Stability and instability of standing waves for a generalized Choquard equation with potential

We are going to study the standing waves for a generalized Choquard equation with potential: $$-i\partial_t u-\Delta u+V(x)u=(|x|^{-\mu}\ast|u|^p)|u|^{p-2}u, \ \ \hbox{in}\ \ \mathbb{R}\times\mathbb{R}^3,$$ where $V(x)$ is a real function, $0<\mu<3$, $2-\mu/3<p<6-\mu$ and $\ast$ stands for convolution. Under suitable assumptions on the potential and appropriate frequency $\omega$ , the stability and instability of the standing waves $u=e^{i \omega t}\varphi(x)$ are investigated .Comment: 2

On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents

We consider the following nonlinear Choquard equation with Dirichlet boundary condition -\Delta u =\left(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy\right)|u|^{2_{\mu}^{\ast}-2}u+\lambda f(u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \Omega, where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $\lambda>0$, $N\geq3$, $0<\mu<N$ and $2_{\mu}^{\ast}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on different types of nonlinearities $f(u)$, we are able to prove some existence and multiplicity results for the equation by variational methods.Comment: 3

On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation

We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation -\Delta u =\left(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy\right)|u|^{2_{\mu}^{\ast}-2}u+\lambda u\4.14mm\mbox{in}\1.14mm \Omega, where $\Omega$ is a bounded domain of $\mathbb{R}^N$, with Lipschitz boundary, $\lambda$ is a real parameter, $N\geq3$, $2_{\mu}^{\ast}=(2N-\mu)/(N-2)$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.Comment: 3

Existence of positive multi-bump solutions for a Schr\"odinger-Poisson system in R3\mathbb{R}^{3}

In this paper we are going to study a class of Schr\"odinger-Poisson system \left\{ \begin{array}{ll} - \Delta u + (\lambda a(x)+1)u+ \phi u = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3},\\ -\Delta \phi=u^2 \mbox{ in } \,\,\, \mathbb{R}^{3}.\\ \end{array} \right. Assuming that the nonnegative function $a(x)$ has a potential well $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint components $\Omega_1, \Omega_2, ....., \Omega_k$ and the nonlinearity $f(t)$ has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by variational methods.Comment: arXiv admin note: text overlap with arXiv:1402.683

Existence of solutions for a nonlocal variational problem in R2\mathbb{R}^2 with exponential critical growth

We study the existence of solution for the following class of nonlocal problem, -\Delta u +V(x)u =\Big( I_\mu\ast F(x,u)\Big)f(x,u) \quad \mbox{in} \quad \mathbb{R}^2, where $V$ is a positive periodic potential, $I_\mu=\frac{1}{|x|^\mu}$, $0<\mu<2$ and $F(x,s)$ is the primitive function of $f(x,s)$ in the variable $s$. In this paper, by assuming that the nonlinearity $f(x,s)$ has an exponential critical growth at infinity, we prove the existence of solutions by using variational methods

Existence and qualitative analysis for nonlinear weighted Choquard equations

The aim of this paper is to classify the solutions of the critical nonlocal equation with weighted nonlocal term -\Delta u =\frac{1}{|x|^{\alpha}}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2_{\alpha,\mu}^{\ast}}} {|x-y|^{\mu}|y|^{\alpha}}dy\right)|u|^{2_{\alpha,\mu}^{\ast}-2}u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^{N} and the subcritical case of the form -\Delta u+u =\frac{1}{|x|^{\alpha}}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{p}} {|x-y|^{\mu}|y|^{\alpha}}dy\right)|u|^{p-2}u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^{N}. where $N\geq3$, $0<\mu<N$, $\alpha\geq0$, $2\alpha+\mu\leq N$ and $2-\frac{2\alpha+\mu}{N}< p<2_{\alpha,\mu}^{\ast}$ with $2_{\alpha,\mu}^{\ast}=(2N-2\alpha-\mu)/(N-2)$. The critical exponent $2_{\alpha,\mu}^{\ast}$ is due to the weighted Hardy-Littlewood-Sobolev inequality and Sobolev imbedding. We prove the existence of positive ground state solutions for the subcritical case by using Schwarz symmetrization and the critical case by a nonlocal version of concentration-compactness principle. We also establish the regularity of positive solutions for these two equations . Finally, we prove the symmetry of positive solutions by the moving plane method in integral forms.Comment: 3

