Related papers: Multiple positive solutions for a Schr\"odinger-Po…

In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem \begin{equation*} \left\{ \begin{array}[c]{ll} -\Delta u - \Delta (u^2)u = |u|^{p-2}u & \mbox{ in } \Omega u= 0 &\mbox{ on }…

In this paper, we investigate the existence of multiple positive solutions to the following multi-critical Schr\"{o}dinger equation \begin{equation} \label{p} \begin{cases} -\Delta u+\lambda V(x)u=\mu…

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}…

We consider the following Schr\"odinger-Bopp-Podolsky system in $\mathbb R^{3}$ $$\left\{ \begin{array}{c} -\varepsilon^{2} \Delta u + V(x)u + \phi u = f(u)\\ -\varepsilon^{2} \Delta \phi + \varepsilon^{4} \Delta^{2}\phi = 4\pi\varepsilon…

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, &…

The existence and $L^{\infty}$ estimate of positive solutions are discussed for the following Schr\"{o}dinger-Poisson system {ll} -\Delta u +(\lambda+\frac{1}{|y|^\alpha})u+\phi (x) u =|u|^{p-1}u, x=(y,z)\in \mathbb{R}^2\times\mathbb{R},…

We prove that, for any real $\lambda$, the system $-\Delta u +\lambda u = u^3-\beta uv^2$, $ -\Delta v+\lambda v =v^3-\beta vu^2$, $ u,v\in H^1_0(\Omega),$ where $\Omega$ is a bounded smooth domain of $R^3$, admits a bounded family of…

In this article we study the existence of solutions to the system \begin{equation*}\left\{ \begin{array}{ll} -\left(a+b\int_{\Omega}|\nabla u|^{2}\right)\Delta u +\phi u= f(x, u) &\text{in }\Omega \hbox{} -\Delta \phi= u^{2} &\text{in…

In this paper we are going to study a class of Schr\"odinger-Poisson system $$ \left\{ \begin{array}{ll} - \Delta u + (\lambda a(x)+1)u+ \phi u = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3},\\ -\Delta \phi=u^2 \mbox{ in } \,\,\,…

Consider the Schr\"odinger--Bopp--Podolsky system \[ \begin{cases} -\epsilon^2\Delta u+(V+K\phi)u=u|u|^{p-1};\newline \Delta^2\phi-\Delta\phi=4\pi K u^2 \end{cases} ~\text{in}~\mathbb{R}^3 \] for sufficiently small $\epsilon>0$, where…

In this paper, we consider the following Schr\"odinger-Poisson system \begin{equation*} \begin{cases} - \Delta u+\lambda V(x)u+ \mu\phi u=|u|^{p-2}u &\text{in $\mathbb{R}^3$},\cr -\Delta \phi=u^{2} &\text{in $\mathbb{R}^3$}, \end{cases}…

We obtain multiple solutions for the zero mass Schr{\"o}dinger-Poisson-Slater equation \[ - \Delta u + \left( \frac{1}{4 \pi | x |} \ast u^2 \right) u = \lambda g (x) | u |^{p - 2} u + | u |^{6 - 2} u \text{, \ \ \ \ } u \in \mathcal{D}^{1,…

We study a nonlinear Schr\"{o}dinger-Poisson system which reduces to the nonlinear and nonlocal equation \[- \Delta u+ u + \lambda^2 \left(\frac{1}{\omega|x|^{N-2}}\star \rho u^2\right) \rho(x) u = |u|^{q-1} u \quad x \in \mathbb R^N, \]…

In this paper we prove the existence of positive solutions of the following singular quasilinear Schr\"{o}dinger equations at critical growth \begin{eqnarray*} -\Delta u-\lambda c(x)u-\kappa\alpha(\Delta(|u|^{2\alpha}))|u|^{2\alpha-2}u =…

We look for solutions to the Schr\"{o}dinger-Poisson-Slater equation $$- \Delta u + \lambda u - \gamma (|x|^{-1} * |u|^2) u - a |u|^{p-2}u = 0 \quad \text{in} \quad \mathbb{R}^3, $$ which satisfy \begin{equation*} \int_{\mathbb{R}^3}|u|^2…

We study the Schr\"{o}dinger-Poisson type system: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+\lambda u+\left( \mu _{11}\phi _{u}-\mu _{12}\phi _{v}\right) u=% \frac{1}{2\pi }\int_{0}^{2\pi }\left\vert u+e^{i\theta }v\right\vert…

Let $N \geq 3$ and $\Omega \subset \mathbb{R}^N$ be $C^2$ bounded domain. We study the existence of positive solution $u \in H^1(\Omega)$ of \begin{align*} \left\{ \begin{array}{l} -\Delta u + \lambda u = \frac{|u|^{2^*(s)-2}u}{|x-x_1|^s} +…

In this paper we consider the problem \begin{equation*} \left \{ \begin{array}{l} -\Delta u \pm \phi u + W'(x,u) = 0\hbox{ in } \mathbb{R}^2,\newline \Delta \phi = u^2 \hbox{ in } \mathbb{R}^2, \end{array} \right. \end{equation*} where $W$…

Given a smooth bounded domain $\Omega\subset \mathbb R^3$, we consider the following nonlinear Schr\"odinger-Poisson type system \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+ \phi u -\abs{u}^{p-2}u = \omega u & \quad \text{in }…

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…