arXiv subjects

Fashun Gao

Publications and source records attributed to Fashun Gao.

11 recordsLinked to original sources

arXiv2022

Construction of infinitely many solutions for a critical Choquard equation via local Pohožaev identities

In this paper, we study a class of the critical Choquard equations with axisymmetric potentials, $$ -Δu+ V(|x'|,x'')u =\Big(|x|^{-4}\ast |u|^{2}\Big)u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^6, $$ where $(x',x'')\in \mathbb{R}^2\times\mathbb{R}^{4}$, $V(|x'|, x'')$ is a bounded nonnegative function in $\mathbb{R}^{+}\times\mathbb{R}^{4}$, and $*$ stands for the standard convolution. The equation is critical in the sense of the Hardy-Littlewood-Sobolev inequality. By applying a finite dimensional reduction argument and developing novel local Pohožaev identities, we prove that if the function $r^2V(r,x'')$ has a topologically nontrivial critical point then the problem admits infinitely many solutions with arbitrary large energies.

math.AP↗

arXiv2022

On elliptic equations with Stein-Weiss type convolution parts

The aim of this paper is to study the critical elliptic equations with Stein-Weiss type convolution parts $$ \displaystyle-Δu =\frac{1}{|x|^α}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2_{α, μ}^{\ast}}}{|x-y|^μ|y|^α}dy\right) |u|^{2_{α, μ}^{\ast}-2}u,~~~x\in\mathbb{R}^{N}, $$ where the critical exponent is due to the weighted Hardy-Littlewood-Sobolev inequality and Sobolev embedding. We develop a nonlocal version of concentration-compactness principle to investigate the existence of solutions and study the regularity, symmetry of positive solutions by moving plane arguments. In the second part, the subcritical case is also considered, the existence, symmetry, regularity of the positive solutions are obtained.

math.AP↗

arXiv2020

High energy positive solutions for a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponents

We study the coupled Hartree system $$ \left\{\begin{array}{ll} -Δu+ V_1(x)u =α_1\big(|x|^{-4}\ast u^{2}\big)u+β\big(|x|^{-4}\ast v^{2}\big)u &\mbox{in}\ \mathbb{R}^N,\\[1mm] -Δv+ V_2(x)v =α_2\big(|x|^{-4}\ast v^{2}\big)v +β\big(|x|^{-4}\ast u^{2}\big)v &\mbox{in}\ \mathbb{R}^N, \end{array}\right. $$ where $N\geq 5$, $β>\max\{α_1,α_2\}\geq\min\{α_1,α_2\}>0$, and $V_1,\,V_2\in L^{N/2}(\mathbb{R}^N)\cap L_{\text{loc}}^{\infty}(\mathbb{R}^N)$ are nonnegative potentials. This system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. For the system with $V_1=V_2=0$ we employ moving sphere arguments in integral form to classify positive solutions and to prove the uniqueness of positive solutions up to translation and dilation, which is of independent interest. Then using the uniqueness property, we establish a nonlocal version of the global compactness lemma and prove the existence of a high energy positive solution for the system assuming that $|V_1|_{L^{N/2}(\mathbb{R}^N)}+|V_2|_{L^{N/2}(\mathbb{R}^N)}>0$ is suitably small.

math.AP↗

arXiv2017

Existence of solutions for critical Choquard equations via the concentration compactness method

In this paper we consider the nonlinear Choquard equation $$ -Δu+V(x)u =\left(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^μ}dy\right)g(x,u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $0<μ<N$, $N\geq3$, $g(x,u)$ is of critical growth due to the Hardy--Littlewood--Sobolev inequality and $G(x,u)=\displaystyle\int^u_0g(x,s)ds$. Firstly, by assuming that the potential $V(x)$ might be sign-changing, we study the existence of Mountain-Pass solution via a concentration-compactness principle for the Choquard equation. Secondly, under the conditions introduced by Benci and Cerami \cite{BC1}, we also study the existence of high energy solution by using a global compactness lemma for the nonlocal Choquard equation.

