 0\lt s\lt 1,2s\lt N\lt 4s , 1 < q < 2 1\lt q\lt 2 , λ > 0 , a > 0 , b > 0 , c > 0 \lambda \gt 0,a\gt 0,b\gt 0,c\gt 0 . First, we prove that the bounded Palais-Smale sequence has a profile decomposition in the fractional Laplacian setting. Then, by utilizing decomposition techniques and variational methods, we acquire that there are two positive normalized solutions for the aforementioned problems.

This article is concerned with the qualitative properties for the Cauchy problem of a non-Newtonian filtration equation with a reaction source term and volumetric moisture content. On the basis of the slowly decaying behavior of volumetric moisture content, we establish new critical exponents that depend on the ratio of coefficients and exponent of volumetric moisture content. Meantime, under appropriate conditions, we show that the solution globally exists for small enough initial data and blows up in finite time for large enough or any nontrivial initial datum.

This article is concerned with the existence of multiple normalized solutions for a class of Choquard equations with a parametric perturbation − ε 2 Δ u + V ( x ) u = λ u + ε − α ( I α * F ( u ) ) f ( u ) , x ∈ R N , ∫ R N ∣ u ∣ 2 d x = a 2 ε N , \left\{\begin{array}{ll}-{\varepsilon }^{2}\Delta u+V\left(x)u=\lambda u+{\varepsilon }^{-\alpha }\left({I}_{\alpha }* F\left(u))f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={a}^{2}{\varepsilon }^{N},\hspace{1.0em}& \end{array}\right. where a > 0 a\gt 0 is a constant, ε > 0 \varepsilon \gt 0 is a parameter, N ≥ 3 N\ge 3 , α ∈ ( 0 , N ) \alpha \in \left(0,N) , λ ∈ R \lambda \in {\mathbb{R}} is unknown and appears as a Lagrange multiplier, f f is a continuous function with L 2 {L}^{2} -subcritical growth, and V : R N → [ 0 , ∞ ) V:{{\mathbb{R}}}^{N}\to \left[0,\infty ) is a continuous function, satisfying del Pino and Felmer’s local conditions. With the help of the penalization method, and Lusternik-Schnirelmann theory, we investigate the relationship between the number of positive normalized solutions and the topology of the set, where the potential V V attains its minimum value if the parameter ε > 0 \varepsilon \gt 0 is small.

Let ( M , ∂ M , g ) \left(M,\partial M,g) be a compact Riemannian manifold with boundary. As a generalization of the Yamabe invariant with boundary Y ( M , ∂ M , g ) Y\left(M,\partial M,g) , we define the k th Yamabe invariant with boundary Y k ( M , ∂ M , g ) {Y}_{k}\left(M,\partial M,g) . We prove some of its properties and study when it can be attained by the generalized metric. We also prove a version of conformal Schwarz lemma on ( M , ∂ M , g ) \left(M,\partial M,g) by using the Yamabe flow with boundary.

In this study, we establish the existence and nonexistence of smooth and peaked solitary wave solutions (or periodic) to the Degasperis-Procesi-Kadomtsev-Petviashvili (DP-KP) equation with a weak transverse effect. We have also shown that DP-KP equation possesses periodic shock waves similar to that of the Degasperis-Procesi equation.

