Related papers: On the Schr\"odinger-Maxwell system involving subl…

We prove the existence of infinitely many solutions $\lambda_1, \lambda_2 \in \mathbb{R}$, $u,v \in H^1(\mathbb{R}^3)$, for the nonlinear Schr\"odinger system \[ \begin{cases} -\Delta u - \lambda_1 u = \mu u^3+ \beta u v^2 & \text{in…

We consider the Schr\"{o}dinger equation $-\Delta u +V(x)u=f(x, u)$, where $V$ is periodic and $f$ is non-periodic, 0 is a boundary point of the continuous spectrum of $A:=-\Delta +V(x)$. We use M. Willem and W. M. Zou's linking theorem and…

In the present paper, we study the following Schr\"{o}dinger-Maxwell equation with combined nonlinearities \begin{align*} \displaystyle - \Delta u+\lambda u+ \left(|x|^{-1}\ast |u|^2\right)u =|u|^{p-2}u +\mu|u|^{q-2}u\quad \text{in} \…

This work is concerned with the existence and multiplicity of solutions for the following class of quasilinear problems $$ -\Delta_{\Phi}u+\phi(|u|)u=f(u)~\text{in} ~\Omega_{\lambda}, u(x)>0 ~\text{in}~\Omega_{\lambda}, u=0~ \mbox{on}…

We prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equation \[ -\triangle u+V\left( \left| x\right| \right) u=g\left( \left| x\right| ,u\right) \quad \textrm{in }\Omega \subseteq…

Using variational methods we prove some results about existence and multiplicity of positive bound states of to the following Schr\"odinger-Poisson system: $$ \left\{ \begin{array}{l} \vspace{2mm} -\Delta u+V(x)u+K(x)\phi(x)u=u^5 -\Delta…

We study the following Schr\"odinger-Poisson system (P_\lambda)\{ll} -\Delta u + (1+\mu g(x))u+\lambda \phi (x) u =|u|^{p-1}u, x\in \mathbb{R}^3,

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…

In the present work we are concerned with the existence of normalized solutions to the following Schr\"odinger-Poisson System $$ \left\{ \begin{array}{ll} -\Delta u + \lambda u + \mu (\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ \ in \ }…

In this paper a nonlinear coupled Schrodinger system in the presence of mixed cubic and superlinear power laws is considered. Focus are made on the steady state solutions of the continuous system for existence and uniqueness by minimizing…

Consider the following nonlinear Schr\"odinger-Bopp-Podolsky system in $\mathbb{R}^3$: $$ \begin{cases} -\varepsilon^2 \Delta u + (V + \phi) u = u |u|^{p-1}; \\ a^2 \Delta^2 \phi - \Delta \phi = 4 \pi u^2, \end{cases} $$ where $a,…

In this paper, we study normalized solutions for the following critical Schr\"odinger-Bopp-Podolsky system: $$-\Delta u + q(x)\phi u = \lambda u + |u|^{p-2}u + \bigl(I_\alpha * |u|^{3+\alpha}\bigr)|u|^{1+\alpha}u,\quad \text{in }…

This paper is devoted to the following class of nonlinear fractional Schr\"odinger equations: \begin{equation*} (-\Delta)^{s} u + V(x)u = f(x,u) + \lambda g(x,u), \quad \text{in}\: \mathbb{R}^N, \end{equation*} where $s\in (0,1)$, $N>2s$,…

In this paper, we are concerned with the coupled nonlinear Schr\"{o}dinger system \begin{align*} \begin{cases} -\varepsilon^{2}\Delta u+a(x)u=\mu_{1}u^{3}+\beta v^{2}u \ \ \ \ \mbox{in}\ \mathbb{R}^{N},\\ -\varepsilon^{2}\Delta…

We study a class of critical Schr\"odinger-Poisson system of the form \begin{equation*} \begin{cases} -\Delta u+\lambda V(x)u+\phi u=\mu |u|^{p-2}u+|u|^{4}u& \quad x\in \mathbb{R}^3,\\ -\Delta \phi=u^2&\quad x\in \mathbb{R}^3,\\ \end{cases}…

The paper considers the Schrodinger-Maxwell system with supercritical nonlinearitie. We prove the existence of at least one non-trivial weak solution. This result is already known for the subcritical case. In this paper, we extend it to the…

Consider the following system of double coupled Schr\"odinger equations arising from Bose-Einstein condensates etc., \begin{equation*} \left\{\begin{array}{l} -\Delta u + u =\mu_1 u^3 + \beta uv^2- \kappa v, -\Delta v + v =\mu_2 v^3 + \beta…

The present study is concerned with the following Schr\"{o}dinger-Poisson system involving critical nonlocal term $$ \left\{ \begin{array}{ll} -\Delta u+u-K(x)\phi |u|^3u=\lambda f(x)|u|^{q-2}u, & x\in\mathbb{R}^3, -\Delta \phi=K(x)|u|^5, &…

In this paper, we consider the singularly perturbed fractional Schr\"{o}dinger equation \begin{equation*} \epsilon^{2\alpha}(-\Delta)^\alpha u+V(x)u=f(u),\quad x\in \mathbb{R}^N, \end{equation*} where $\epsilon>0$ is a small parameter,…

In this paper we focus our attention on the following nonlinear fractional Schr\"odinger equation with magnetic field \begin{equation*} \varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{N},…