Volume 12, Issue 1

In this article, we study the existence of multiple solutions to a generalized p ( ⋅ ) p\left(\cdot ) -Laplace equation with two parameters involving critical growth. More precisely, we give sufficient “local” conditions, which mean that growths between the main operator and nonlinear term are locally assumed for p ( ⋅ ) p\left(\cdot ) -sublinear, p ( ⋅ ) p\left(\cdot ) -superlinear, and sandwich-type cases. Compared to constant exponent problems (e.g., p p -Laplacian and ( p , q ) \left(p,q) -Laplacian), this characterizes the study of variable exponent problems. We show this by applying variants of the mountain pass theorem for p ( ⋅ ) p\left(\cdot ) -sublinear and p ( ⋅ ) p\left(\cdot ) -superlinear cases and constructing critical values defined by a minimax argument in the genus theory for sandwich-type case. Moreover, we also obtain a nontrivial nonnegative solution for sandwich-type case changing the role of parameters. Our work is a generalization of several existing works in the literature.

In this paper, we deal with the following critical Choquard-Kirchhoff problem on the Heisenberg group of the form: M ( ‖ u ‖ 2 ) ( − Δ H u + V ( ξ ) u ) = ∫ H N ∣ u ( η ) ∣ Q λ ∗ ∣ η − 1 ξ ∣ λ d η ∣ u ∣ Q λ ∗ − 2 u + μ f ( ξ , u ) , M\left(\Vert u{\Vert }^{2})\left(-{\Delta }_{{\mathbb{H}}}u\left+V\left(\xi )u)=\left(\mathop{\int }\limits_{{{\mathbb{H}}}^{N}}\frac{| u\left(\eta ){| }^{{Q}_{\lambda }^{\ast }}}{| {\eta }^{-1}\xi {| }^{\lambda }}{\rm{d}}\eta \right)| u{| }^{{Q}_{\lambda }^{\ast }-2}u+\mu f\left(\xi ,u), where M M is the Kirchhoff function, Δ H {\Delta }_{{\mathbb{H}}} is the Kohn Laplacian on the Heisenberg group H N {{\mathbb{H}}}^{N} , f f is a Carathéodory function, μ > 0 \mu \gt 0 is a parameter and Q λ ∗ = 2 Q − λ Q − 2 {Q}_{\lambda }^{\ast }=\frac{2Q-\lambda }{Q-2} is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We first establish a new version of the concentration-compactness principle for the Choquard equation on the Heisenberg group. Then, combining with the mountain pass theorem, we obtain the existence of nontrivial solutions to the aforementioned problem in the case of nondegenerate and degenerate cases.

We study the local behavior of bounded local weak solutions to a class of anisotropic singular equations of the kind ∑ i = 1 s ∂ i i u + ∑ i = s + 1 N ∂ i ( A i ( x , u , ∇ u ) ) = 0 , x ∈ Ω ⊂ ⊂ R N for 1 ≤ s ≤ ( N − 1 ) , \mathop{\sum }\limits_{i=1}^{s}{\partial }_{ii}u+\mathop{\sum }\limits_{i=s+1}^{N}{\partial }_{i}({A}_{i}\left(x,u,\nabla u))=0,\hspace{1.0em}x\in \Omega \subset \hspace{-0.3em}\subset \hspace{0.33em}{{\mathbb{R}}}^{N}\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}1\le s\le \left(N-1), where each operator A i {A}_{i} behaves directionally as the singular p p -Laplacian, 1 < p < 2 1\lt p\lt 2 . Throughout a parabolic approach to expansion of positivity we obtain the interior Hölder continuity and some integral and pointwise Harnack inequalities.

In this article, we establish a viscosity method for random homogenization of an obstacle problem with nondivergence structure. We study the asymptotic behavior of the viscosity solution u ε {u}_{\varepsilon } of fully nonlinear equations in a perforated domain with the stationary ergodic condition. By capturing the behavior of the homogeneous solution, analyzing the characters of the corresponding obstacle problem, and finding the capacity-like quantity through the construction of appropriate barriers, we prove that the limit profile u u of u ε {u}_{\varepsilon } satisfies a homogenized equation without obstacles.

We study the homogeneous Dirichlet problem for the parabolic equations u t − div ( A ( z , ∣ ∇ u ∣ ) ∇ u ) = F ( z , u , ∇ u ) , z = ( x , t ) ∈ Ω × ( 0 , T ) , {u}_{t}-{\rm{div}}\left({\mathcal{A}}\left(z,| \nabla u| )\nabla u)=F\left(z,u,\nabla u),\hspace{1.0em}z=\left(x,t)\in \Omega \times \left(0,T), with the double phase flux A ( z , ∣ ∇ u ∣ ) ∇ u = ( ∣ ∇ u ∣ p ( z ) − 2 + a ( z ) ∣ ∇ u ∣ q ( z ) − 2 ) ∇ u {\mathcal{A}}\left(z,| \nabla u| )\nabla u=(| \nabla u{| }^{p\left(z)-2}+a\left(z)| \nabla u{| }^{q\left(z)-2})\nabla u and the nonlinear source F F . The initial function belongs to a Musielak-Orlicz space defined by the flux. The functions a a , p p , and q q are Lipschitz-continuous, a ( z ) a\left(z) is nonnegative, and may vanish on a set of nonzero measure. The exponents p p , and q q satisfy the balance conditions 2 N N + 2 < p − ≤ p ( z ) ≤ q ( z ) < p ( z ) + r ∗ 2 \frac{2N}{N+2}\lt {p}^{-}\le p\left(z)\le q\left(z)\lt p\left(z)+\frac{{r}^{\ast }}{2} with r ∗ = r ∗ ( p − , N ) {r}^{\ast }={r}^{\ast }\left({p}^{-},N) , p − = min Q ¯ T p ( z ) {p}^{-}={\min }_{{\overline{Q}}_{T}}\hspace{0.33em}p\left(z) . It is shown that under suitable conditions on the growth of F ( z , u , ∇ u ) F\left(z,u,\nabla u) with respect to the second and third arguments, the problem has a solution u u with the following properties: u t ∈ L 2 ( Q T ) , ∣ ∇ u ∣ p ( z ) + δ ∈ L 1 ( Q T ) for every 0 ≤ δ < r ∗ , ∣ ∇ u ∣ s ( z ) , a ( z ) ∣ ∇ u ∣ q ( z ) ∈ L ∞ ( 0 , T ; L 1 ( Ω ) ) with s ( z ) = max { 2 , p ( z ) } . \begin{array}{l}{u}_{t}\in {L}^{2}\left({Q}_{T}),\hspace{1.0em}| \nabla u{| }^{p\left(z)+\delta }\in {L}^{1}\left({Q}_{T})\hspace{1.0em}\hspace{0.1em}\text{for every}\hspace{0.1em}\hspace{0.33em}0\le \delta \lt {r}^{\ast },\\ | \nabla u{| }^{s\left(z)},\hspace{0.33em}a\left(z)| \nabla u{| }^{q\left(z)}\in {L}^{\infty }\left(0,T;\hspace{0.33em}{L}^{1}\left(\Omega ))\hspace{1em}{\rm{with}}\hspace{0.33em}s\left(z)=\max \left\{2,p\left(z)\right\}.\end{array} Uniqueness is proven under stronger assumptions on the source F F . The same results are established for the equations with the regularized flux A ( z , ( ε 2 + ∣ ∇ u ∣ 2 ) 1 / 2 ) ∇ u {\mathcal{A}}(z,{({\varepsilon }^{2}+| \nabla u{| }^{2})}^{1\text{/}2})\nabla u , ε > 0 \varepsilon \gt 0 .

This paper is concerned with a parabolic-elliptic chemotaxis model with density-suppressed motility and general logistic source in an n -dimensional smooth bounded domain with Neumann boundary conditions. Under the minimal conditions for the density-suppressed motility function, we explore how strong the logistic damping can warrant the global boundedness of solutions and further establish the asymptotic behavior of solutions on top of the conditions.