This study is concerned with the existence of normalized solutions for the Choquard equations with critical nonlinearities − Δ u + λ u = f ( u ) + ( I α ∗ ∣ u ∣ 2 α * ) ∣ u ∣ 2 α * − 2 u , in R N , ∫ R N ∣ u ∣ 2 d x = a 2 , \left\{\begin{array}{l}-\Delta u+\lambda u=f\left(u)+\left({I}_{\alpha }\ast {| u| }^{{2}_{\alpha }^{* }}){| u| }^{{2}_{\alpha }^{* }-2}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={a}^{2},\hspace{1.0em}\end{array}\right. where N > 2 N\gt 2 , α ∈ ( 0 , N ) \alpha \in \left(0,N) , a > 0 a\gt 0 , and I α ( x ) {I}_{\alpha }\left(x) is the Riesz potential given by I α ( x ) = A α ∣ x ∣ N − α with A α = Γ N − α 2 2 α π N 2 Γ α 2 , {I}_{\alpha }\left(x)=\frac{{A}_{\alpha }}{{| x| }^{N-\alpha }}\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{A}_{\alpha }=\frac{\Gamma \left(\phantom{\rule[-0.68em]{}{0ex}},\frac{N-\alpha }{2}\right)}{{2}^{\alpha }{\pi }^{\tfrac{N}{2}}\Gamma \left(\phantom{\rule[-0.68em]{}{0ex}},\frac{\alpha }{2}\right)}, and 2 α * = N + α N − 2 {2}_{\alpha }^{* }=\frac{N+\alpha }{N-2} is the Hardy-Littlewood-Sobolev critical exponent and f f is a subcritical nonlinearity. In the case that f f is L 2 {L}^{2} -supercritical growth, by means of the Pohozaev manifold method and mountain pass theorem, we obtain a couple of the normalized solution; while in the case f ( u ) = μ ∣ u ∣ q − 2 u f\left(u)=\mu {| u| }^{q-2}u with 2 < q < 2 + 4 N 2\lt q\lt 2+\frac{4}{N} , being L 2 {L}^{2} -subcritical growth and μ > 0 \mu \gt 0 a parameter, we employ the truncation technique and the genus theory to prove the multiplicity of normalized solutions.

The aim of this article is to consider a three-dimensional Cauchy problem for the parabolic-elliptic system arising from biological transport networks. For such problem, we first establish the global existence, uniqueness, and uniform boundedness of the strong solution by estimating the derivative of the diagonal permeability tensor with respect to time variable. Moreover, for the diffusion coefficient appropriately large, we demonstrate that the corresponding stationary problem admits a strong solution and that the solution of the Cauchy problem will stabilize to its stationary counterpart in infinite time with a time-decay rate.

In this study, we consider the initial boundary value problem of the planar magnetohydrodynamics (MHD) system when the viscous coefficients and heat conductivity depend on the temperature, which are assumed to be proportional to θ α {\theta }^{\alpha } , α ≥ 0 \alpha \ge 0 , and θ β {\theta }^{\beta } , β ≥ 0 \beta \ge 0 , respectively, and magnetic diffusivity coefficient depends on the specific volume. We prove the existence and uniqueness of the global-in-time classical solution with general large initial data provided that α \alpha is sufficiently small and there is no restriction on the parameter β \beta . Moreover, the nonlinearly exponential stability of the solution is obtained. As a result, we extend the works given by Sun et al. for the full compressible Navier-Stokes equations and by Li and Shang for the full compressible MHD equations.

Our purpose is to establish nonexistence results concerning complete noncompact mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products, under mild constraints on the warping and soliton functions. Applications to self-shrinkers in the Euclidean space, as well as to mean curvature flow solitons in other important warped product models such as the Schwarzschild and Reissner-Nordström spaces, and Robertson-Walker spacetimes such as the Einstein-de Sitter spacetime, are also given. Furthermore, we study the nonexistence of entire solutions to the mean curvature flow equation. Our approach is based on a suitable conformal change of metric jointly with a maximum principle for complete noncompact Riemannian manifolds with polynomial volume growth due to Alías et al.

This study proposed and analyzed a vector-borne reaction–diffusion–advection model with vector-bias mechanism and heterogeneous parameters in one-dimensional habitat. The basic reproduction number R 0 {{\mathfrak{R}}}_{0} in connection with principal eigenvalue of elliptic eigenvalue problem is characterized as the role of determining the threshold dynamics of the system. The main objective of this study is to investigate the asymptotic profiles and monotonicity of R 0 {{\mathfrak{R}}}_{0} with respect to diffusion rates and advection rates under certain conditions. Through exploring the level set of R 0 {{\mathfrak{R}}}_{0} , we also find that there exists a unique surface separating the dynamics. Our results also reveal that the infected hosts and vectors will aggregate at the downstream end if the ratio of advection rates and diffusion rates is sufficiently large.