math.AP↗

arXiv2017

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 $$ -\vr^2Δu +V(x)u =\Big(\int_{\R^N} \frac{G(u(y))}{|x-y|^μ}dy\Big)g(u) \quad \mbox{in $\R^N$}, $$ where $0<μ<N$, $N\geq3$, $\vr$ 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.

math.AP↗

arXiv2017

Singularly perturbed critical Choquard equations

In this paper we study the semiclassical limit for the singularly perturbed Choquard equation $$ -\vr^2Δu +V(x)u =\vr^{μ-3}\Big(\int_{\R^3} \frac{Q(y)G(u(y))}{|x-y|^μ}dy\Big)Q(x)g(u) \quad \mbox{in $\R^3$}, $$ where $0<μ<3$, $\vr$ is a positive parameter, $V,Q$ are two continuous real function on $\R^3$ and $G$ is the primitive of $g$ which is of critical growth due to the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on the nonlinearity $g$, we first establish the existence of ground states for the critical Choquard equation with constant coefficients in $\R^3$. Next we establish existence and multiplicity of semi-classical solutions and characterize the concentration behavior by variational methods.

math.AP↗

arXiv2017

On the critical Choquard equation with potential well

In this paper we are interested in the following nonlinear Choquard equation $$ -Δu+(λV(x)-β)u =\big(|x|^{-μ}\ast |u|^{2_μ^{\ast}}\big)|u|^{2_μ^{\ast}-2}u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $λ,β\in\mathbb{R}^+$, $0<μ 0$ is a constant such that the operator $-Δ+λV(x)-β$ is non-degenerate, we prove the existence of ground state solutions which localize near the potential well int $V^{-1}(0)$ for $λ$ large enough and also characterize the asymptotic behavior of the solutions as the parameter $λ$ goes to infinity. Furthermore, for any $0<β<β_{1}$, we are able to find the existence of multiple solutions by the Lusternik-Schnirelmann category theory, where $β_{1}$ is the first eigenvalue of $-Δ$ on $Ω$ with Dirichlet boundary condition.

math.AP↗

arXiv2017

A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality

In this paper we are concerned with the following nonlinear Choquard equation $$-Δu+V(x)u =\left(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^μ}dy\right)g(x,u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $N\geq4$, $0<μ<N$ and $G(x,u)=\displaystyle\int^u_0g(x,s)ds$. If $0$ lies in a gap of the spectrum of $-Δ+V$ and $g(x,u)$ is of critical growth due to the Hardy-Littlewood-Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in \cite{AC, KS, MS2}.

math.AP↗

arXiv2016

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

We consider the following nonlinear Choquard equation with Dirichlet boundary condition $$-Δu =\left(\int_Ω\frac{|u|^{2_μ^{\ast}}}{|x-y|^μ}dy\right)|u|^{2_μ^{\ast}-2}u+λf(u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} Ω, $$ where $Ω$ is a smooth bounded domain of $\mathbb{R}^N$, $λ>0$, $N\geq3$, $0<μ<N$ and $2_μ^{\ast}$ 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.

math.AP↗

arXiv2016

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 $$-Δu =\left(\int_Ω\frac{|u|^{2_μ^{\ast}}}{|x-y|^μ}dy\right)|u|^{2_μ^{\ast}-2}u+λu\4.14mm\mbox{in}\1.14mm Ω, $$ where $Ω$ is a bounded domain of $\mathbb{R}^N$, with Lipschitz boundary, $λ$ is a real parameter, $N\geq3$, $2_μ^{\ast}=(2N-μ)/(N-2)$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.

math.AP↗

arXiv2015

Groundstates for nonlinear fractional Choquard equations with general nonlinearities

We study the following nonlinear Choquard equation driven by a fractional Laplacian: $$ (-Δ)^{s}u+ u =(|x|^{-μ}\ast F(u))f(u)|{4.14mm}{in}|{1.14mm} \mathbb{R}^N, $$ with $N\geq3$, $s\in(0,1)$ and $μ\in(0,N)$. By Supposing that the nonlinearities satisfy the general Berestycki-Lions type conditions \cite{BL}, we are able to prove the existence of groundstates for this equation by variational methods.

math.AP↗