We study the one-dimensional Kirchhoff-type equation − ( b + a ‖ u ′ ‖ 2 ) u ″ ( x ) = λ u ( x ) p , x ∈ I ≔ ( − 1 , 1 ) , u ( x ) > 0 , x ∈ I , u ( ± 1 ) = 0 , -\left(b+a\Vert u^{\prime} {\Vert }^{2}){u}^{^{\prime\prime} }\left(x)=\lambda u{\left(x)}^{p},\hspace{1em}x\in I:= \left(-1,1),\hspace{1em}u\left(x)\gt 0,\hspace{1em}x\in I,\hspace{1em}u\left(\pm 1)=0, where ‖ u ′ ‖ = ∫ I u ′ ( x ) 2 d x 1 / 2 \Vert u^{\prime} \Vert ={\left({\int }_{I}u^{\prime} {\left(x)}^{2}{\rm{d}}x\right)}^{1\text{/}2} , a > 0 , b > 0 , p > 0 a\gt 0,b\gt 0,p\gt 0 are given constants and λ > 0 \lambda \gt 0 is a bifurcation parameter. We establish the exact solution u λ ( x ) {u}_{\lambda }\left(x) and complete shape of the bifurcation curves λ = λ ( ξ ) \lambda =\lambda \left(\xi ) , where ξ ≔ ‖ u λ ‖ ∞ \xi := \Vert {u}_{\lambda }{\Vert }_{\infty } . We also study the nonlinear eigenvalue problem − ‖ u ′ ‖ p − 1 u ″ ( x ) = μ u ( x ) p , x ∈ I , u ( x ) > 0 , x ∈ I , u ( ± 1 ) = 0 , -\Vert u^{\prime} {\Vert }^{p-1}{u}^{^{\prime\prime} }\left(x)=\mu u{\left(x)}^{p},x\in I,\hspace{1em}u\left(x)\gt 0,x\in I,\hspace{1em}u\left(\pm 1)=0, where p > 1 p\gt 1 is a given constant and μ > 0 \mu \gt 0 is an eigenvalue parameter. We obtain the first eigenvalue and eigenfunction of this problem explicitly by using a simple time map method.

We study the existence and nonexistence of the standing wave solution for the generalized Jackiw-Pi model by using variational method. Depending on interaction strength λ \lambda , we have three different situations. The existence and nonexistence of the standing wave solution correspond to 1 < λ 1\lt \lambda and 0 < λ < 1 0\lt \lambda \lt 1 , respectively. We have the explicit solution of self-dual equation for the borderline λ = 1 \lambda =1 .

We study a model for a fluid showing viscoelastic and viscoplastic behavior, which describes the flow in terms of the fluid velocity and a symmetric deviatoric stress tensor. This stress tensor is transported via the Zaremba-Jaumann rate, and it is subject to two dissipation processes: one induced by a nonsmooth convex potential and one by stress diffusion. We show short-time existence of strong solutions as well as their uniqueness in a class of Leray-Hopf-type weak solutions satisfying the tensorial component in the sense of an evolutionary variational inequality. The global-in-time existence of such generalized solutions has been established in a previous work. We further study the limit when stress diffusion vanishes. In this case, the above notion of generalized solutions is no longer suitable, and we introduce the concept of energy-variational solutions, which is based on an inequality for the relative energy. We derive general properties of energy-variational solutions and show their existence by passing to the nondiffusive limit in the relative energy inequality satisfied by generalized solutions for nonzero stress diffusion.

In this paper, we investigate the buckling problem of the drifting Laplacian of arbitrary order on a bounded connected domain in complete smooth metric measure spaces (SMMSs) supporting a special function, and successfully obtain a general inequality for its eigenvalues. By applying this general inequality, if the complete SMMSs considered satisfy some curvature constraints, we can obtain a universal inequalities for eigenvalues of this buckling problem.

The aim of this article is to study nonlinear problem driven by the p ( t ) p\left(t) -Laplacian with nonsmooth potential. We establish the existence of homoclinic solutions by using variational principle for locally Lipschitz functions and the properties of the generalized Lebesgue-Sobolev space under two cases of the nonsmooth potential: periodic and nonperiodic, respectively. The resulting problem engages two major difficulties: first, due to the appearance of the variable exponent, commonly known methods and techniques for studying constant exponent equations fail in the setting of problems involving variable exponents. Another difficulty we must overcome is verifying the link geometry and certifying boundedness of the Palais-Smale sequence. To our best knowledge, our theorems appear to be the first such result about homoclinic solution for differential inclusion system involving the p ( t ) p\left(t) -Laplacian.

Time-fractional partial differential equations are nonlocal-in-time and show an innate memory effect. Previously, examples like the time-fractional Cahn-Hilliard and Fokker-Planck equations have been studied. In this work, we propose a general framework of time-fractional gradient flows and we provide a rigorous analysis of well-posedness using the Faedo-Galerkin approach. Furthermore, we investigate the monotonicity of the energy functional of time-fractional gradient flows. Interestingly, it is still an open problem whether the energy is dissipating in time. This property is essential for integer-order gradient flows and many numerical schemes exploit this steepest descent characterization. We propose an augmented energy functional, which includes the history of the solution. Based on this new energy, we prove the equivalence of a time-fractional gradient flow to an integer-order one. This correlation guarantees the dissipating character of the augmented energy. The state function of the integer-order gradient flow acts on an extended domain similar to the Caffarelli-Silvestre extension for the fractional Laplacian. Additionally, we present a numerical scheme for solving time-fractional gradient flows, which is based on kernel compressing methods and reduces the problem to a system of ordinary differential equations. We illustrate the behavior of the original and augmented energy in the case of the Ginzburg-Landau energy.

In this paper, we consider stochastic dynamics of a two-dimensional stochastic differential equation with additive noise. When the strength of the noise is zero, this equation undergoes a Bautin bifurcation. We obtain the main conclusions including the existence and uniqueness of the solution, synchronization of system and property of the random equilibrium, where going through some processes like deducing the stationary probability density of the equation and calculating the Lyapunov exponent. For better understanding of the effect under noise, we make a clear comparison between the stochastic system and the deterministic one and make precise numerical simulations to show the slight changes at Bautin bifurcation point. Furthermore, we take a real model as an example to present the application of our theoretical results.

By introducing and solving a new cross-constrained variational problem, a one-to-one correspondence from the prescribed mass to frequency of soliton is established for the generalized Davey-Stewartson system in two-dimensional space. Orbital stability of small soiltons depending on frequencies is proved. Multisolitons with different speeds are constructed by stable small solitons.

A nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate dependent viscosity is considered. This problem is nonlinear and nonlocal in time and inverse to the nonlinear heat equation. The provided mathematical analysis includes the proof of the existence, uniqueness, regularity, and stability of the velocity and the pressure slope for a given flux carrier and of the exponential decay of the solution as the time variable goes to infinity for the exponentially decaying flux.

In this article, we study a Hardy-Sobolev critical elliptic system involving coupled perturbation terms: (0.1) − Δ u + V 1 ( x ) u = η 1 η 1 + η 2 ∣ u ∣ η 1 − 2 u ∣ v ∣ η 2 ∣ x ′ ∣ + α α + β Q ( x ) ∣ u ∣ α − 2 u ∣ v ∣ β , − Δ v + V 2 ( x ) v = η 2 η 1 + η 2 ∣ v ∣ η 2 − 2 v ∣ u ∣ η 1 ∣ x ′ ∣ + β α + β Q ( x ) ∣ v ∣ β − 2 v ∣ u ∣ α , \left\{\begin{array}{c}-\Delta u+{V}_{1}\left(x)u=\frac{{\eta }_{1}}{{\eta }_{1}+{\eta }_{2}}\frac{{| u| }^{{\eta }_{1}-2}u{| v| }^{{\eta }_{2}}}{| x^{\prime} | }+\frac{\alpha }{\alpha +\beta }Q\left(x)| u{| }^{\alpha -2}u| v{| }^{\beta },\\ -\Delta v+{V}_{2}\left(x)v=\frac{{\eta }_{2}}{{\eta }_{1}+{\eta }_{2}}\frac{{| v| }^{{\eta }_{2}-2}v{| u| }^{{\eta }_{1}}}{| x^{\prime} | }+\frac{\beta }{\alpha +\beta }Q\left(x){| v| }^{\beta -2}v{| u| }^{\alpha },\end{array}\right. where n ≥ 3 n\ge 3 , 2 ≤ m < n 2\le m\lt n , x ≔ ( x ′ , x ″ ) ∈ R m × R n − m x:= \left(x^{\prime} ,{x}^{^{\prime\prime} })\in {{\mathbb{R}}}^{m}\times {{\mathbb{R}}}^{n-m} , η 1 , η 2 > 1 {\eta }_{1},{\eta }_{2}\gt 1 , and η 1 + η 2 = 2 ( n − 1 ) n − 2 {\eta }_{1}+{\eta }_{2}=\frac{2\left(n-1)}{n-2} , α , β > 1 \alpha ,\beta \gt 1 and α + β < 2 n n − 2 \alpha +\beta \lt \frac{2n}{n-2} , and V 1 ( x ) , V 2 ( x ) , Q ( x ) ∈ C ( R n ) {V}_{1}\left(x),{V}_{2}\left(x),Q\left(x)\in C\left({{\mathbb{R}}}^{n}) . Observing that (0.1) is doubly coupled, we first develop two efficient tools (i.e., a refined Sobolev inequality and a variant of the “Vanishing” lemma). On the previous tools, we will establish a global compactness result (i.e., a complete description for the Palais-Smale sequences of the corresponding energy functional) and some existence result for (0.1) via variational method. Our strategy turns out to be very concise because we avoid the use of Levy concentration functions and truncation techniques.

This paper deals with the global well-posedness and blow-up phenomena for a strongly damped semilinear wave equation with time-varying source and singular dissipative terms under the null Dirichlet boundary condition. On the basis of cut-off technique, multiplier method, contraction mapping principle, and the modified potential well method, we establish the local well-posedness and obtain the threshold between the existence and nonexistence of the global solution (including the critical case). Meanwhile, with the aid of modified differential inequality technique, the blow-up result of the solutions with arbitrarily positive initial energy and the lifespan of the blow-up solutions are derived.

