Band 22, Heft 1

We consider Monge-Ampère type equations involving the gradient that are elliptic in the framework of convex functions. Through suitable symmetrization we find sharp estimates to solutions of such equations. An overdetermined problem related to our Monge-Ampère type operators is also considered and we show that such a problem may admit a solution only when the underlying domain is a ball.

In this article, we are concerned with the following prescribed curvature problem involving polyharmonic operator on S N {{\mathbb{S}}}^{N} : D m u = K ( ∣ y ∣ ) u m ∗ − 1 , u > 0 in S N , u ∈ H m ( S N ) , {D}^{m}u=K\left(| y| ){u}^{{m}^{\ast }-1},\hspace{1.0em}u\gt 0\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{S}}}^{N},\hspace{1.0em}u\in {H}^{m}\left({{\mathbb{S}}}^{N}), where K ( ∣ y ∣ ) K\left(| y| ) is a positive function, m ∗ = 2 N N − 2 m {m}^{\ast }=\frac{2N}{N-2m} is the Sobolev embedding critical exponent, N > 2 m + 2 N\gt 2m+2 . D m {D}^{m} is the 2 m 2m order differential operator given by D m = ∏ l = 1 m − Δ g + 1 4 ( N − 2 l ) ( N + 2 l − 2 ) , {D}^{m}=\mathop{\prod }\limits_{l=1}^{m}\left(-{\Delta }_{g}+\frac{1}{4}\left(N-2l)\left(N+2l-2)\right), where Δ g {\Delta }_{g} is the Laplace-Beltrami operator on S N {{\mathbb{S}}}^{N} , S N {{\mathbb{S}}}^{N} is the unit sphere with Riemann metric g g . We first establish two kinds of local Pohozaev identities for polyharmonic operator, then we prove that the positive bubbling solution constructed in the study of Guo and Li is non-degenerate.

We study the following fractional logarithmic Schrödinger equation: ( − Δ ) s u + V ( x ) u = u log u 2 , x ∈ R N , {\left(-\Delta )}^{s}u+V\left(x)u=u\log {u}^{2},\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ≥ 1 N\ge 1 , ( − Δ ) s {\left(-\Delta )}^{s} denotes the fractional Laplace operator, 0 < s < 1 0\lt s\lt 1 and V ( x ) ∈ C ( R N ) V\left(x)\in {\mathcal{C}}\left({{\mathbb{R}}}^{N}) . Under different assumptions on the potential V ( x ) V\left(x) , we prove the existence of positive ground state solution and least energy sign-changing solution for the equation. It is known that the corresponding variational functional is not well defined in H s ( R N ) {H}^{s}\left({{\mathbb{R}}}^{N}) , and inspired by Cazenave ( Stable solutions of the logarithmic Schrödinger equation , Nonlinear Anal. 7 (1983), 1127–1140), we first prove that the variational functional is well defined in a subspace of H s ( R N ) {H}^{s}\left({{\mathbb{R}}}^{N}) . Then, by using minimization method and Lions’ concentration-compactness principle, we prove that the existence results.

In this article, we investigate the boundary blow-up problem Δ ∞ h u = f ( x , u ) , in Ω , u = ∞ , on ∂ Ω , \left\{\begin{array}{ll}{\Delta }_{\infty }^{h}u=f\left(x,u),& {\rm{in}}\hspace{0.33em}\Omega ,\\ u=\infty ,& {\rm{on}}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Δ ∞ h u = ∣ D u ∣ h − 3 ⟨ D 2 u D u , D u ⟩ {\Delta }_{\infty }^{h}u=| Du\hspace{-0.25em}{| }^{h-3}\langle {D}^{2}uDu,Du\rangle is the highly degenerate operator related to infinity Laplacian which comes from the absolutely minimizing Lipschitz extension and has a close relationship with the random game named tug-of-war. When the function f f satisfies the Keller-Osserman-type condition, we establish the existence of the boundary blow-up viscosity solution. Moreover, for the separable case f ( x , u ) = b ( x ) g ( u ) f\left(x,u)=b\left(x)g\left(u) , we establish the asymptotic estimate of the blow-up solution near the boundary under some regular conditions of the domain. Based on the asymptotic estimate and comparison principle, we obtain the uniqueness of the large viscosity solution. During this procedure, we also study the non-existence of the large solution. For the separable case, we show that the Keller-Osserman-type condition is sufficient and necessary for the existence of the boundary blow-up viscosity solution.