In this article, we study the following Kirchhoff equation with combined nonlinearities: − a + b ∫ R 4 ∣ ∇ u ∣ 2 d x Δ u + λ u = μ ∣ u ∣ q − 2 u + ∣ u ∣ 2 u , in R 4 , ∫ R 4 ∣ u ∣ 2 d x = c 2 , \left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{4}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u+\lambda u=\mu {| u| }^{q-2}u+{| u| }^{2}u,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{4},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{4}}{| u| }^{2}{\rm{d}}x={c}^{2},\end{array}\right. where a , b , c > 0 a,b,c\gt 0 , μ , λ ∈ R \mu ,\lambda \in {\mathbb{R}} , 2 < q < 4 2\lt q\lt 4 . Under different assumptions on b , c > 0 b,c\gt 0 and μ ∈ R \mu \in {\mathbb{R}} , we prove some existence, nonexistence, and asymptotic behavior of the obtained normalized solutions. When μ > 0 \mu \gt 0 :(i) for 2 < q < 3 2\lt q\lt 3 , we obtain the existence of a local minimizer ground-state solution and a mountain-pass-type solution, (ii) for q = 3 q=3 and 3 < q < 4 3\lt q\lt 4 , we obtain the existence of a mountain-pass type ground-state solution respectively, under different assumptions. When μ < 0 \mu \lt 0 and 2 < q < 4 2\lt q\lt 4 , we prove the nonexistence result of the aforementioned problem. We also investigate the asymptotic behavior of the normalized ground-state solutions, when μ → 0 + \mu \to {0}^{+} and b → 0 + b\to {0}^{+} , respectively.

Compressible Euler equations with space-dependent damping in high dimensions R n ( n = 2 , 3 ) {{\bf{R}}}^{n}\hspace{0.33em}\hspace{0.33em}\left(n=2,3) are considered in this article. Assuming that the small initial velocity and small perturbation of the initial density have compact support, we establish finite-time blow-up results for the Euler system, by combining energy estimate and new test functions constructed by the solutions of the following linear elliptic partial differential equations system: − G 1 ( x ) + ∇ ⋅ G 2 → ( x ) = 0 , − G 2 → ( x ) + ∇ G 1 ( x ) = μ G 2 → ( x ) ( 1 + ∣ x ∣ ) λ . \left\{\begin{array}{l}-{G}_{1}\left(x)+\nabla \cdot \overrightarrow{{G}_{2}}\left(x)=0,\\ -\overrightarrow{{G}_{2}}\left(x)+\nabla {G}_{1}\left(x)=\frac{\mu \overrightarrow{{G}_{2}}\left(x)}{{(1+| x| )}^{\lambda }}.\end{array}\right. This result generalizes the one in the literature from 1 − D 1-D to high dimension R n ( n = 2 , 3 ) {{\bf{R}}}^{n}\hspace{0.33em}\hspace{0.33em}\left(n=2,3) .

