Related papers: Existence and concentration of solution for a non-…

In this article we are interested in the following non-linear Schr\"odinger equation with non-local regional diffusion $$ (-\Delta)_{\rho_\epsilon}^{\alpha}u + u = f(u) \hbox{ in } \mathbb{R}^n, \quad u \in H^\alpha(\mathbb{R}^n),…

In this paper, we study the following nonlinear Schr\"odinger-Poisson type equation \begin{equation*} \begin{cases} -\varepsilon^2\Delta u+V(x)u+K(x)\phi u=f(u)&\text{in}\ \mathbb{R}^3,\\ -\varepsilon^2\Delta \phi=K(x)u^2&\text{in}\…

In this article we are interested in the nonlocal regional Schr\"odinger equation with critical exponent \begin{eqnarray*} &\epsilon^{2\alpha} (-\Delta)_{\rho}^{\alpha}u + u = \lambda u^q + u^{2_{\alpha}^{*}-1} \mbox{ in } \mathbb{R}^{N},…

The aim of this paper is to investigate the existence, multiplicity and concentration of positive solutions for the following nonlocal system of fractional Schr\"odinger equations \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}…

In this paper we study the existence, multiplicity and concentration behavior of solutions for the following critical fractional Schr\"odinger system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-\Delta)^{s}u+V(x)…

Goal of this paper is to study positive semiclassical solutions of the nonlinear Schr\"odinger equation $$ \varepsilon^{2s}(- \Delta)^s u+ V(x) u= f(u), \quad x \in \mathbb{R}^N,$$ where $s \in (0,1)$, $N \geq 2$, $V \in…

We study the following singularly perturbed nonlocal Schr\"{o}dinger equation $$ -\vr^2\Delta u +V(x)u =\vr^{\mu-2}\Big[\frac{1}{|x|^{\mu}}\ast F(u)\Big]f(u) \quad \mbox{in} \quad \R^2, $$ where $V(x)$ is a continuous real function on…

In this paper we study the following class of fractional relativistic Schr\"odinger equations: \begin{equation*} \left\{ \begin{array}{ll} (-\Delta+m^{2})^{s}u + V(\varepsilon x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in…

We deal with the existence of positive solutions for the following fractional Schr\"odinger equation $$ \varepsilon ^{2s} (-\Delta)^{s} u + V(x) u = f(u) \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$,…

We study the Choquard equation involving mixed local and nonlocal operators \[-\varepsilon^{2}\Delta u+\varepsilon^{2s}(-\Delta)^{s}u+V(x)u=\varepsilon^{\mu-2}\left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u)\quad \text{in }\R^{2},\] where…

We consider the equation $$-\Delta u+u=Q_\varepsilon(x)|u|^{p-2}u,\qquad u\in H^1(\mathbb{R}^N),$$ where $Q_\varepsilon$ takes the value $1$ on each ball $B_\varepsilon(y)$, $y\in\mathbb{Z}^N$, and the value $-1$ elsewhere. We establish the…

This paper is concerned with the following fractional Schr\"{o}dinger equations involving critical exponents: \begin{eqnarray*} (-\Delta)^{\alpha}u+V(x)u=k(x)f(u)+\lambda|u|^{2_{\alpha}^{*}-2}u\quad\quad \mbox{in}\ \mathbb{R}^{N},…

In this article we are concerned with the following logarithmic Schr\"{o}dinger equation $$ \left\{ \begin{array}{lc} -{\epsilon}^2\Delta u+ V(x)u=u \log u^2, & \mbox{in} \,\, \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\…

This paper concerns the existence of multiple solutions for the fractional logarithmic Schr\"odinger-Possion system of the form \begin{equation*} \begin{cases} {\varepsilon}^{2\alpha} (-\Delta )^{\alpha}u+V(x) u+\phi u=u \log u^{2}+u^{q-1},…

This article concerns the fractional elliptic equations \begin{equation*}(-\Delta)^{s}u+\lambda V(x)u=f(u), \quad u\in H^{s}(\mathbb{R}^N), \end{equation*}where $(-\Delta)^{s}$ ($s\in (0\,,\,1)$) denotes the fractional Laplacian, $\lambda…

In this paper we deal with the following nonlocal systems of fractional Schr\"odinger equations \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-\Delta)^{s}u+V(x)u=Q_{u}(u, v)+\gamma H_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}\\…

This article focuses on the existence and non-existence of solutions for the following system of local and nonlocal type \begin{equation*} \left\{ \begin{aligned} -\partial_{xx}u + (-\Delta)_{y}^{s_{1}} u + u - u^{2_{s_{1}}^{}-1} = \kappa…

We consider the following fractional Schr\"odinger-Poisson type equation with magnetic fields \begin{equation*} \varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u \quad \mbox{ in }…

We study the concentration phenomenon for solutions of the fractional nonlinear Schr\"{o}dinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation…

In this paper, we study the following fractional Schr\"{o}dinger-Poisson system involving competing potential functions \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=K(x)f(u)+Q(x)|u|^{2_s^{\ast}-2}u,…