This article focuses on the singularity formation of smooth solutions for a one-dimensional nonlinear degenerate hyperbolic-parabolic coupled system originating from the Poiseuille flow of nematic liquid crystals. Without assuming that the wave speed of the hyperbolic equation is a positive function, we show that its smooth solution will break down in finite time even for an arbitrarily small initial energy. Based on an estimate of the solution for the heat equation, we use the method of characteristics to control the wave speed and its derivative so that the wave speed does not degenerate and its derivative does not change sign in a period of time.

This article studies the stability of a stationary solution to the three-dimensional Navier-Stokes equations in a bounded domain, where surface tension effects are taken into account. More precisely, this article considers the stability of equilibrium figure of uniformly rotating viscous incompressible fluid in R 3 {{\mathbb{R}}}^{3} , which are rotationally symmetric about a certain axis. It is proved that this stability result can be obtained by the positivity of the second variation of the energy functional associated with the equation that determines an equilibrium figure, provided that initial data are close to an equilibrium state. The unique global solution is constructed in the L p {L}^{p} -in-time and L q {L}^{q} -in-space setting with ( p , q ) ∈ ( 2 , ∞ ) × ( 3 , ∞ ) \left(p,q)\in \left(2,\infty )\times \left(3,\infty ) satisfying 2 / p + 3 / q < 1 2\hspace{0.1em}\text{/}p+3\text{/}\hspace{0.1em}q\lt 1 , where the solution becomes real analytic, jointly in time and space. It is also proved that the solution converges exponentially to the equilibrium.

We are concerned with the stabilization of the wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic and frictional effects. Here, one of the novelties is: the viscoelastic and frictional damping together effect only in a part of domain, not in entire domain, which is only assumed to meet the piecewise multiplier geometric condition that their summed interior and boundary measures can be arbitrarily small. Furthermore, there is no other additional restriction for the location of the viscoelastic-effect region. That is, it is dropped that the viscoelastic-effect region includes a part of the system boundary, which is the fundamental condition in almost all previous literature even if when two types of damping together cover the entire system domain. The other distinct novelty is: in this article we remove the fundamental condition that the derivative of the relaxation function is controlled by relaxation function itself, which is a necessity in the previous literature to obtain the optimal uniform decay rate. Under such weak conditions, we successfully establish a series of decay theorems, which generalize and extend essentially the previous related stability results for viscoelastic model regardless of local damping case, entire damping case and mixed-type damping case.

In this article, we consider the following Choquard system in R N N ≥ 1 {{\mathbb{R}}}^{N}N\ge 1 − Δ u + u = 2 p p + q ( I α ∗ ∣ v ∣ q ) ∣ u ∣ p − 2 u , − Δ v + v = 2 q p + q ( I α ∗ ∣ u ∣ p ) ∣ v ∣ q − 2 v , u ( x ) → 0 , v ( x ) → 0 as ∣ x ∣ → ∞ , \left\{\begin{array}{l}-\Delta u+u=\frac{2p}{p+q}({I}_{\alpha }\ast | v{| }^{q})| u{| }^{p-2}u,\\ -\Delta v+v=\frac{2q}{p+q}({I}_{\alpha }\ast | u{| }^{p})| v{| }^{q-2}v,\\ u\left(x)\to 0,v\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 + α N < p , q < N + α N − 2 \frac{N+\alpha }{N}\lt p,q\lt \frac{N+\alpha }{N-2} , 2 ∗ α {2}_{\ast }^{\alpha } denotes N + α N − 2 \frac{N+\alpha }{N-2} if N ≥ 3 N\ge 3 and 2 ∗ α ≔ ∞ {2}_{\ast }^{\alpha }:= \infty if N = 1 , 2 N=1,2 , I α {I}_{\alpha } is a Riesz potential. By analyzing the asymptotic behavior of Riesz potential energy, we prove that minimal action sign-changing solutions have an odd symmetry with respect to the a hyperplane when α \alpha is either close to 0 or close to N N . Our results can be regarded as a generalization of the results by Ruiz et al.

In this article, we consider a four-dimensional singular differential system that can describe the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein condensates. On the basis of the topological degree theory and some analysis methods, we prove that such a system has two distinct families of periodic solutions and two distinct families of quasi-periodic solutions. Some results in the literature are generalized and improved.

In this article, we study a class of nonlinear fractional Laplace problems with a parameter and superlinear nonlinearity ( − Δ ) s u = λ u + f ( x , u ) , in Ω , u = 0 , in R N \ Ω . \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}{\left(-\Delta )}^{s}u=\lambda u+f\left(x,u),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right.\end{array}\right. Multiplicity of nontrivial solutions is obtained when the parameter is near the eigenvalue of the fractional Laplace operator without Ambrosetti and Rabinowitz condition for the nonlinearity. Our methods are the combination of minimax method, bifurcation theory, and Morse theory.

In the present article, we are concerned with the following problem: v t = Δ v + ∣ x ∣ β e v , x ∈ R N , t > 0 , v ( x , 0 ) = v 0 ( x ) , x ∈ R N , \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}{v}_{t}=\Delta v+| x{| }^{\beta }{e}^{v},\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\hspace{0.33em}t\gt 0,\\ v\left(x,0)={v}_{0}\left(x),\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right. where N ≥ 3 N\ge 3 , 0 < β < 2 0\lt \beta \lt 2 , and v 0 {v}_{0} is a continuous function in R N {{\mathbb{R}}}^{N} . We prove the existence and asymptotic behavior of forward self-similar solutions in the case where v 0 {v}_{0} decays at the rate − ( 2 + β ) log ∣ x ∣ -\left(2+\beta )\log | x| as ∣ x ∣ → ∞ | x| \to \infty . Particularly, we obtain the optimal decay bound for initial value v 0 {v}_{0} .

This article is directed to the study of the following Schrödinger-Poisson-Slater type equation: − ε 2 Δ u + V ( x ) u + ε − α ( I α ∗ ∣ u ∣ 2 ) u = λ ∣ u ∣ p − 1 u in R N , -{\varepsilon }^{2}\Delta u+V\left(x)u+{\varepsilon }^{-\alpha }\left({I}_{\alpha }\ast | u{| }^{2})u=\lambda | u{| }^{p-1}u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where ε , λ > 0 \varepsilon ,\lambda \gt 0 are parameters, N ⩾ 2 N\geqslant 2 , ( α + 6 ) / ( α + 2 ) < p < 2 ∗ − 1 \left(\alpha +6)\hspace{0.1em}\text{/}\hspace{0.1em}\left(\alpha +2)\lt p\lt {2}^{\ast }-1 , I α {I}_{\alpha } is the Riesz potential with 0 < α < N 0\lt \alpha \lt N , and V ∈ C ( R N , R ) V\in {\mathcal{C}}\left({{\mathbb{R}}}^{N},{\mathbb{R}}) . By using variational methods, we prove that there is a positive ground state solution for the aforementioned equation concentrating at a global minimum of V V in the semi-classical limit, and then we found that this solution satisfies the property of exponential decay. Finally, the multiplicity and concentration behavior of positive solutions for the aforementioned problem is investigated by the Ljusternik-Schnirelmann theory. Our article improves and extends some existing results in several directions.

The validity of Korn’s first inequality in the fractional setting in bounded domains has been open. We resolve this problem by proving that in fact Korn’s first inequality holds in the case p s > 1 ps\gt 1 for fractional W 0 s , p ( Ω ) {W}_{0}^{s,p}\left(\Omega ) Sobolev fields in open and bounded C 1 {C}^{1} -regular domains Ω ⊂ R n \Omega \subset {{\mathbb{R}}}^{n} . Also, in the case p s < 1 ps\lt 1 , for any open bounded C 1 {C}^{1} domain Ω ⊂ R n \Omega \subset {{\mathbb{R}}}^{n} , we construct counterexamples to the inequality, i.e., Korn’s first inequality fails to hold in bounded domains. The proof of the inequality in the case p s > 1 ps\gt 1 follows a standard compactness approach adopted in the classical case, combined with a Hardy inequality, and a recently proven Korn second inequality by Mengesha and Scott [ A Fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations , Commun. Math. Sci.  20 (2022), no. 2, 405–423]. The counterexamples constructed in the case p s < 1 ps\lt 1 are interpolations of a constant affine rigid motion inside the domain away from the boundary and of the zero field close to the boundary.

This article presents new local and global gradient estimates of Li-Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under natural lower bounds on the associated Bakry-Émery Ricci curvature tensor and find utility in proving fairly general Harnack inequalities and Liouville-type theorems to name a few. The results here unify, extend and improve various existing results in the literature for special nonlinearities already of huge interest and applications. Some consequences are presented and discussed.