On the critical Choquard equation with potential well

In this paper we are interested in the following nonlinear Choquard equation -\Delta u+(\lambda V(x)-\beta)u =\big(|x|^{-\mu}\ast |u|^{2_{\mu}^{\ast}}\big)|u|^{2_{\mu}^{\ast}-2}u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, where $\lambda,\beta\in\mathbb{R}^+$, $0<\mu<N$, $N\geq4$, $2_{\mu}^{\ast}=(2N-\mu)/(N-2)$ is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality and the nonnegative potential function $V\in \mathcal{C}(\mathbb{R}^N,\mathbb{R})$ such that \Omega :=\mbox{int} V^{-1}(0) is a nonempty bounded set with smooth boundary. If $\beta>0$ is a constant such that the operator $-\Delta +\lambda V(x)-\beta$ is non-degenerate, we prove the existence of ground state solutions which localize near the potential well int $V^{-1}(0)$ for $\lambda$ large enough and also characterize the asymptotic behavior of the solutions as the parameter $\lambda$ goes to infinity. Furthermore, for any $0<\beta<\beta_{1}$, we are able to find the existence of multiple solutions by the Lusternik-Schnirelmann category theory, where $\beta_{1}$ is the first eigenvalue of $-\Delta$ on $\Omega$ with Dirichlet boundary condition.Comment: 2

Semiclassical states for Choquard type equations with critical growth: critical frequency case

In this paper we are interested in the existence of semiclassical states for the Choquard type equation -\vr^2\Delta u +V(x)u =\Big(\int_{\R^N} \frac{G(u(y))}{|x-y|^\mu}dy\Big)g(u) \quad \mbox{in $\R^N$}, where $0<\mu<N$, $N\geq3$, \vr is a positive parameter and $G$ is the primitive of $g$ which is of critical growth due to the Hardy--Littlewood--Sobolev inequality. The potential function $V(x)$ is assumed to be nonnegative with $V(x)=0$ in some region of $\R^N$, which means it is of the critical frequency case. Firstly we study a Choquard equation with double critical exponents and prove the existence and multiplicity of semiclassical solutions by the Mountain-Pass Theorem and the genus theory. Secondly we consider a class of critical Choquard equation without lower perturbation, by establishing a global Compactness lemma for the nonlocal Choquard equation, we prove the multiplicity of high energy semiclassical states by the Lusternik--Schnirelman theory

Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity

In this paper, we study a class of nonlinear Choquard type equations involving a general nonlinearity. By using the method of penalization argument, we show that there exists a family of solutions having multiple concentration regions which concentrate at the minimum points of the potential $V$. Moreover, the monotonicity of $f(s)/s$ and the so-called Ambrosetti-Rabinowitz condition are not required.Comment: 20 page

Multi-bump solutions for Choquard equation with deepening potential well

We study the existence of multi-bump solutions to Choquard equation \begin{array}{ll} -\Delta u + (\lambda a(x)+1)u=\displaystyle\big(\frac{1}{|x|^{\mu}}\ast |u|^p\big)|u|^{p-2}u \mbox{ in } \,\,\, \R^3, \end{array} where $\mu \in (0,3), p\in(2, 6-\mu)$, $\lambda$ is a positive parameter and the nonnegative function $a(x)$ has a potential well $\Omega:=int (a^{-1}(0))$ consisting of $k$ disjoint bounded components $\Omega:=\cup_{j=1}^{k}\Omega_j$. We prove that if the parameter $\lambda$ is large enough then the equation has at least $2^{k}-1$ multi-bump solutions.Comment: 26page