We investigate the multiplicity of solutions for a quasilinear scalar field equation with a nonhomogeneous differential operator defined by S u ≔ − div ϕ u 2 + ∣ ∇ u ∣ 2 2 ∇ u + ϕ u 2 + ∣ ∇ u ∣ 2 2 u , Su:= -\hspace{0.1em}\text{div}\hspace{0.1em}\left\{\phi \left(\frac{{u}^{2}+{| \nabla u| }^{2}}{2}\right)\nabla u\right\}+\phi \left(\frac{{u}^{2}+{| \nabla u| }^{2}}{2}\right)u, where ϕ : [ 0 , + ∞ ) → R \phi :\left[0,+\infty )\to {\mathbb{R}} is a positive continuous function. This operator is introduced by Stuart [Two positive solutions of a quasilinear elliptic Dirichlet problem, Milan J. Math.   79 (2011), 327–341] and depends on not only ∇ u \nabla u but also u u . This particular quasilinear term generally appears in the study of nonlinear optics model, which describes the propagation of self-trapped beam in a cylindrical optical fiber made from a self-focusing dielectric material. When the reaction term is concave-convex nonlinearities, by using the Nehari manifold and doing a fine analysis associated on the fibering map, we obtain that the equation admits at least one positive energy solution and negative energy solution, which is also the ground state solution of the equation. We overcome two main difficulties which are caused by the nonhomogeneity of the differential operator S S : (i) the almost everywhere convergence of the gradient for the minimizing sequence { u n } \left\{{u}_{n}\right\} ; (ii) seeking the reasonable restrictions about S S .

Let Δ k {\Delta }_{k} be the Dunkl generalized Laplacian operator associated with a root system R R of R N {{\mathbb{R}}}^{N} , N ≥ 2 N\ge 2 , and a nonnegative multiplicity function k k defined on R R and invariant by the finite reflection group W W . In this study, we study the existence and nonexistence of weak solutions to the semilinear inequality − Δ k u + λ ∣ x ∣ 2 u ≥ ∣ u ∣ p -{\Delta }_{k}u+\frac{\lambda }{{| x| }^{2}}u\ge {| u| }^{p} in R N \ B 1 ¯ {{\mathbb{R}}}^{N}\backslash \overline{{B}_{1}} under the boundary condition u ≥ 0 u\ge 0 on ∂ B 1 \partial {B}_{1} , where p > 1 p\gt 1 , λ ≥ − ( N − 2 + 2 γ ) 2 ⁄ 4 \lambda \ge -{(N-2+2\gamma )}^{2}/4 , and B 1 {B}_{1} is the open unit ball of R N {{\mathbb{R}}}^{N} . Namely, we show that the dividing line with respect to existence and nonexistence is given by a critical exponent that depends on λ \lambda , N N , and γ ( k ) \gamma \left(k) , where γ ( k ) = ∑ α ∈ R + k ( α ) \gamma \left(k)={\sum }_{\alpha \in {R}^{+}}k\left(\alpha ) and R + {R}^{+} is the positive subsystem.

In this article, we mainly establish the local extra fractional differentiability (Besov regularity) of weak solutions for the following divergence nonlinear elliptic equations of p p -Laplacian type: div A ( D u , x ) = div F , \hspace{0.1em}\text{div}\hspace{0.1em}A(Du,x)=\hspace{0.1em}\text{div}\hspace{0.1em}{\bf{F}}, where A A is a Carathéodory function with p − 1 p-1 growth for 1 < p < 2 1\lt p\lt 2 . The standard example for the aforementioned equations is the classical elliptic p p -Laplacian equation div ( ∣ D u ∣ p − 2 D u ) = div F , for 1 < p < 2 . \hspace{0.1em}\text{div}\hspace{0.1em}({| Du| }^{p-2}Du)=\hspace{0.1em}\text{div}\hspace{0.1em}{\bf{F}},\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}1\lt p\lt 2. Remarkably, the case 1 < p < 2 1\lt p\lt 2 is very different from the case p ≥ 2 p\ge 2 since the modulus of ellipticity in the elliptic p p -Laplacian equation tends to infinity when ∣ D u ∣ → 0 | Du| \to 0 .

In this article, we study the following Choquard equation: − Δ u + u = ( I α ⋆ u 2 ) u , x ∈ R 3 , -\Delta u+u=\left({{\rm{I}}}_{\alpha }\star {u}^{2})u,\hspace{1.0em}x\in {{\mathbb{R}}}^{3}, where I α {{\rm{I}}}_{\alpha } is the Riesz potential and α \alpha is sufficiently close to 2. By investigating the limit profile of ground states of the equation as α → 2 \alpha \to 2 , we prove the uniqueness and nondegeneracy of ground states.

