相关论文: Mixed order conformally invariant system with expo…

In this paper, without any assumption on $v$ and under the extremely mild assumption $u(x)= O(|x|^{K})$ as $|x|\rightarrow+\infty$ for some $K\gg1$ arbitrarily large, we classify solutions of the following conformally invariant system with…

In this paper, without any assumption on $v$ and under extremely mild assumption $u(x)=O(|x|^{K})$ at $\infty$ for some $K\gg1$ arbitrarily large, we prove classification of solutions to the following conformally invariant system with mixed…

In this paper, we classify the solution of the following mixed-order conformally invariant system with coupled nonlinearity in $ \mathbb{R}^4$: \begin{equation}\left\{ \begin{aligned} & -\Delta u(x) = u^{p_1}(x) e^{q_1v(x)}, \quad x\in…

We consider the following nonlinear Schr\"odinger equations with critical growth: \begin{equation} - \Delta u + V(|y|)u=u^{\frac{N+2}{N-2}},\quad u>0 \ \ \mbox{in} \ \mathbb {R}^N, \end{equation} where $V(|y|)$ is a bounded positive radial…

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,…

We study the existence of positive solutions for the system of fractional elliptic equations of the type, \begin{equation*} \begin{array}{rl} (-\Delta)^{\frac{1}{2}} u &=\frac{p}{p+q}\lambda f(x)|u|^{p-2}u|v|^q + h_1(u,v)…

In this article we study the nonlocal equation \begin{align} (-\Delta)^{\frac{n}{2}}u=(n-1)!e^{nu}\quad \text{in $\mathbb{R}^n$}, \quad\int_{\mathbb{R}^n}e^{nu}dx<\infty, \notag \end{align} which arises in the conformal geometry. Inspired…

This paper is concerned with the following planar Schr\"{o}dinger-Poisson system \begin{equation*} \begin{cases} -\triangle{u}+V(x)u+\phi{(x)}|u|^{p-2}u=f(x,u),&\text{in $\mathbb{R}^{2}$}, \triangle{\phi}=|u|^{p},&\text{in…

The higher order Kirchhoff type equation $$\int_{\mathbb{R}^{2m}}(|\nabla^m u|^2 +\sum_{\gamma=0}^{m-1}a_{\gamma}(x)|\nabla^{\gamma}u|^2)dx \left((-\Delta)^m u+\sum_{\gamma=0}^{m-1}(-1)^\gamma \nabla^\gamma\cdot(a_\gamma (x)\nabla^\gamma…

We obtain the existence of ground state solution for the nonlocal problem $$ m\left(\int_{\mathbb{R}^2}(|\nabla u|^2 + b(x)u^2) \textrm{d}x\right)(-\Delta u + b(x)u) = A(x)f(u) \ \ \ \textrm{in} \ \ \ \mathbb{R}^2, $$ where $m$ is a…

This paper is devoted to the magnetic nonlinear Schr\"{o}dinger equation \[ \Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u=f(| u|^{2})u \text{ in } \mathbb{R}^{2}, \] where $\varepsilon>0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow…

In this paper we study the multiplicity of normalized solutions to the following nonlinear Schr\"{o}dinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-\Delta u=\lambda u+\mu |u|^{q-2}u+f(u), \quad \quad \hbox{in…

This paper deals with the existence and multiplicity of solutions for a class of Kirchhoff type elliptic system involving the Trudinger-Moser exponential growth nonlinearities. We first study the existence of solutions for the following…

In this paper, we study the existence of normalized solutions to the following nonlinear Schr\"{o}dinger systems with exponential growth \begin{align*} \left\{ \begin{aligned} &-\Delta u+\lambda_{1}u=H_{u}(u,v), \quad \quad \hbox{in…

In this work we study the existence of nontrivial solution for the following class of multivalued elliptic problems $$ -\Delta u+V(x)u-\epsilon h(x)\in \partial_t F(x,u) \quad \text{in} \quad \mathbb{R}^2, \eqno{(P)} $$ where $\epsilon>0$,…

In this paper, we study the following nonlocal nonautonomous Hamiltonian system on whole $\mathbb R$ $$ \left\{\begin{array}{ll} (-\Delta)^\frac12~ u +u=Q(x) g(v)&\quad\mbox{in } \mathbb R,\\ (-\Delta)^\frac12~ v+v = P(x)f(u)&\quad\mbox{in…

In this paper, we study the following Schr\"{o}dinger-Born-infeld system with a general nonlinearity $$ \left\{ \begin{array}{ll} -\triangle u+u+\phi u=f(u)+\mu|u|^4u\,\,&\mbox{in}\,\,\R^3,\\…

We study existence and multiplicity of nontrivial solutions of the following problem $$ \left\{ \begin{array}{rcll} -\Delta_p u+(-\Delta_p)^{s} u & = & \lambda|u|^{q-2}u+|u|^{p^{\ast}-2}u & \mbox{ in }\Omega,\\ u & = & 0 & \mbox{ on }…

We study the existence of solutions of the following nonlinear Schr\"odinger equation $$ -\Delta u+V(x)u-\frac{(N-2)^2}{4|x|^2}u=f(x,u) $$ where $V:\mathbb{R}^N\to\mathbb{R}$ and $f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$ are periodic…

In this paper we study the normalized solutions of the following critical growth Choquard equation with mixed local and non-local operators: \begin{equation*} \begin{array}{rcl} -\Delta u +(-\Delta)^s u & = & \lambda u +\mu |u|^{p-2}u…