Related papers: Solutions for a critical elliptic system with peri…

In this paper, we analyze the existence of solution for a fractional elliptic system coupled by critical nonlinearities and endowed with mixed Dirichlet-Neumann boundary conditions. By means of variational methods and an…

We are looking for solutions to nonlinear Schr\"odinger-type equations of the form $$ (-\Delta)^{\alpha / 2} u (x) + V(x) u(x) = h (x,u(x)), \quad x \in \mathbb{R}^N, $$ where $V : \mathbb{R}^N \rightarrow \mathbb{R}$ is an external…

We consider the following fractional Schr\"{o}dinger equation involving critical exponent: \begin{equation*} \left\{\begin{array}{ll} (-\Delta)^s u+V(|y'|,y'')u=u^{2^*_s-1} \ \hbox{ in } \ \mathbb{R}^N, \\ u>0, \ y \in \mathbb{R}^N,…

We establish quantitative asymptotic behaviors for nonnegative solutions of the critical semilinear equation $-\Delta u=u^{\frac{n+2}{n-2}}$ with isolated boundary singularities, where $n\ge 3$ is the dimension.

In this paper, we will study the following PDE in $\mathbb{R}^N$ involving multiple Hardy-Sobolev critical exponents: $$ \begin{cases} \Delta u+\sum_{i=1}^{l}\lambda_i \frac{u^{2^*(s_i)-1}}{|x|^{s_i}}+u^{2^*-1}=0\;\hbox{in}\;\mathbb{R}^N,…

We consider the following nonlinear Schrodinger equation [{l} \Delta u-(1+\delta V)u+f(u)=0 in \R^N, u>0 in \R^N, u\in H^1(\R^N).] where $V$ is a potential satisfying some decay condition and $ f(u)$ is a superlinear nonlinearity satisfying…

We study the following gradient elliptic system with Neumann boundary conditions \begin{equation*} -\Delta u + \lambda_1 u = u^3 + \beta uv^2, \ -\Delta v + \lambda_2 v = v^3 + \beta u^2 v \ \text{in } \Omega,\qquad \frac{\partial…

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…

In this article, we study the following fractional Laplacian equation with critical growth and singular nonlinearity $$\quad (-\Delta)^s u = \lambda a(x) u^{-q} + u^{2^*_s-1}, \quad u>0 \; \text{in}\; \Omega,\quad u = 0 \; \mbox{in}\;…

We study the existence of nontrivial bound state solutions to the following system of coupled nonlinear time-independent Schr\"odinger equations $$ - \Delta u_j+ \lambda_j u_j =\mu_j u_j^3+ \sum_{k=1;k\neq j}^N\beta_{jk} u_ju_k^2,\quad…

In this paper we deal with the cubic Schr\"odinger system $ -\Delta u_i = \sum_{j=1}^n \beta_{ij}u_j^2 u_i$, $u_1,\dots,u_n \geq 0$ in $\mathbb{R}^N (N\leq 3)$, where $\beta=(\beta_{i,j})_{ij}$ is a symmetric matrix with real coefficients…

We look for a solutions to a nonlinear, fractional Schr\"odinger equation $$(-\Delta)^{\alpha / 2}u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where potential $V$ is coercive or $V=V_{per} + V_{loc}$ is a sum of periodic…

This paper is devoted to the study of the existence of positive solutions for a problem related to a higher order fractional differential equation involving a nonlinear term depending on a fractional differential operator,…

We construct a new family of entire solutions for the nonlinear Schr\"odinger equation \begin{align*} \begin{cases} -\Delta u+ V(y ) u = u^p, \quad u>0, \quad \text{in}~ \mathbb{R}^N, \\[2mm] u \in H^1(\mathbb{R}^N), \end{cases}…

We study the existence of positive bound states for the nonlinear elliptic system \[ \begin{cases} - \Delta u_i + \lambda_i u_i = \sum_{j=1}^d \beta_{ij} u_j^2 u_i & \text{in $\Omega$} \\ u_1 =\cdots = u_d=0 & \text{on $\partial \Omega$},…

The aim of this paper is to prove the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint: \begin{align*} \left\{ \begin{array}{ll}…

We consider the system of coupled elliptic equations \[ \begin{cases} -\Delta u - \lambda_1 u = \mu_1 u^3+ \beta u v^2 \\ -\Delta v- \lambda_2 v = \mu_2 v^3 +\beta u^2 v \end{cases} \text{in $\mathbb{R}^3$}, \] and study the existence of…

In this paper, we study the existence of positive solution for the following class of fractional elliptic equation $$ \epsilon^{2s} (-\Delta)^{s}{u}+V(z)u=\lambda |u|^{q-2}u+|u|^{2^{*}_{s}-2}u\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where…

In this paper, we study the existence of nontrivial solutions of the Dirichlet boundary value problem for the following elliptic system: \begin{equation} \left\{ \begin{aligned} -\Delta u & = au + bv + f(x,u,v); &\quad\mbox{ for…

In this paper, we are concerned with the following type of elliptic problems: $$ (-\Delta)^{\alpha} u+a(x) u=\frac{|u|^{2^*_{s}-2}u}{|x|^s}+k(x)|u|^{q-2}u, u\,\in\,H^\alpha({\mathbb R}^N), $$ where $2<q< 2^*$, $0<\alpha<1$, $0<s<2\alpha$,…