For the following quasilinear Choquard-type equation: − Δ u − Δ ( u 2 ) u + V ( x ) u = ( I μ * ∣ u ∣ p ) ∣ u ∣ p − 2 u , x ∈ R N , -\Delta u-\Delta \left({u}^{2})u+V\left(x)u=\left({I}_{\mu }* {| u| }^{p}){| u| }^{p-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ≥ 3 , 0 < μ < N N\ge 3,0\lt \mu \lt N , V ( x ) = a − b 1 + ∣ x ∣ 2 V\left(x)=a-\frac{b}{1+{| x| }^{2}} , 1 < a < + ∞ 1\lt a\lt +\infty , 0 < b < 1 2 0\lt b\lt \frac{1}{2} , 2 ( N + μ ) N < p < 2 ( N + μ ) N − 2 \frac{2\left(N+\mu )}{N}\lt p\lt \frac{2\left(N+\mu )}{N-2} , and I μ {{I}}_{\mu } is the Riesz potential. Our work is finding the positive solutions and the ground-state solutions. Using a change of variables method, we overcome the difficulties which the quasilinear term may bring us and consider the corresponding functional with variational arguments. Then, we establish the nonexistence results via the Pohožaev identity.

In this study, we obtain the weighted Hardy-Adams inequality of any even dimension n ≥ 4 n\ge 4 . Namely, for u ∈ C 0 ∞ ( B n ) u\in {C}_{0}^{\infty }\left({{\mathbb{B}}}^{n}) with ∫ B n ∣ ∇ n 2 u ∣ 2 d x − ∏ k = 1 n ⁄ 2 ( 2 k − 1 ) 2 ∫ B n u 2 ( 1 − ∣ x ∣ 2 ) n d x ≤ 1 , \mathop{\int }\limits_{{{\mathbb{B}}}^{n}}{| {\nabla }^{\frac{n}{2}}u| }^{2}{\rm{d}}x-\mathop{\prod }\limits_{k=1}^{n/2}{\left(2k-1)}^{2}\mathop{\int }\limits_{{{\mathbb{B}}}^{n}}\frac{{u}^{2}}{{\left(1-{| x| }^{2})}^{n}}{\rm{d}}x\le 1, then the following Hardy-Adams inequalities hold: ∫ B n exp 2 2 − n − ϑ β 0 n 2 , n ( u φ ϑ ) 2 ( 1 − ∣ x ∣ 2 ) 2 ϑ + n − 4 d x ≤ C n , ϑ , ϑ ∈ 3 − n 2 , 0 \begin{array}{r}\mathop{\displaystyle \int }\limits_{{{\mathbb{B}}}^{n}}\exp \left({2}^{2-n-{\vartheta }}{\beta }_{0}\left(\frac{n}{2},n\right){\left(u{\varphi }_{{\vartheta }})}^{2}\right){(1-{| x| }^{2})}^{2{\vartheta }+n-4}{\rm{d}}x\le {C}_{n,{\vartheta }},\hspace{0.33em}\hspace{0.33em}{\vartheta }\in \left(\frac{3-n}{2},0\right]\end{array} and ∫ B n Φ 2 2 2 − n − ϑ β 0 n 2 , n ( u φ ϑ ) 2 ( 1 − ∣ x ∣ 2 ) 2 ϑ + n − 4 d x ≤ C n , ϑ , ϑ ∈ ( 2 − n , 3 − n 2 ] , \mathop{\int }\limits_{{{\mathbb{B}}}^{n}}{\Phi }_{2}\left({2}^{2-n-{\vartheta }}{\beta }_{0}\left(\frac{n}{2},n\right){\left(u{\varphi }_{{\vartheta }})}^{2}\right){(1-{| x| }^{2})}^{2{\vartheta }+n-4}{\rm{d}}x\le {C}_{n,{\vartheta }},\hspace{0.33em}\hspace{0.33em}{\vartheta }\in \left(2-n,\frac{3-n}{2}\right], where Φ 2 ( t ) = e t − ∑ 1 j = 0 t j j ! {\Phi }_{2}\left(t)={e}^{t}-\mathop{\sum ^{1}}\limits_{j=0}\frac{{t}^{j}}{j\&#x0021;} , φ ϑ ( x ) = 2 1 − ∣ x ∣ 2 n + ϑ − 2 2 {\varphi }_{{\vartheta }}\left(x)={\left(,\frac{2}{1-{| x| }^{2}}\right)}^{\tfrac{n+{\vartheta }-2}{2}} and C n , ϑ {C}_{n,{\vartheta }} is a positive constant independent of u u . The crucial method in this article is borrowed from Lu and Yang [Adv. Math. 319 (2017)] and Li, Lu and Yang [Adv. Math. 333 (2018)]. We also apply set level method which was first developed by Lam and Lu [J. Diff. Equa. 255 (2013)]. The new ingredient here is to utilize the relationship between Δ H {\Delta }_{{\mathbb{H}}} and the Laplace-Beltrami operator Δ ϑ {\Delta }_{{\vartheta }} , where Δ ϑ {\Delta }_{{\vartheta }} is developed by Liu and Peng [Indiana Univ. Math. J. 3 (2009), 1457–1492]. Furthermore, the Hardy-Adams inequality in this article is the weighted version of the result of Li et al. [Adv. Math. 333 (2018), 350–385].