The chemotaxis–Stokes system n t + u ⋅ ∇ n = ∇ ⋅ ( D ( n ) ∇ n ) − ∇ ⋅ ( n S ( x , n , c ) ⋅ ∇ c ) , c t + u ⋅ ∇ c = Δ c − n c , u t = Δ u + ∇ P + n ∇ Φ , ∇ ⋅ u = 0 , \left\{\begin{array}{l}{n}_{t}+u\cdot \nabla n=\nabla \cdot (D\left(n)\nabla n)-\nabla \cdot (nS\left(x,n,c)\cdot \nabla c),\\ {c}_{t}+u\cdot \nabla c=\Delta c-nc,\\ {u}_{t}=\Delta u+\nabla P+n\nabla \Phi ,\hspace{1.0em}\nabla \cdot u=0,\end{array}\right. is considered in a smoothly bounded convex domain Ω ⊂ R 3 \Omega \subset {{\mathbb{R}}}^{3} , with given suitably regular functions D : [ 0 , ∞ ) → [ 0 , ∞ ) D:{[}0,\infty )\to {[}0,\infty ) , S : Ω ¯ × [ 0 , ∞ ) × ( 0 , ∞ ) → R 3 × 3 S:\overline{\Omega }\times {[}0,\infty )\times \left(0,\infty )\to {{\mathbb{R}}}^{3\times 3} and Φ : Ω ¯ → R \Phi :\overline{\Omega }\to {\mathbb{R}} such that D > 0 D\gt 0 on ( 0 , ∞ ) \left(0,\infty ) . It is shown that if with some nondecreasing S 0 : ( 0 , ∞ ) → ( 0 , ∞ ) {S}_{0}:\left(0,\infty )\to \left(0,\infty ) we have ∣ S ( x , n , c ) ∣ ≤ S 0 ( c ) c 1 2 for all ( x , n , c ) ∈ Ω ¯ × [ 0 , ∞ ) × ( 0 , ∞ ) , | S\left(x,n,c)| \le \frac{{S}_{0}\left(c)}{{c}^{\tfrac{1}{2}}}\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}\left(x,n,c)\in \overline{\Omega }\times {[}0,\infty )\times \left(0,\infty ), then for all M > 0 M\gt 0 there exists L ( M ) > 0 L\left(M)\gt 0 such that whenever liminf n → ∞ D ( n ) > L ( M ) and liminf n ↘ 0 D ( n ) n > 0 , \mathop{\mathrm{liminf}}\limits_{n\to \infty }D\left(n)\gt L\left(M)\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\mathop{\mathrm{liminf}}\limits_{n\searrow 0}\frac{D\left(n)}{n}\gt 0, for all sufficiently regular initial data ( n 0 , c 0 , u 0 ) \left({n}_{0},{c}_{0},{u}_{0}) fulfilling ‖ c 0 ‖ L ∞ ( Ω ) ≤ M \Vert {c}_{0}{\Vert }_{{L}^{\infty }\left(\Omega )}\le M an associated no-flux/no-flux/Dirichlet initial-boundary value problem admits a global bounded weak solution, classical if additionally D ( 0 ) > 0 D\left(0)\gt 0 . When combined with previously known results, this particularly implies global existence of bounded solutions when D ( n ) = n m − 1 D\left(n)={n}^{m-1} , n ≥ 0 n\ge 0 , with arbitrary m > 1 m\gt 1 , but beyond this asserts global boundedness also in the presence of diffusivities which exhibit arbitrarily slow divergence to + ∞ +\infty at large densities and of possibly singular chemotactic sensitivities.