In this article, we formulate and perform a strict analysis of a reaction–diffusion mosquito-borne disease model with total human populations stabilizing at H ( x ) in a spatially heterogeneous environment. By utilizing some fundamental theories of the dynamical system, we establish the threshold-type results of the model relying on the basic reproduction number. Specifically, we explore the mutual impacts of the spatial heterogeneity and diffusion coefficients on the basic reproduction number and investigate the existence, uniqueness, and global attractivity of the nontrivial steady state by utilizing the arguments of asymptotically autonomous semiflows. For the case that all parameters are independent of space, the global attractivity of the nontrivial steady state is achieved by the Lyapunov function.

We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N {{\mathbb{R}}}^{N} . We assume that the nonlinear term satisfies the locally super- ( m 1 , m 2 ) \left({m}_{1},{m}_{2}) condition, that is, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ G x\in G , where G G is a domain in R N {{\mathbb{R}}}^{N} , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ R N x\in {{\mathbb{R}}}^{N} . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [ Nontrivial solutions for Schrodinger equation with local super-quadratic conditions , J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [ Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces , J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.

In this article, we study the following system: − Δ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = λ f ( u ) + ∣ u ∣ 4 u , in R 3 , Δ ϕ + β Δ 4 ϕ = 4 π ( ω + ϕ ) u 2 , in R 3 , \left\{\begin{array}{ll}-\Delta u+V\left(x)u-\left(2\omega +\phi )\phi u=\lambda f\left(u)+| u{| }^{4}u,& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{{\mathbb{R}}}^{3},\\ \Delta \phi +\beta {\Delta }_{4}\phi =4\pi \left(\omega +\phi ){u}^{2},& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{{\mathbb{R}}}^{3},\\ \end{array}\right. where f ( u ) f\left(u) is without any growth and Ambrosetti-Rabinowitz condition. We use a cut-off function and Moser iteration to obtain the existence of nontrivial solution. Finally, as a by-product of our approaches, we obtain the same result for the Klein-Gordon-Maxwell system.

In this article, we develop a continuous periodic switching model depicting Wolbachia infection frequency dynamics in mosquito populations by releasing Wolbachia-infected mosquitoes, which is different from the discrete modeling efforts in the literature. We obtain sufficient conditions on the existence of a unique and exactly two periodic solutions and analyze the stability of each periodic solution, respectively. We also provide a brief discussion and several numerical examples to illustrate our theoretical results.

We deal with localized semiclassical states for singularly perturbed Kirchhoff-type equation as follows: − ε 2 a + ε b ∫ R 3 ∣ ∇ v ∣ 2 d x Δ v + V ( x ) v = P ( x ) f ( v ) , x ∈ R 3 , -\left({\varepsilon }^{2}a+\varepsilon b\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}| \nabla v{| }^{2}{\rm{d}}x\right)\Delta v+V\left(x)v=P\left(x)f\left(v),\hspace{1em}x\in {{\mathbb{R}}}^{3}, where V , P ∈ C 1 ( R 3 , R ) V,P\in {C}^{1}\left({{\mathbb{R}}}^{3},{\mathbb{R}}) and bounded away from zero. By applying the penalization approach together with the Nehari manifold approach in the studies of Szulkin and Weth, we obtain the existence of an infinite sequence of localized solutions of higher topological type. In addition, we also give a concrete set as the concentration position of these localized solutions. It is noted that, in our main results, f f only belongs to C ( R , R ) C\left({\mathbb{R}},{\mathbb{R}}) and does not satisfy the Ambrosetti-Rabinowitz-type condition.

This work is concerned with the nonexistence of nontrivial nonnegative weak solutions for a quasilinear parabolic differential inequality with weighted nonlocal source term in the whole space, which involves weighted polytropic filtration operator or generalized mean curvature operator. We establish the new critical Fujita exponents containing the first and second types. The key ingredient of the technique in proof is the test function method developed by Mitidieri and Pohozaev. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.

This article deals with approximations of center manifolds for delay stochastic differential equations with additive noise. We first prove the existence and smoothness of random center manifolds for these approximation equations. Then we show that the C k {C}^{k} invariant center manifolds of the system with colored noise approximate that of the original system.

In this article, we study the existence of periodic solutions to a class of distributed delay differential equations. We transform the search for periodic solutions with the special symmetry of a delay differential equation to the problem of finding periodic solutions of an associated Hamiltonian system. Using the critical point theory and the pseudo-index theory, we obtain some sufficient conditions for the multiplicity of periodic solutions. This is the first time that critical point theory has been used to study the existence of periodic solutions to distributed delay differential equations.

In this article, we study the existence of ground state solutions for the Schrödinger-Poisson-Slater type equation with the Coulomb-Sobolev critical growth: − Δ u + 1 4 π ∣ x ∣ ∗ ∣ u ∣ 2 u = ∣ u ∣ u + μ ∣ u ∣ p − 2 u , in R 3 , -\Delta u+\left(\frac{1}{4\pi | x| }\ast | u{| }^{2}\right)u=| u| u+\mu | u{| }^{p-2}u,\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3}, where μ > 0 \mu \gt 0 and 3 < p < 6 3\lt p\lt 6 . With the help of the Nehari-Pohozaev method, we obtain a ground-state solution for the above equation by employing compactness arguments.

In this article, we study the following Klein-Gordon-Maxwell system: − Δ u − ( 2 ω + ϕ ) ϕ u = g ( u ) , in R 3 , Δ ϕ = ( ω + ϕ ) u 2 , in R 3 , \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}-\Delta u-\left(2\omega +\phi )\phi u=g\left(u),\hspace{1.0em}{\rm{in}}\hspace{1em}{{\mathbb{R}}}^{3},\hspace{1.0em}\\ \Delta \phi =\left(\omega +\phi ){u}^{2},\hspace{1.0em}{\rm{in}}\hspace{1em}{{\mathbb{R}}}^{3},\hspace{1.0em}\end{array}\right. where ω \omega is a constant that stands for the phase; u u and ϕ \phi are unknowns and g g satisfies the Berestycki-Lions condition [ Nonlinear scalar field equations. I. Existence of a ground state , Arch. Rational Mech. Anal. 82 (1983), 313–345; Nonlinear scalar field equations. II. Existence of infinitelymany solutions , Arch. Rational Mech. Anal. 82 (1983), 347–375]. The Klein-Gordon-Maxwell system is a model describing solitary waves for the nonlinear Klein-Gordon equation interacting with an electromagnetic field. By using variational methods and some analysis techniques, the existence of positive solution and multiple solutions can be obtained. Moreover, we study the properties of decay estimates and asymptotic behavior for the positive solution.

This article is concerned with the study of the initial value problem for the three-dimensional viscous Boussinesq system in the thin domain Ω ≔ R 2 × ( 0 , R ) \Omega := {{\mathbb{R}}}^{2}\times \left(0,R) . We construct a global finite energy Sobolev regularity solution ( v , ρ ) ∈ H s ( Ω ) × H s ( Ω ) \left({\bf{v}},\rho )\in {H}^{s}\left(\Omega )\times {{\mathbb{H}}}^{s}\left(\Omega ) with the small initial data in the Sobolev space H s + 2 ( Ω ) × H s + 2 ( Ω ) {H}^{s+2}\left(\Omega )\times {{\mathbb{H}}}^{s+2}\left(\Omega ) . Some features of this article are the following: (i) we do not require the initial data to be axisymmetric; (ii) the Sobolev exponent s s can be an arbitrary big positive integer; and (iii) the explicit asymptotic expansion formulas of Sobolev regular solution is given. The key point of the proof depends on the structure of the perturbation system by means of a suitable initial approximation function of the Nash-Moser iteration scheme.

In this article, we are concerned with normalized solutions for the p p -Laplacian equation with a trapping potential and L r {L}^{r} -supercritical growth, where r = p r=p or 2 . 2. The solutions correspond to critical points of the underlying energy functional subject to the L r {L}^{r} -norm constraint, namely, ∫ R N ∣ u ∣ r d x = c {\int }_{{{\mathbb{R}}}^{N}}| u{| }^{r}{\rm{d}}x=c for given c > 0 . c\gt 0. When r = p , r=p, we show that such problem has a ground state with positive energy for c c small enough. When r = 2 , r=2, we show that such problem has at least two solutions both with positive energy, which one is a ground state and the other one is a high-energy solution.

We consider the nonlinear elliptic–parabolic boundary value problem involving the Dirichlet-to-Neumann operator of p-Laplace type at the critical Sobolev exponent. We first obtain the existence and asymptotic estimates of the global solution, and the sufficient conditions of finite time blowup of the solution by using the energy method. Second, we improve the regularity of solution by Moser-type iteration. Finally, we analyze the long-time asymptotic behavior of the global solution. Moreover, with the help of the concentration compactness principle, we present a precise description of the concentration phenomenon of the solution in the forward time infinity.

This article considers the Cauchy problem for compressible Euler system in R {\bf{R}} with damping, in which the coefficient depends on the space variable. Assuming the initial density has a small perturbation around a constant state and both the small perturbation and the small initial velocity field are compact supported, finite-time blow-up result will be established. This result reveals the fact that if the space-dependent damping coefficient decays fast enough in the far field (belongs to L 1 ( R ) {L}^{1}\left({\bf{R}}) ), then the damping is non-effective to the long-time behavior of the solution.