We prove an existence result for a p -Laplacian problem set in the whole Euclidean space and exhibiting a critical term perturbed by a singular, convective reaction. The approach used combines variational methods, truncation techniques, and concentration compactness arguments, together with set-valued analysis and fixed point theory. De Giorgi’s technique, a priori gradient estimates, and nonlinear regularity theory are employed to obtain local C 1 , α {C}^{1,\alpha } regularity of solutions, as well as their pointwise decay at infinity. The result is new even in the non-singular case, also for the Laplacian.

In this article, we investigate the general qusi-one-dimensional nozzle flows governed by non-isentropic compressible Euler system. First, the steady states of the subsonic and supersonic flows are analyzed. Then, the existence, stability, and uniqueness of the subsonic temporal periodic solutions around the steady states are proved by constructing a new iterative format technically. Besides, further regularity and stability of the obtained temporal periodic solutions are obtained, too. The main difficulty in the proof is coming from derivative loss, which is caused by the diagonalization. Observing that the entropy is conserved along the second characteristic curve, we overcome this difficulty by transforming the derivative of entropy with respect to x x into a derivative along the direction of first or third characteristic. The results demonstrate that dissipative boundary feedback control can stabilize the non-isentropic compressible Euler equations in qusi-one-dimensional nozzles.

This study deals with the existence of nodal solutions for the following gauged nonlinear Schrödinger equation with zero mass: − Δ u + h u 2 ( ∣ x ∣ ) ∣ x ∣ 2 + ∫ ∣ x ∣ + ∞ h u ( s ) s u 2 ( s ) d s u = ∣ u ∣ p − 2 u , x ∈ R 2 , -\Delta u+\left(\frac{{h}_{u}^{2}\left(| x| )}{{| x| }^{2}}+\underset{| x| }{\overset{+\infty }{\int }}\frac{{h}_{u}\left(s)}{s}{u}^{2}\left(s){\rm{d}}s\right)u={| u| }^{p-2}u,\hspace{1.0em}x\in {{\mathbb{R}}}^{2}, where p > 6 p\gt 6 and h u ( s ) = 1 2 ∫ 0 s r u 2 ( r ) d r {h}_{u}\left(s)=\frac{1}{2}{\int }_{0}^{s}r{u}^{2}\left(r){\rm{d}}r . By variational methods, we prove that for any integer k ≥ 0 k\ge 0 , the above equation has a nodal solution w k {w}_{k} which changes sign exactly k k times. Moreover, we also prove that w k {w}_{k} belongs to L 2 ( R 2 ) {L}^{2}\left({{\mathbb{R}}}^{2}) provided p > 10 p\gt 10 .