We obtain asymptotic mean-value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. The families of equations that we consider include well-known operators such as Pucci, Issacs, and k -Hessian operators.

In this article, we prove the upper weighted critical exponents for some embeddings from weighted Sobolev spaces of radial functions into weighted Lebesgue spaces. We also consider the lower critical exponent for certain embedding.

This paper deals with the following weakly coupled nonlinear Schrödinger system − Δ u 1 + a 1 ( x ) u 1 = ∣ u 1 ∣ 2 p − 2 u 1 + b ∣ u 1 ∣ p − 2 ∣ u 2 ∣ p u 1 , x ∈ R N , − Δ u 2 + a 2 ( x ) u 2 = ∣ u 2 ∣ 2 p − 2 u 2 + b ∣ u 2 ∣ p − 2 ∣ u 1 ∣ p u 2 , x ∈ R N , \left\{\begin{array}{ll}-\Delta {u}_{1}+{a}_{1}\left(x){u}_{1}=| {u}_{1}{| }^{2p-2}{u}_{1}+b| {u}_{1}{| }^{p-2}| {u}_{2}{| }^{p}{u}_{1},& x\in {{\mathbb{R}}}^{N},\\ -\Delta {u}_{2}+{a}_{2}\left(x){u}_{2}=| {u}_{2}{| }^{2p-2}{u}_{2}+b| {u}_{2}{| }^{p-2}| {u}_{1}{| }^{p}{u}_{2},& x\in {{\mathbb{R}}}^{N},\end{array}\right. where N ≥ 1 N\ge 1 , b ∈ R b\in {\mathbb{R}} is a coupling constant, 2 p ∈ ( 2 , 2 ∗ ) 2p\in \left(2,{2}^{\ast }) , 2 ∗ = 2 N / ( N − 2 ) {2}^{\ast }=2N\hspace{0.1em}\text{/}\hspace{0.1em}\left(N-2) if N ≥ 3 N\ge 3 and + ∞ +\infty if N = 1 , 2 N=1,2 , a 1 ( x ) {a}_{1}\left(x) and a 2 ( x ) {a}_{2}\left(x) are two positive functions. Assuming that a i ( x ) ( i = 1 , 2 ) {a}_{i}\left(x)\hspace{0.33em}\left(i=1,2) satisfies some suitable conditions, by constructing creatively two types of two-dimensional mountain-pass geometries, we obtain a positive synchronized solution for ∣ b ∣ > 0 | b| \gt 0 small and a positive segregated solution for b < 0 b\lt 0 , respectively. We also show that when 1 < p < min { 2 , 2 ∗ / 2 } 1\lt p\lt {\rm{\min }}\left\{2,{2}^{\ast }\hspace{0.1em}\text{/}\hspace{0.1em}2\right\} , the positive solutions are not unique if b > 0 b\gt 0 is small. The asymptotic behavior of the solutions when b → 0 b\to 0 and b → − ∞ b\to -\infty is also studied.

In this article, we introduce a class of geometric optics measure-the L q Σ {L}_{q}\hspace{0.33em}\Sigma -reflector measure which arises from an L q {L}_{q} extension of the Σ \Sigma -reflector measure. And we ask a Minkowski-type problem for this class of measure, called the L q Σ {L}_{q}\hspace{0.33em}\Sigma -reflector problem. It is shown that the foundations of such measure have been laid by Caffarelli et al. ( Reflector problem in endowed with non-Euclidean norm , Arch. Ration. Mech. Anal. 193 (2009), no. 2, 445–473) (for the Σ \Sigma -reflector measure). Inspired by Alexandrov, we present some variational arguments and existence results of solutions to the L q Σ {L}_{q}\hspace{0.33em}\Sigma -reflector problem.