In this article, we study the following general Kirchhoff type equation: − M ∫ R 3 ∣ ∇ u ∣ 2 d x Δ u + u = a ( x ) f ( u ) in R 3 , -M\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}| \nabla u{| }^{2}{\rm{d}}x\right)\Delta u+u=a\left(x)f\left(u)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3}, where inf R + M > 0 {\inf }_{{{\mathbb{R}}}^{+}}M\gt 0 and f f is a superlinear subcritical term. By using the Pohozǎev manifold, we obtain the existence of high energy solutions of the aforementioned equation without the well-known Ambrosetti-Rabinowitz type condition.

In this article, we formulate a model describing the evolution of thickness of a grounded shallow ice sheet. The thickness of the ice sheet is constrained to be nonnegative. This renders the problem under consideration an obstacle problem. A rigorous analysis shows that the model is thus governed by a set of variational inequalities that involve nonlinearities in the time derivative and in the elliptic term, and that it admits solutions, whose existence is established by means of a semi-discrete scheme and the penalty method.

We present a survey concerning the convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier-Stokes equations to solutions of the Euler equations. After considering the Cauchy problem, particular attention is given to the convergence under Navier slip-type boundary conditions. We show that, in the presence of flat boundaries (typically, the half-space case), convergence holds, uniformly in time, with respect to the initial data’s norm. In spite of this result (and of a similar result for arbitrary two-dimensional domains), strong inviscid limit results are proved to be false in general domains, in correspondence to a very large family of smooth initial data. In Section 6, we present a result in this direction.

In the present work, we establish a blow-up criterion for viscoelastic wave equations with nonlinear damping, logarithmic source, delay in the velocity, and acoustic boundary conditions. Due to the nonlinear damping term, we cannot apply the concavity method introduced by Levine. Thus, we use the energy method to show that the solution with negative initial energy blows up after finite time. Furthermore, we investigate the upper and lower bounds of the blow-up time.

In this article, we study the asymptotic behaviour at infinity for viscosity solutions to a singular Monge-Ampère equation in half space from affine geometry. In particular, we extend the Liouville theorem for smooth solutions to the case of viscosity solutions by a completely different method from the smooth case.

The current article is concerned with the traveling fronts for a class of double degenerate equations with delay. We first show that the traveling fronts decay algebraically at one end, while those may decay exponentially or algebraically at the other end, which depend on the wave speed of traveling fronts. Based on the asymptotical behavior, the uniqueness and stability of traveling fronts are then proved. Of particular interest is the effect of the lower order term and higher order term on the critical speed. We mention that, under the double degenerate case, the nonlinear reaction is less competitive due to the appearance of degeneracy. This yields that the critical speed depends on the lower order term and higher order term, which is different from the nondegenerate case.

We consider a nonlinear singular Dirichlet problem driven by the ( p , q ) \left(p,q) -Laplacian and a reaction where the singular term u − η {u}^{-\eta } is multiplied by a strictly positive Carathéodory function f ( z , u ) f\left(z,u) . By using a topological approach, based on the Leray-Schauder alternative principle, we show the existence of a smooth positive solution.

In this article, we consider anisotropic parabolic systems of p p -Laplace type. The model case is the parabolic p i {p}_{i} -Laplace system u t − ∑ i = 1 n ∂ ∂ x i ( ∣ D i u ∣ p i − 2 D i u ) = 0 {u}_{t}-\mathop{\sum }\limits_{i=1}^{n}\frac{\partial }{\partial {x}_{i}}({| {D}_{i}u| }^{{p}_{i}-2}{D}_{i}u)=0 with exponents p i ≥ 2 {p}_{i}\ge 2 . Under the assumption that the exponents are not too far apart, i.e., the difference p max − p min {p}_{\max }-{p}_{\min } is sufficiently small, we establish a higher integrability result for weak solutions. This extends a result, which was only known for the elliptic setting, to the parabolic setting.

We consider analytic families of planar vector fields depending analytically on the parameters in Λ \Lambda that guarantee the existence of a (may be degenerate and with characteristic directions) monodromic singularity. We characterize the structure of the asymptotic Dulac series of the Poincaré map associated to the singularity when the family possesses a Puiseux inverse integrating factor in terms of its multiplicity and index. This characterization is only valid in a restricted monodromic parameter space Λ \ Λ ∗ \Lambda \backslash {\Lambda }^{\ast } associated to the nonexistence of local curves with zero angular speed. As a byproduct, we are able to study the center-focus problem (under the assumption of the existence of some Cauchy principal values) in very degenerated cases where no other tools are available. We illustrate the theory with several nontrivial examples.

We study the asymptotic behavior of solutions for n n -component Ginzburg-Landau equations as ε → 0 \varepsilon \to 0 . We prove that the minimizers converge locally in any C k {C}^{k} -norm to a solution of a system of generalized harmonic map equations.

We are concerned with the following Kirchhoff-type equation with exponential critical nonlinearities − a + b ∫ R 2 ∣ ∇ u ∣ 2 d x Δ u + ( h ( x ) + μ V ( x ) ) u = K ( x ) f ( u ) in R 2 , -\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{2}}| \nabla u{| }^{2}{\rm{d}}x\right)\Delta u+\left(h\left(x)+\mu V\left(x))u=K\left(x)f\left(u)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{2}, where a , b , μ > 0 a,b,\mu \gt 0 , the potential V V has a bounded set of zero points and decays at infinity as ∣ x ∣ − γ | x{| }^{-\gamma } with γ ∈ ( 0 , 2 ) \gamma \in \left(0,2) , the weight K K has finite singular points and may have exponential growth at infinity. By using the truncation technique and working in some weighted Sobolev space, we obtain the existence of a mountain pass solution for μ > 0 \mu \gt 0 large and the concentration behavior of solutions as μ → + ∞ \mu \to +\infty .

In this article, we study certain critical Schrödinger-Kirchhoff-type systems involving the fractional p p -Laplace operator on a bounded domain. More precisely, using the properties of the associated functional energy on the Nehari manifold sets and exploiting the analysis of the fibering map, we establish the multiplicity of solutions for such systems.

In this paper, we study multiplicity results for double phase problems of Kirchhoff type with right-hand sides that include a parametric singular term and a nonlinear term of subcritical growth. Under very general assumptions on the data, we prove the existence of at least two weak solutions that have different energy sign. Our treatment is based on the fibering method in form of the Nehari manifold. We point out that we cover both the nondegenerate as well as the degenerate Kirchhoff case in our setting.

We prove Li-Yau-Kröger-type bounds for Neumann-type eigenvalues of the biharmonic operator on bounded domains in a Euclidean space. We also prove sharp estimates for lower order eigenvalues of a biharmonic Steklov problem and of the Laplacian, which directly implies two sharp Reilly-type inequalities for the corresponding first nonzero eigenvalue.

In this article, we study the following Kirchhoff equation: (0.1) − ( a + b ‖ ∇ u ‖ L 2 ( R 3 ) 2 ) Δ u + V ( ∣ x ∣ ) u = f ( u ) in R 3 , -(a+b\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2})\Delta u+V\left(| x| )u=f\left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}, where a , b > 0 a,b\gt 0 , V V is a positive radial potential function, and f ( u ) f\left(u) is an asymptotically cubic term. The nonlocal term b ‖ ∇ u ‖ L 2 ( R 3 ) 2 Δ u b\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta u is 3-homogeneous in the sense that b ‖ ∇ t u ‖ L 2 ( R 3 ) 2 Δ ( t u ) = t 3 b ‖ ∇ u ‖ L 2 ( R 3 ) 2 Δ u b\Vert \nabla tu{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta \left(tu)={t}^{3}b\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta u , so it competes complicatedly with the asymptotically cubic term f ( u ) f\left(u) , which is totally different from the super-cubic case. By using the Miranda theorem and classifying the domain partitions, via the gluing method and variational method, we prove that for each positive integer k k , equation (0.1) has a radial nodal solution U k , 4 b {U}_{k,4}^{b} , which has exactly k + 1 k+1 nodal domains. Moreover, we show that the energy of U k , 4 b {U}_{k,4}^{b} is strictly increasing in k k , and for any sequence { b n } → 0 + , \left\{{b}_{n}\right\}\to {0}_{+}, up to a subsequence, U k , 4 b n {U}_{k,4}^{{b}_{n}} converges strongly to U k , 4 0 {U}_{k,4}^{0} in H 1 ( R 3 ) {H}^{1}\left({{\mathbb{R}}}^{3}) , where U k , 4 0 {U}_{k,4}^{0} also has k + 1 k+1 nodal domains exactly and solves the classical Schrödinger equation: − a Δ u + V ( ∣ x ∣ ) u = f ( u ) in R 3 . -a\Delta u+V\left(| x| )u=f\left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}. Our results extend the ones in Deng et al. from the super-cubic case to the asymptotically cubic case.