In this study, we are interested in the regularization of an ill-posed problem generated by a generalized elliptic equation in an abstract framework. The regularization strategy is based on the modified quasi-boundary-valued method, which allows us to construct a stable solution depending on a small parameter ε \varepsilon . To justify the theoretical results obtained, we present a few numerical examples to demonstrate the accuracy of the approximate solution and the effectiveness of the method used in our investigation.

We study heat equations with generalized fractional Laplacian, which is defined by the spectral theory. Here we develop the existence theory for those equations. Also, we present some numerical simulations for our problems.

In this article, we are concerned with the existence of nontrivial solutions to the Choquard equation − Δ u + α ( x ) u = ( ∣ x ∣ − μ ∗ ∣ u ∣ q ) ∣ u ∣ q − 2 u in R N ( N ≥ 2 ) , -\Delta u+\alpha \left(x)u=\left({| x| }^{-\mu }\ast {| u| }^{q}){| u| }^{q-2}u\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\hspace{1em}\left(N\ge 2), with recurrent potential α \alpha , where 0 < μ < N 0\lt \mu \lt N and 2 N − μ N < q < 2 N − μ N − 2 \frac{2N-\mu }{N}\lt q\lt \frac{2N-\mu }{N-2} . Our results include some classical cases where α \alpha is constant and α \alpha is periodic, as well as some new cases, such as α \alpha being almost periodic and α \alpha being only bounded and uniformly continuous.

The Maxwell-Bloch system of equations with inhomogeneous broadening is studied, and the local and global well-posedness of the corresponding initial-boundary value problem is established by taking advantage of the integrability of the system and making use of the corresponding inverse scattering transform (IST). A key ingredient in the analysis is the L 2 {L}^{2} -Sobolev bijectivity of the direct and IST established by Xin Zhou for the focusing Zakharov-Shabat problem.

In this article, we propose a model describing laminar shallow water flows with two velocities to handle an uneven bottom. The model is established by taking the different velocities into account. The source terms generated from the bottom topography prevent us from solving the Riemann problem directly. We first derive the elementary waves including the stationary wave where the global entropy condition is used to ensure uniqueness. Then, we analyze the resonance phenomenon and coalescence of waves by classifying the initial data into different regions. Finally, the Riemann problem is resolved explicitly on a case-by-case basis.

Considering the prevalence of asymptomatic individuals during the spread of disease, this article develops a model of degenerate reaction diffusion Cholera with asymptomatic individuals. First, the well-posedness of model is studied, including the global existence of solutions and the existence of attractor. Second, the basic reproduction number ℛ 0 {{\mathcal{ {\mathcal R} }}}_{0} is defined to determine whether the disease is vanishing or persistent. In particular, we also analyze the asymptotic behavior of the endemic steady state when the diffusion rate of susceptible or asymptomatic individuals tends to 0 or infinity. Finally, by fitting the theoretical results with some numerical simulations, we find that the spatial distribution of disease and local epidemic risk are less affected by the mobility of susceptible populations, whereas the mobility of asymptomatic or symptomatic populations significantly affects the spatial and temporal distribution of infected populations. In addition, we found that the proportion of asymptomatic individuals to infected individuals is also a key factor in disease epidemics, and how to quickly diagnose asymptomatic individuals for disease control and prevention should be of a particular concern.