In the present paper, we study the existence of the normalized solutions for the following coupled elliptic system with quadratic nonlinearity − Δ u − λ 1 u = μ 1 ∣ u ∣ u + β u v in R N , − Δ v − λ 2 v = μ 2 ∣ v ∣ v + β 2 u 2 in R N , \left\{\begin{array}{ll}-\Delta u-{\lambda }_{1}u={\mu }_{1}| u| u+\beta uv\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ -\Delta v-{\lambda }_{2}v={\mu }_{2}| v| v+\frac{\beta }{2}{u}^{2}\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array}\right. where u , v u,v satisfying the additional condition ∫ R N u 2 d x = a 1 , ∫ R N v 2 d x = a 2 . \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={a}_{1},\hspace{1em}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{v}^{2}{\rm{d}}x={a}_{2}. On the one hand, we prove the existence of minimizer for the system with L 2 {L}^{2} -subcritical growth ( N ≤ 3 N\le 3 ). On the other hand, we prove the existence results for different ranges of the coupling parameter β > 0 \beta \gt 0 with L 2 {L}^{2} -supercritical growth ( N = 5 N=5 ). Our argument is based on the rearrangement techniques and the minimax construction.

The purpose of this article is to establish sharp conditions for the existence of normalized solutions to a class of scalar field equations involving mixed fractional Laplacians with different orders. This study includes the case when one operator is local and the other one is non-local. This type of equation arises in various fields ranging from biophysics to population dynamics. Due to the importance of these applications, this topic has very recently received an increasing interest. In this article, we provide a complete description of the existence/non-existence of ground state solutions using constrained variational approaches. This study addresses the mass subcritical, critical and supercritical cases. Our model presents some difficulties due to the “conflict” between the different orders and requires a novel analysis, especially in the mass supercritical case. We believe that our results will open the door to other valuable contributions in this important field.

In the present article, we study multiplicity of semi-classical solutions of a Yukawa-coupled massive Dirac-Klein-Gordon system with the general nonlinear self-coupling, which is either subcritical or critical growth. The number of solutions obtained is described by the ratio of maximum and behavior at infinity of the potentials. We use the variational method that relies upon a delicate cutting off technique. It allows us to overcome the lack of convexity of the nonlinearities.

In this article, we investigate the existence of multiple positive solutions to the following multi-critical Schrödinger equation: (0.1) − Δ u + λ V ( x ) u = μ ∣ u ∣ p − 2 u + ∑ i = 1 k ( ∣ x ∣ − ( N − α i ) ∗ ∣ u ∣ 2 i ∗ ) ∣ u ∣ 2 i ∗ − 2 u in R N , u ∈ H 1 ( R N ) , \left\{\begin{array}{l}-\Delta u+\lambda V\left(x)u=\mu | u{| }^{p-2}u+\mathop{\displaystyle \sum }\limits_{i=1}^{k}\left(| x{| }^{-\left(N-{\alpha }_{i})}\ast | u{| }^{{2}_{i}^{\ast }})| u{| }^{{2}_{i}^{\ast }-2}u\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\hspace{0.33em}\in {H}^{1}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}\end{array}\right. where λ , μ ∈ R + , N ≥ 4 \lambda ,\mu \in {{\mathbb{R}}}^{+},N\ge 4 , and 2 i ∗ = N + α i N − 2 {2}_{i}^{\ast }=\frac{N+{\alpha }_{i}}{N-2} with N − 4 < α i < N N-4\lt {\alpha }_{i}\lt N , i = 1 , 2 , … , k i=1,2,\ldots ,k are critical exponents and 2 < p < 2 min ∗ = min { 2 i ∗ : i = 1 , 2 , … , k } 2\lt p\lt {2}_{\min }^{\ast }={\rm{\min }}\left\{{2}_{i}^{\ast }:i=1,2,\ldots ,k\right\} . Suppose that Ω = int V − 1 ( 0 ) ⊂ R N \Omega ={\rm{int}}\hspace{0.33em}{V}^{-1}\left(0)\subset {{\mathbb{R}}}^{N} is a bounded domain, we show that for λ \lambda large, problem (0.1) possesses at least cat Ω ( Ω ) {{\rm{cat}}}_{\Omega }\left(\Omega ) positive solutions.