Our aim of this article is to study qualitative properties of Dirichlet problems involving the Hardy-Leray operator ℒ V ≔ − Δ + V {{\mathcal{ {\mathcal L} }}}_{V}:= -\Delta +V , where V ( x ) = ∑ i = 1 m μ i ∣ x − A i ∣ 2 V\left(x)={\sum }_{i=1}^{m}\frac{{\mu }_{i}}{{| x-{A}_{i}| }^{2}} , with μ i ≥ − ( N − 2 ) 2 4 {\mu }_{i}\ge -\frac{{\left(N-2)}^{2}}{4} being the Hardy-Leray potential containing the polars’ set A m = { A i : i = 1 , … , m } {{\mathcal{A}}}_{m}=\left\{{A}_{i}:i=1,\ldots ,m\right\} in R N {{\mathbb{R}}}^{N} ( N ≥ 2 N\ge 2 ). Since the inverse-square potentials are critical with respect to the Laplacian operator, the coefficients { μ i } i = 1 m {\left\{{\mu }_{i}\right\}}_{i=1}^{m} and the locations of polars { A i } \left\{{A}_{i}\right\} play an important role in the properties of solutions to the related Poisson problems subject to zero Dirichlet boundary conditions. Let Ω \Omega be a bounded domain containing A m {{\mathcal{A}}}_{m} . First, we obtain increasing Dirichlet eigenvalues: ℒ V u = λ u in Ω , u = 0 on ∂ Ω , {{\mathcal{ {\mathcal L} }}}_{V}u=\lambda u\hspace{1.0em}{\rm{in}}\hspace{0.33em}\Omega ,\hspace{1.0em}u=0\hspace{1.0em}{\rm{on}}\hspace{0.33em}\partial \Omega , and the positivity of the principle eigenvalue depends on the strength μ i {\mu }_{i} and polars’ setting. When the spectral does not contain the origin, we then consider the weak solutions of the Poisson problem ( E ) ℒ V u = ν in Ω , u = 0 on ∂ Ω , \left(E)\hspace{1.0em}\hspace{1.0em}{{\mathcal{ {\mathcal L} }}}_{V}u=\nu \hspace{1em}{\rm{in}}\hspace{0.33em}\Omega ,\hspace{1.0em}u=0\hspace{1em}{\rm{on}}\hspace{0.33em}\partial \Omega , when ν \nu belongs to L p ( Ω ) {L}^{p}\left(\Omega ) , with p > 2 N N + 2 p\gt \frac{2N}{N+2} in the variational framework, and we obtain a global weighted L ∞ {L}^{\infty } estimate when p > N 2 p\gt \frac{N}{2} . When the principle eigenvalue is positive and ν \nu is a Radon measure, we build a weighted distributional framework to show the existence of weak solutions of problem ( E ) \left(E) . Moreover, via this weighted distributional framework, we can obtain a sharp assumption of ν ∈ C γ ( Ω ¯ \ A m ) \nu \in {{\mathcal{C}}}^{\gamma }\left(\bar{\Omega }\setminus {{\mathcal{A}}}_{m}) for the existence of isolated singular solutions for problem ( E ) \left(E) .

We are concerned with the incompressible limit of global-in-time strong solutions with arbitrary large initial velocity for the three-dimensional compressible viscoelastic equations. The incompressibility is achieved by the large value of the volume viscosity, which is different from the low Mach number limit. To obtain the uniform estimates, we establish the estimates for the potential part and the divergence-free part of the velocity. As the volume viscosity goes to infinity, the dispersion associated with the pressure waves tends to disappear, but the large volume viscosity provides a strong dissipation on the potential part of the velocity forcing the flow to be almost incompressible.

In this article, we show continuity of commutators of Calderón-Zygmund operators [ b , T ] \left[b,T] with BMO functions in generalized Orlicz-Morrey spaces M Φ , φ ( R n ) {M}^{\Phi ,\varphi }\left({{\mathbb{R}}}^{n}) . We give necessary and sufficient conditions for the boundedness of the genuine Calderón-Zygmund operators T T and for their commutators [ b , T ] \left[b,T] on generalized Orlicz-Morrey spaces, respectively.

In this article, we develop a new set of results based on a non-local gradient jointly inspired by the Riesz s s -fractional gradient and peridynamics, in the sense that its integration domain depends on a ball of radius δ > 0 \delta \gt 0 (horizon of interaction among particles, in the terminology of peridynamics), while keeping at the same time the singularity of the Riesz potential in its integration kernel. Accordingly, we define a functional space suitable for non-local models in calculus of variations and partial differential equations. Our motivation is to develop the proper functional analysis framework to tackle non-local models in continuum mechanics , which requires working with bounded domains, while retaining the good mathematical properties of Riesz s s -fractional gradients. This functional space is defined consistently with Sobolev and Bessel fractional ones: we consider the closure of smooth functions under the natural norm obtained as the sum of the L p {L}^{p} norms of the function and its non-local gradient. Among the results showed in this investigation, we highlight a non-local version of the fundamental theorem of calculus (namely, a representation formula where a function can be recovered from its non-local gradient), which allows us to prove inequalities in the spirit of Poincaré, Morrey, Trudinger, and Hardy as well as the corresponding compact embeddings. These results are enough to show the existence of minimizers of general energy functionals under the assumption of convexity. Equilibrium conditions in this non-local situation are also established, and those can be viewed as a new class of non-local partial differential equations in bounded domains.

We prove an existence result for obstacle problems related to convection-diffusion parabolic equations with singular coefficients in the convective term. Our operator is not coercive, the obstacle function is time-dependent irregular, and the coefficients in the lower-order term belong to a borderline mixed Lebesgue-Marcinkiewicz space.

We study general problems modeling electrostatic microelectromechanical systems devices ( P λ ) φ ( r , − u ′ ( r ) ) = λ ∫ 0 r f ( s ) g ( u ( s ) ) d s , r ∈ ( 0 , 1 ) , 0 < u ( r ) < 1 , r ∈ ( 0 , 1 ) , u ( 1 ) = 0 , \left\{\begin{array}{ll}\varphi (r,-u^{\prime} \left(r))=\lambda \underset{0}{\overset{r}{\displaystyle \int }}\frac{f\left(s)}{g\left(u\left(s))}{\rm{d}}s,\hspace{1.0em}& r\in \left(0,1),\\ 0\lt u\left(r)\lt 1,\hspace{1.0em}& r\in \left(0,1),\\ u\left(1)=0,\hspace{1.0em}\end{array}\right. where φ \varphi , g g , and f f are some functions on [ 0 , 1 ] \left[0,1] and λ > 0 \lambda \gt 0 is a parameter. We obtain results on the existence and regularity of a touchdown solution to ( P λ {P}_{\lambda } ) and find upper and lower bounds on the respective pull-in voltage. In the particular case, when φ ( r , v ) = r α ∣ v ∣ β v \varphi \left(r,v)={r}^{\alpha }{| v| }^{\beta }v , i.e., when the associated differential equation involves the operator r − γ ( r α ∣ u ′ ∣ β u ′ ) ′ {r}^{-\gamma }\left({r}^{\alpha }{| u^{\prime} | }^{\beta }u^{\prime} )^{\prime} , we obtain an exact asymptotic behavior of the touchdown solution in a neighborhood of the origin.

In this article, under some weaker assumptions on a > 0 a\gt 0 and f f , the authors aim to study the existence of nontrivial radial solutions and nonexistence of nontrivial solutions for the following Schrödinger-Poisson system with zero mass potential − Δ u + ϕ u = − a ∣ u ∣ p − 2 u + f ( u ) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 , \left\{\begin{array}{ll}-\Delta u+\phi u=-a{| u| }^{p-2}u+f\left(u),& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi ={u}^{2},& x\in {{\mathbb{R}}}^{3},\end{array}\right. where p ∈ 2 , 12 5 p\in \left(2,\frac{12}{5}\right) . In particular, as a corollary for the following system: − Δ u + ϕ u = − ∣ u ∣ p − 2 u + ∣ u ∣ q − 2 u , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 , \left\{\begin{array}{ll}-\Delta u+\phi u=-{| u| }^{p-2}u+{| u| }^{q-2}u,& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi ={u}^{2},& x\in {{\mathbb{R}}}^{3},\end{array}\right. a sufficient and necessary condition is obtained on the existence of nontrivial radial solutions.

We address a quasi-stationary fluid–structure interaction problem coupled with cell reactions and growth, which comes from the plaque formation during the stage of the atherosclerotic lesion in human arteries. The blood is modeled by the incompressible Navier-Stokes equation, while the motion of vessels is captured by a quasi-stationary equation of nonlinear elasticity. The growth happens when both cells in fluid and solid react, diffuse and transport across the interface, resulting in the accumulation of foam cells, which are exactly seen as the plaques. Via a fixed-point argument, we derive the local well-posedness of the nonlinear system, which is sustained by the analysis of decoupled linear systems.