This work focuses on the Kirchhoff-Schrödinger-Poisson-type system with singular term and critical Sobolev nonlinearity as follows: − a + b ∫ Ω ∣ ∇ u ∣ p d x Δ p u + ϕ ∣ u ∣ q − 2 u = λ u − γ + ∣ u ∣ p ∗ − 2 u in Ω , − Δ ϕ = ∣ u ∣ q in Ω , u = ϕ = 0 on ∂ Ω , \left\{\begin{array}{ll}-\left(a+b\mathop{\displaystyle \int }\limits_{\Omega }{| \nabla u| }^{p}{\rm{d}}x\right){\Delta }_{p}u+\phi {| u| }^{q-2}u=\lambda {u}^{-\gamma }+{| u| }^{{p}^{\ast }-2}u\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -\Delta \phi ={| u| }^{q}\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=\phi =0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Ω \Omega is a bounded domain in R N {{\mathbb{R}}}^{N} with Lipschitz boundary ∂ Ω \partial \Omega , 0 < γ < 1 0\lt \gamma \lt 1 , Δ p u = div ( ∣ ∇ u ∣ p − 2 ∇ u ) {\Delta }_{p}u={\rm{div}}\left({| \nabla u| }^{p-2}\nabla u) , 1 < p < q < p * 2 1\lt p\lt q\lt \frac{{p}^{* }}{2} , p * = N p ⁄ N − p {p}^{* }=Np/N-p is the critical Sobolev exponent, and λ > 0 \lambda \gt 0 . With the Nehari manifold approach, the above problem is discovered to have at least one weak solution. Furthermore, the singular term and critical nonlinearity arise concurrently, which is the main innovation and difficulty of this article. To some extent, we generalize the previous results.

In this article, we introduce the concept of energy-variational solutions for a class of nonlinear dissipative evolutionary equations, which turns out to be especially suited to treat viscoelastic fluid models. Under certain convexity assumptions, the existence of such solutions can be shown constructively by an adapted minimizing movement scheme in a general framework. Weak-strong uniqueness follows by a suitable relative energy inequality. Our main motivation is to apply the general framework to viscoelastic fluid models. Therefore, we give a short overview on different versions of such models and their derivation. The abstract result is applied to two of these viscoelastic fluid models in full detail. In the conclusion, we comment on further applications of the general theory and its possible impact.

In the article, we are concerned with the three-dimensional anisotropic Boussinesq equations with the velocity dissipation in x 2 {x}_{2} and x 3 {x}_{3} directions and the thermal diffusion in only x 3 {x}_{3} direction. When the spatial domain is the whole space R 3 {{\mathbb{R}}}^{3} , the global well-posedness and stability problem for the partially dissipated Boussinesq system remain the extremely challenging open problems. Attention here focuses on the periodic domain Ω = R × T 2 \Omega ={\mathbb{R}}\times {{\mathbb{T}}}^{2} . We aim at establishing the stability for the problem of perturbations near hydrostatic equilibrium and the large-time behavior of the perturbed solution. We first obtain the global existence of some symmetric fluids in H 2 ( Ω ) {H}^{2}\left(\Omega ) for small initial data. Then the exponential decay rates for the oscillations u ˜ \widetilde{u} and θ \theta in H 1 ( Ω ) {H}^{1}\left(\Omega ) and the homogeneous Sobolev space H v 2 ˙ ( Ω ) \dot{{H}_{v}^{2}}\left(\Omega ) are also shown. The proof is based on a key observation that we can decompose the velocity u u into the average u ¯ \overline{u} on T 2 {{\mathbb{T}}}^{2} and the corresponding oscillation u ˜ \widetilde{u} . This enables us to establish the strong Poincaré-type inequalities on u ˜ \widetilde{u} , u 3 , θ {u}_{3},\theta and some anisotropic inequalities, which ensure the establishment of the closed priori estimates. In addition, we also prove the oscillations in one direction u ˜ ( 2 ) , u ˜ ( 3 ) {\widetilde{u}}^{\left(2)},{\widetilde{u}}^{\left(3)} in H 1 ( Ω ) {H}^{1}\left(\Omega ) decay to zero exponentially.

In this article, using the least action principle and minimax methods in critical point theory, some existence and multiplicity results for periodic solutions of second-order Hamiltonian systems are obtained.