This paper deals with the existence of solutions of a class of contact mean-field game systems of first order consisting of a contact Hamilton-Jacobi equation and a continuity equation. Evans found a connection between Hamilton-Jacobi equations and continuity equations from the weak KAM point of view, where the coupling term is zero. Inspired by his work, we prove the main existence result by analyzing the properties of the Mather set for contact Hamiltonian systems.

This paper is concerned with the n -dimensional strongly coupled parabolic systems with triangular form in the cylinder Ω × ( 0 , T ] \Omega \times (0,T] . We investigate L 2 {L}^{2} and Hölder regularity of the derivatives of weak solutions ( u 1 , u 2 ) \left({u}_{1},{u}_{2}) for the systems in the following two cases: one is that the boundedness of u 1 {u}_{1} and u 2 {u}_{2} has not been shown in existence result of solutions; the other is that the boundedness of u 1 {u}_{1} or u 2 {u}_{2} has been shown in existence result of solutions. By using difference ratios and Steklov averages methods and various estimates, we prove that if ( u 1 , u 2 ) \left({u}_{1},{u}_{2}) is a weak solution of the system, then for any Ω ′ ⊂ ⊂ Ω \Omega ^{\prime} \subset \hspace{-0.3em}\subset \hspace{0.33em}\Omega and t ′ ∈ ( 0 , T ) t^{\prime} \in \left(0,T) , u 1 , u 2 {u}_{1},{u}_{2} belong to C α ′ , α ′ / 2 ( Ω ¯ ′ × [ t ′ , T ] ) {C}^{\alpha ^{\prime} ,\alpha ^{\prime} \text{/}2}\left(\bar{\Omega }^{\prime} \times \left[t^{\prime} ,T]) and W 2 2 , 1 ( Ω ′ × ( t ′ , T ] ) {W}_{2}^{2,1}\left(\Omega ^{\prime} \times (t^{\prime} ,T]) under certain conditions, and u 1 , u 2 {u}_{1},{u}_{2} belong to C 2 + α ′ , 1 + α ′ / 2 ( Ω ¯ ′ × [ t ′ , T ] ) {C}^{2+\alpha ^{\prime} ,1+\alpha ^{\prime} \text{/}2}\left(\bar{\Omega }^{\prime} \times \left[t^{\prime} ,T]) under stronger assumptions. Applications of these results are given to two ecological models with cross-diffusion.

In this manuscript, we analyze the exponential stability of a strongly coupled semilinear system of Klein-Gordon type, posed in an inhomogeneous medium Ω \Omega , subject to local dampings of different natures distributed around a neighborhood of the boundary according to the geometric control condition (GCC). The first one is of the type viscoelastic and is distributed around a neighborhood ω \omega of the boundary ∂ Ω \partial \Omega of Ω \Omega , according to the GCC. The second one is a frictional damping and we consider it hurting the GCC. The third dissipation, which acts only in the second equation (according to the GCC), is of the frictional type. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase space. We also prove the exponential decay for the linear problem associated with this same system, and in this case, no restrictions are made with respect to dimension of the space nor with respect to the limitation of the initial data in the phase space.