We establish fine bounds for best constants of the fractional subcritical Sobolev embeddings W 0 s , p ( Ω ) ↪ L q ( Ω ) , {W}_{0}^{s,p}(\Omega )\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{q}(\Omega ), where N ≥ 1 N\ge 1 , 0 < s < 1 0\lt s\lt 1 , p = 1 , 2 p=1,2 , 1 ≤ q < p s ∗ = N p N − s p 1\le q\lt {p}_{s}^{\ast }=\frac{Np}{N-sp} , and Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded smooth domain or the whole space R N {{\mathbb{R}}}^{N} . Our results cover the borderline case p = 1 p=1 , the Hilbert case p = 2 p=2 , N > 2 s N\gt 2s , and the so-called Sobolev limiting case N = 1 N=1 , s = 1 2 s=\frac{1}{2} , and p = 2 p=2 , where a sharp asymptotic estimate is given by means of a limiting procedure. We apply the obtained results to prove existence and non-existence of solutions for a wide class of nonlocal partial differential equations.

In the present work, we find the Lie point symmetries of the Ricci flow on an n -dimensional manifold, and we introduce a method in order to reutilize these symmetries to obtain the Lie point symmetries of particular metrics. We apply this method to retrieve the Lie point symmetries of the Einstein equations (seen as a “static” Ricci flow) and of some particular types of metrics of interest, such as, on warped products of manifolds. Finally, we use the symmetries found to obtain invariant solutions of the Ricci flow for the particular families of metrics considered.

The article investigates a second-order nonlinear difference equation of Emden-Fowler type Δ 2 u ( k ) ± k α u m ( k ) = 0 , {\Delta }^{2}u\left(k)\pm {k}^{\alpha }{u}^{m}\left(k)=0, where k k is the independent variable with values k = k 0 , k 0 + 1 , … k={k}_{0},{k}_{0}+1,\ldots \hspace{0.33em} , u : { k 0 , k 0 + 1 , … } → R u:\left\{{k}_{0},{k}_{0}+1,\ldots \hspace{0.33em}\right\}\to {\mathbb{R}} is the dependent variable, k 0 {k}_{0} is a fixed integer, and Δ 2 u ( k ) {\Delta }^{2}u\left(k) is its second-order forward difference. New conditions with respect to parameters α ∈ R \alpha \in {\mathbb{R}} and m ∈ R m\in {\mathbb{R}} , m ≠ 1 m\ne 1 , are found such that the equation admits a solution asymptotically represented by a power function that is asymptotically equivalent to the exact solution of the nonlinear second-order differential Emden-Fowler equation y ″ ( x ) ± x α y m ( x ) = 0 . {y}^{^{\prime\prime} }\left(x)\pm {x}^{\alpha }{y}^{m}\left(x)=0. Two-term asymptotic representations are given not only for the solution itself but also for its first- and second-order forward differences as well. Previously known results are discussed, and illustrative examples are considered.

In this work, we consider the topological gradient method to deal with an inverse problem associated with three-dimensional Stokes equations. The problem consists in detecting unknown point forces acting on fluid from measurements on the boundary of the domain. We present an asymptotic expansion of the considered cost function using the topological sensitivity analysis method. A detection algorithm is then presented using the developed formula. Some numerical tests are presented to show the efficiency of the presented algorithm.

We consider the nonlocal quasilinear elliptic problem: − Δ m u ( x ) = H ( x ) ( ( I α * ( Q f ( u ) ) ) ( x ) ) β g ( u ( x ) ) in Ω , -{\Delta }_{m}u\left(x)=H\left(x){(\left({I}_{\alpha }* \left(Qf\left(u)))\left(x))}^{\beta }g\left(u\left(x))\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega , where Ω \Omega is a smooth domain in R N {{\mathbb{R}}}^{N} , β ≥ 0 \beta \ge 0 , I α {I}_{\alpha } , 0 < α < N 0\lt \alpha \lt N , stands for the Riesz potential, f , g : [ 0 , a ) → [ 0 , ∞ ) f,g:\left[0,a)\to \left[0,\infty ) , 0 < a ≤ ∞ 0\lt a\le \infty , are monotone nondecreasing functions with f ( s ) , g ( s ) > 0 f\left(s),g\left(s)\gt 0 for s > 0 s\gt 0 , and H , Q : Ω → R H,Q:\Omega \to {\mathbb{R}} are nonnegative measurable functions. We provide explicit quantitative pointwise estimates on positive weak supersolutions. As an application, we obtain bounds on extremal parameters of the related nonlinear eigenvalue problems in bounded domains for various nonlinearities f f and g g such as e u , ( 1 + u ) p {e}^{u},{\left(1+u)}^{p} , and ( 1 − u ) − p {\left(1-u)}^{-p} , p > 1 p\gt 1 . We also discuss the Liouville-type results in unbounded domains.

The aim of this article is twofold. First, we study the uniform complex time heat kernel estimates of e − z ( − Δ ) α 2 {e}^{-z{\left(-\Delta )}^{\frac{\alpha }{2}}} for α > 0 , z ∈ C + \alpha \gt 0,z\in {{\mathbb{C}}}^{+} . To this end, we establish the asymptotic estimates for P ( z , x ) P\left(z,x) with z z satisfying 0 < ω ≤ ∣ θ ∣ < π 2 0\lt \omega \le | \theta | \lt \frac{\pi }{2} followed by the uniform complex time heat kernel estimates. Second, we studied the uniform complex time estimates of the analytic semigroup generated by H = ( − Δ ) α 2 + V H={\left(-\Delta )}^{\tfrac{\alpha }{2}}+V , where V V belongs to higher-order Kato class.

In this article, we consider the Cauchy problem for a semi-linear wave equation with time-dependent damping and memory nonlinearity. Conditions for global existence are presented in the energy space H 1 ( R n ) × L 2 ( R n ) {H}^{1}\left({{\mathbb{R}}}^{n})\times {L}^{2}\left({{\mathbb{R}}}^{n}) , n ≥ 1 n\ge 1 .

In this article, we study the fractional critical Choquard equation with a nonlocal perturbation: ( − Δ ) s u = λ u + α ( I μ * ∣ u ∣ q ) ∣ u ∣ q − 2 u + ( I μ * ∣ u ∣ 2 μ , s * ) ∣ u ∣ 2 μ , s * − 2 u , in R N , {\left(-{\Delta })}^{s}u=\lambda u+\alpha \left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{q}){| u| }^{q-2}u+\left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{{2}_{\mu ,s}^{* }}){| u| }^{{2}_{\mu ,s}^{* }-2}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, having prescribed mass ∫ R N u 2 d x = c 2 , \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={c}^{2}, where s ∈ ( 0 , 1 ) , N > 2 s , 0 < μ < N , α > 0 , c > 0 s\in \left(0,1),N\gt 2s,0\lt \mu \lt N,\alpha \gt 0,c\gt 0 , and I μ ( x ) {I}_{\mu }\left(x) is the Riesz potential given by I μ ( x ) = A μ ∣ x ∣ μ with A μ = Γ μ 2 2 N − μ π N ⁄ 2 Γ N − μ 2 , {I}_{\mu }\left(x)=\frac{{A}_{\mu }}{{| x| }^{\mu }}\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{A}_{\mu }=\frac{\Gamma \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{\mu }{2}\right)}{{2}^{N-\mu }{\pi }^{N/2}\Gamma \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{N-\mu }{2}\right)}, and 2 N − μ N < q < 2 μ , s * = 2 N − μ N − 2 s \frac{2N-\mu }{N}\lt q\lt {2}_{\mu ,s}^{* }=\frac{2N-\mu }{N-2s} is the fractional Hardy-Littlewood-Sobolev critical exponent. Under the L 2 {L}^{2} -subcritical perturbation α ( I μ * ∣ u ∣ q ) ∣ u ∣ q − 2 u \alpha \left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{q}){| u| }^{q-2}u with exponent 2 N − μ N < q < 2 N − μ + 2 s N \frac{2N-\mu }{N}\lt q\lt \frac{2N-\mu +2s}{N} , we obtain the existence of normalized ground states and mountain-pass-type solutions. Meanwhile, for the L 2 {L}^{2} -critical and L 2 {L}^{2} -supercritical cases 2 N − μ + 2 s N ≤ q < 2 N − μ N − 2 s \frac{2N-\mu +2s}{N}\le q\lt \frac{2N-\mu }{N-2s} , we also prove that the equation has ground states of mountain-pass-type.

In this article, we establish some reverse weighted Hardy-Littlewood-Sobolev inequalities on the Heisenberg group. We then show the existence of extremal functions for the above inequalities by combining the subcritical approach and the renormalization method.

This article is concerned with the stability and dynamics for the weak damped Lamé system with nonlinear time-varying delay in a bounded domain. Under some appropriate assumptions, the global well-posedness and asymptotic stability are shown in the case where the delay coefficient is upper dominated by the damping one. Moreover, the finite dimensional global and exponential attractors have also been presented by relying on quasi-stability arguments. The results in this article is an extension of Ma, Mesquita, and Seminario-Huertas’s recent work [ Smooth dynamics of weakly damped Lamé systems with delay , SIAM J. Math. Anal. 53 (2021), no. 4, 3759–3771].