This article is devoted to the study of the combined effects of logarithmic and critical nonlinearities for the Kirchhoff-Poisson system − M ∫ Ω ∣ ∇ H u ∣ 2 d ξ Δ H u + μ ϕ u = λ ∣ u ∣ q − 2 u ln ∣ u ∣ 2 + ∣ u ∣ 2 u in Ω , − Δ H ϕ = u 2 in Ω , u = ϕ = 0 on ∂ Ω , \left\{\begin{array}{ll}-M\left(\mathop{\displaystyle \int }\limits_{\Omega }| {\nabla }_{H}u{| }^{2}{\rm{d}}\xi \right){\Delta }_{H}u+\mu \phi u=\lambda | u{| }^{q-2}u\mathrm{ln}| u{| }^{2}+| u{| }^{2}u& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -{\Delta }_{H}\phi ={u}^{2}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=\phi =0& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Δ H {\Delta }_{H} is the Kohn-Laplacian operator in the first Heisenberg group H 1 {{\mathbb{H}}}^{1} , Ω \Omega is a smooth bounded domain of H 1 {{\mathbb{H}}}^{1} , q ∈ ( 2 θ , 4 ) q\in \left(2\theta ,4) , μ ∈ R \mu \in {\mathbb{R}} , and λ > 0 \lambda \gt 0 are some real parameters. Under suitable assumptions on the Kirchhoff function M M , which cover the degenerate case, we prove the existence of nontrivial solutions for the above problem when λ > 0 \lambda \gt 0 is sufficiently large. Moreover, our results are new even in the Euclidean case.

In this paper, we consider the following critical fractional magnetic Choquard equation: ε 2 s ( − Δ ) A ∕ ε s u + V ( x ) u = ε α − N ∫ R N ∣ u ( y ) ∣ 2 s , α ∗ ∣ x − y ∣ α d y ∣ u ∣ 2 s , α ∗ − 2 u + ε α − N ∫ R N F ( y , ∣ u ( y ) ∣ 2 ) ∣ x − y ∣ α d y f ( x , ∣ u ∣ 2 ) u in R N , \begin{array}{rcl}{\varepsilon }^{2s}{\left(-\Delta )}_{A/\varepsilon }^{s}u+V\left(x)u& =& {\varepsilon }^{\alpha -N}\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u(y){| }^{{2}_{s,\alpha }^{\ast }}}{| x-y\hspace{-0.25em}{| }^{\alpha }}{\rm{d}}y\right)| u\hspace{-0.25em}{| }^{{2}_{s,\alpha }^{\ast }-2}u\\ & & +{\varepsilon }^{\alpha -N}\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{F(y,| u(y){| }^{2})}{| x-y\hspace{-0.25em}{| }^{\alpha }}{\rm{d}}y\right)\hspace{0.08em}f\left(x,| u\hspace{-0.25em}{| }^{2})u\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array} where ε > 0 \varepsilon \gt 0 , s ∈ ( 0 , 1 ) s\in \left(0,1) , α ∈ ( 0 , N ) \alpha \in \left(0,N) , N > max { 2 μ + 4 s , 2 s + α ∕ 2 } N\gt {\rm{\max }}\left\{2\mu +4s,2s+\alpha /2\right\} , 2 s , α ∗ = 2 N − α N − 2 s {2}_{s,\alpha }^{\ast }=\frac{2N-\alpha }{N-2s} is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, ( − Δ ) A s {\left(-\Delta )}_{A}^{s} stands for the fractional Laplacian with periodic magnetic field A A of C 0 , μ {C}^{0,\mu } -class with μ ∈ ( 0 , 1 ] \mu \in (0,1] and V V is a continuous potential and allows to be sign-changing. Under some mild assumptions imposed on V V and f f , we establish the existence of at least one ground state solution.

In this article, we investigate the existence of least energy sign-changing solutions for the following Schrödinger-Poisson system − Δ u + V ( x ) u + K ( x ) ϕ u = f ( u ) , x ∈ R 3 , − Δ ϕ = K ( x ) u 2 , x ∈ R 3 , \left\{\begin{array}{ll}-\Delta u+V\left(x)u+K\left(x)\phi u=f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi =K\left(x){u}^{2},\hspace{1.0em}& x\in {{\mathbb{R}}}^{3},\\ \hspace{1.0em}\end{array}\right. where the functions V ( x ) , K ( x ) V\left(x),K\left(x) have finite limits as ∣ x ∣ → ∞ | x| \to \infty satisfying some mild assumptions. By combining variational methods with the global compactness lemma, we obtain a least energy sign-changing solution with exactly two nodal domains, and its energy is strictly larger than twice that of least energy solutions.

The main purpose of this article is to establish the L p {L}^{p} Hardy’s identities and inequalities for Dunkl operator on any finite balls and the entire space R N {{\mathbb{R}}}^{N} . We also prove Hardy’s identities and inequalities on certain domains with distance function to the boundary ∂ Ω \partial \Omega . In particular, we use the notion of Bessel pairs introduced in Ghoussoub and Moradifam to extend Hardy’s identities for the classical gradients obtained by Lam et al., Duy et al., Flynn et al. to Dunkl gradients introduced by Dunkl. Our Hardy’s identities with explicit Bessel pairs significantly improve many existing Hardy’s inequalities for Dunkl operators.

For studying the evolution of the transverse deflection of an extensible beam derived from the connection mechanics, we investigate the initial boundary value problem of nonlinear extensible beam equation with linear strong damping term, nonlinear weak damping term, and nonlinear source term. The key idea of our analysis is to describe the invariant manifold via Nehari manifold. To establish the results of global well-posedness of solution, we consider the problem at three different initial energy levels, i.e., subcritical initial energy level, critical initial energy level, and arbitrarily high initial energy level. We first obtain the local existence of the solution by using the contraction mapping principle. Then, in the framework of potential well, we obtain global existence, nonexistence, and asymptotic behavior of solution for both subcritical initial energy level and critical initial energy level. In the end, we establish the global nonexistence of solution for the problem with linear weak damping and strong damping at the arbitrarily high initial energy level.

In this paper, we study Hessian equations with the prescribed contact angle boundary value or oblique derivative boundary value and finally derive the a priori global gradient estimate for the admissible solutions.

In this article, we establish a Gaffney type inequality, in W ℓ , p {W}^{\ell ,p} -Sobolev spaces, for differential forms on sub-Riemannian contact manifolds without boundary, having bounded geometry (hence, in particular, we have in mind noncompact manifolds). Here, p ∈ ] 1 , ∞ [ p\in ]1,\infty {[} and ℓ = 1 , 2 \ell =1,2 depending on the order of the differential form we are considering. The proof relies on the structure of the Rumin’s complex of differential forms in contact manifolds, on a Sobolev-Gaffney inequality proved by Baldi-Franchi in the setting of the Heisenberg groups and on some geometric properties that can be proved for sub-Riemannian contact manifolds with bounded geometry.

In the present article, we investigate the following Hénon-Lane-Emden elliptic system: − Δ u = ∣ x ∣ a v p , x ∈ R N , − Δ v = ∣ x ∣ b u q , x ∈ R N , \left\{\begin{array}{ll}-\Delta u={| x| }^{a}{v}^{p},& x\in {{\mathbb{R}}}^{N},\\ -\Delta v={| x| }^{b}{u}^{q},& x\in {{\mathbb{R}}}^{N},\end{array}\right. where N ≥ 2 N\ge 2 , p p , q > 0 q\gt 0 , a a , b ∈ R b\in {\mathbb{R}} . We partially prove the Hénon-Lane-Emden conjecture in the case of four and five dimensions. More specifically, we show that there is no nonnegative nontrivial classical solution for the Hénon-Lane-Emden elliptic system when a a , b > − 2 b\gt -2 and the parameter pair ( p , q p,q ) meets p q > 1 , N + a p + 1 + N + b q + 1 > N − 2 , pq\gt 1,\hspace{1.0em}\frac{N+a}{p+1}+\frac{N+b}{q+1}\gt N-2, and additionally p , q < 4 / 3 p,q\lt 4\hspace{0.1em}\text{/}\hspace{0.1em}3 if N = 4 N=4 or p , q < 10 / 9 p,q\lt 10\hspace{0.1em}\text{/}\hspace{0.1em}9 if N = 5 N=5 .