In this article, we study the long-time dynamical behavior of the solution for a class of semilinear edge-degenerate parabolic equations on manifolds with edge singularities. By introducing a family of potential well and compactness method, we reveal the dependence between the initial data and the long-time dynamical behavior of the solution. Specifically, we give the threshold condition for the initial data, which makes the solution exist globally or blowup in finite-time with subcritical, critical, and supercritical initial energy, respectively. Moreover, we also discussed the long-time behavior of the global solution, the estimate of blowup time, and blowup rate. Our results show that the relationship between the initial data and the long-time behavior of the solution can be revealed in the weighted Sobolev spaces for nonlinear parabolic equations on manifolds with edge singularities.

This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity u p {u}^{p} in a bounded domain Ω \Omega with the homogeneous Neumann boundary condition and positive initial values. In the case of p > 1 p\gt 1 , we prove the blowup of solutions u ( x , t ) u\left(x,t) in the sense that ‖ u ( ⋅ , t ) ‖ L 1 ( Ω ) \Vert u\left(\hspace{0.33em}\cdot \hspace{0.33em},t){\Vert }_{{L}^{1}\left(\Omega )} tends to ∞ \infty as t t approaches some value, by using a comparison principle for the corresponding ordinary differential equations and constructing special lower solutions. Moreover, we provide an upper bound for the blowup time. In the case of 0 < p < 1 0\lt p\lt 1 , we establish the global existence of solutions in time based on the Schauder fixed-point theorem.

We prove improved differentiability results for relaxed minimisers of vectorial convex functionals with ( p , q ) \left(p,q) -growth, satisfying a Hölder-growth condition in x x . We consider both Dirichlet and Neumann boundary data. In addition, we obtain a characterisation of regular boundary points for such minimisers. In particular, in case of homogeneous boundary conditions, this allows us to deduce partial boundary regularity of relaxed minimisers on smooth domains for radial integrands. We also obtain some partial boundary regularity results for non-homogeneous Neumann boundary conditions.

We consider a parametric (with two parameters μ , λ > 0 \mu ,\lambda \gt 0 ) Dirichlet problem driven by the double phase differential operator and a reaction which has the competing effect of a singular term and of a superlinear perturbation. We prove a bifurcation-type result in the parameter λ > 0 \lambda \gt 0 , when the other parameter μ > 0 \mu \gt 0 is large.

In this article, we investigate the inverse problem of identification of a discontinuous parameter and a discontinuous boundary datum to an elliptic inclusion problem involving a double phase differential operator, a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. First, we apply a surjectivity theorem for multivalued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone mapping to examine the existence of a nontrivial solution to the double phase obstacle problem, which exactly relies on the first eigenvalue of the Steklov eigenvalue problem for the p p -Laplacian. Then, a nonlinear inverse problem driven by the double phase obstacle equation is considered. Finally, by introducing the parameter-to-solution-map, we establish a continuous result of Kuratowski type and prove the solvability of the inverse problem.

In this article, we will develop an analytical approach to construct the global bounded weak solutions to the initial-boundary value problem of a three-dimensional chemotaxis-Stokes system with porous medium cell diffusion Δ n m \Delta {n}^{m} for m ≥ 65 63 m\ge \frac{65}{63} and general sensitivity. In particular, this extended the precedent results which asserted global solvability within the larger range m > 7 6 m\gt \frac{7}{6} for general sensitivity (M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. 54 (2015), 3789–3828) or m > 9 8 m\gt \frac{9}{8} for scalar sensitivity (M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differ. Equ. 264 (2018), 6109–6151). Our proof is based on a new observation on the quasi-energy-type functional and on an induction argument.

The Cauchy problem for three-dimensional (3D) isentropic compressible radiation hydrodynamic equations is considered. When both shear and bulk viscosity coefficients depend on the mass density ρ \rho in a power law ρ δ {\rho }^{\delta } (with 0 < δ < 1 0\lt \delta \lt 1 ), based on some elaborate analysis of this system’s intrinsic singular structures, we establish the local-in-time well-posedness of regular solution with arbitrarily large initial data and far field vacuum in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions. Note that due to the appearance of the vacuum, the momentum equations are degenerate both in the time evolution and viscous stress tensor, which, along with the strong coupling between the fluid and the radiation field, make the study on corresponding well-posedness challenging. For proving the existence, we first introduce an enlarged reformulated structure by considering some new variables, which can transfer the degeneracies of the radiation hydrodynamic equations to the possible singularities of some special source terms, and then carry out some singularly weighted energy estimates carefully designed for this reformulated system.

In this paper, we concern about a modified version of the Keller-Segel model. The Keller-Segel is a system of partial differential equations used for modeling Chemotaxis in which chemical substances impact the movement of mobile species. For considering memory effects on the model, we replace the classical derivative with respect to time variable by the time-fractional derivative in the sense of Caputo. From this modification, we focus on the well-posedness of the Cauchy problem associated with such the model. Precisely, when the spatial variable is considered in the space R d {{\mathbb{R}}}^{d} , a global solution is obtained in a critical homogeneous Besov space with the assumption that the initial datum is sufficiently small. For the bounded domain case, by using a discrete spectrum of the Neumann Laplace operator, we provide the existence and uniqueness of a mild solution in Hilbert scale spaces. Moreover, the blowup behavior is also studied. To overcome the challenges of the problem and obtain all the aforementioned results, we use the Banach fixed point theorem, some special functions like the Mainardi function and the Mittag-Leffler function, as well as their properties.

In this article, we study the evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity V = 1 α ψ ( x ) κ α V=\frac{1}{\alpha }\psi \left(x){\kappa }^{\alpha } for α < 0 \alpha \lt 0 or α > 1 \alpha \gt 1 , where x ∈ [ 0 , 2 m π ] x\in \left[0,2m\pi ] is the tangential angle at the point on evolving curves. For − 1 ≤ α < 0 -1\le \alpha \lt 0 , we show the flow exists globally and the rescaled flow has a full-time convergence. For α < − 1 \alpha \lt -1 or α > 1 \alpha \gt 1 , we show only type I singularity arises in the flow, and the rescaled flow has subsequential convergence, i.e. for any time sequence, there is a time subsequence along which the rescaled curvature of evolving curves converges to a limit function; furthermore, if the anisotropic function ψ \psi and the initial curve both have some symmetric structure, the subsequential convergence could be refined to be full-time convergence.

In this article, we study the large-time behavior of combination of the rarefaction waves with viscous contact wave for a one-dimensional compressible Navier-Stokes system whose transport coefficients depend on the temperature. It is shown that if the adiabatic exponent γ is suitably close to 1, the unique solution global in time to ideal polytropic gas exists and asymptotically tends toward the combination of a viscous contact wave with rarefaction waves under large initial perturbation. New and subtle analysis is developed to overcome difficulties due to the smallness of γ – 1 to derive heat kernel estimates. Moreover, our results extend the studies in a previous work [F. M. Huang, J. Li, and A. Matsumura, Arch. Ration. Mech. Anal. 197 (2010), no. 1, 89–116].

In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form (0.1) − Δ u + V ( x ) u = f ( x , u ) + λ a ( x ) ∣ u ∣ q − 2 u , x ∈ R 2 , -\Delta u+V\left(x)u=f\left(x,u)+\lambda a\left(x)| u{| }^{q-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{2}, where λ > 0 \lambda \gt 0 , q ∈ ( 1 , 2 ) q\in \left(1,2) , a ∈ L 2 / ( 2 − q ) ( R 2 ) a\in {L}^{2\text{/}\left(2-q)}\left({{\mathbb{R}}}^{2}) , V ( x ) V\left(x) , and f ( x , t ) f\left(x,t) are 1-periodic with respect to x x , and f ( x , t ) f\left(x,t) has critical exponential growth at t = ∞ t=\infty . By combining the variational methods, Trudinger-Moser inequality, and some new techniques with detailed estimates for the minimax level of the energy functional, we prove the existence of a nontrivial solution for the aforementioned equation under some weak assumptions. Our results show that the presence of the concave term (i.e. λ > 0 \lambda \gt 0 ) is very helpful to the existence of nontrivial solutions for equation (0.1) in one sense.

The article deals with the existence of a pair of nontrivial nonnegative and nonpositive solutions for a nonlinear weighted quasilinear equation in R N {{\mathbb{R}}}^{N} , which involves a double-phase general variable exponent elliptic operator A {\bf{A}} . More precisely, A {\bf{A}} has behaviors like ∣ ξ ∣ q ( x ) − 2 ξ {| \xi | }^{q\left(x)-2}\xi if ∣ ξ ∣ | \xi | is small and like ∣ ξ ∣ p ( x ) − 2 ξ {| \xi | }^{p\left(x)-2}\xi if ∣ ξ ∣ | \xi | is large. Existence is proved by the Cerami condition instead of the classical Palais-Smale condition, so that the nonlinear term f ( x , u ) f\left(x,u) does not necessarily have to satisfy the Ambrosetti-Rabinowitz condition.