In this article, we are concerned with the existence of weak C 1 , 1 {C}^{1,1} solution of the k k -Hessian equation on a closed Hermitian manifold under the optimal assumption of the function in the right-hand side of the equation. The key points are to show the weak C 1 , 1 {C}^{1,1} estimates. We prove a Cherrier-type inequality to obtain the C 0 {C}^{0} estimate, and the complex Hessian estimate is proved by using an auxiliary function, which was motivated by Hou et al. and Tosatti and Weinkove. Our result generalizes the Kähler case proved by Dinew et al.

This article is concerned with the Dirichlet eigenvalue problem associated with the ∞ \infty -Laplacian in metric spaces. We establish a direct partial differential equation approach to find the principal eigenvalue and eigenfunctions in a proper geodesic space without assuming any measure structure. We provide an appropriate notion of solutions to the ∞ \infty -eigenvalue problem and show the existence of solutions by adapting Perron’s method. Our method is different from the standard limit process via the variational eigenvalue formulation for p p -Laplacian in the Euclidean space.

This article studies point concentration phenomena of nonlinear Schrödinger equations with magnetic potentials and constant electric potentials. The existing results show that a common magnetic field has no effect on the locations of point concentrations, as long as the electric potential is not a constant. This article finds out the role of the magnetic fields in the locations of point concentrations when the electric potential is a constant.

We continue the work started in a previous article and introduce a general setting in which we define nets of nonlinear operators whose domains are some set of functions defined in a locally compact topological group. We analyze the behavior of such nets and detect the fairest assumption, which are needed for the nets to converge with respect to the uniform convergence and in the setting of Orlicz spaces. As a consequence, we give results of convergence in this frame, study some important special cases, and provide graphical representations.

We study the existence of solutions for the quasilinear Schrödinger equation with the critical exponent and steep potential well. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals satisfy the geometric conditions of the Mountain Pass Theorem for suitable assumptions. The existence of a ground state solution is obtained, and its concentration behavior is also considered.

The Chaplygin gas model is both interesting and important in the theory of gas dynamics and conservation laws, all the characteristic families of which are linearly degenerate. Majda conjectured that the shock formation never happens for smooth data. In this article, we prove the conjecture for the two space dimensional axisymmetric case. Different from previous approaches to study wave equations with different speeds, we reformulate the problem in the Lagrangian coordinates and consider a single wave equation with variable coefficients. This not only gives a simpler proof but also enables us to treat the case with large angular velocities.

Existence of two solutions to a parametric singular quasi-linear elliptic problem is proved. The equation is driven by the Φ \Phi -Laplacian operator, and the reaction term can be nonmonotone. The main tools employed are the local minimum theorem and the Mountain pass theorem, together with the truncation technique. Global C 1 , τ {C}^{1,\tau } regularity of solutions is also investigated, chiefly via a priori estimates and perturbation techniques.

In this article, we establish existence and multiplicity results for first-order Stieltjes differential equations satisfying a periodic boundary condition or an initial value condition. No monotonicity condition involving the right-hand side f f is imposed at the discontinuity points of the derivator g g . Our results rely on the fixed point index theory and new notions of strict lower and upper solutions. An application to a population model with an extreme event is presented to study the persistence of a species.

The concentration-compactness principle of Lions type in Euclidean space relies on the Pólya-Szegö inequality, which is not available in non-Euclidean settings. The first proof of the concentration-compactness principle in non-Euclidean setting, such as the Heisenberg group, was given by Li et al. by using a symmetrization-free nonsmooth truncation argument. In this article, we study the concentration-compactness principle of second-order Adams’ inequality in Lorentz-Sobolev space W 0 2 L 2 , q ( Ω ) {W}_{0}^{2}{L}^{2,q}(\Omega ) for all 1 < q < ∞ 1\lt q\lt \infty . Due to the absence of the Pólya-Szegö inequality with respect to the second-order derivatives, we will use a symmetrization-free argument to study the concentration-compactness principle of second-order Adams’ inequality in Lorentz-Sobolev space. Furthermore, we show the sharpness of result by constructing a test function sequence. Our result is even new in the first-order case when q > 2 q\gt 2 .