Related papers: Positive solutions for the Schr\"{o}dinger-Poisson…

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

We study the following Schr\"odinger-Poisson system (P_\lambda){ll} -\Delta u + V(x)u+\lambda \phi (x) u =Q(x)u^{p}, x\in \mathbb{R}^3 \\ -\Delta\phi = u^2, \lim\limits_{|x|\to +\infty}\phi(x)=0, u>0, where $\lambda\geqslant0$ is a…

In this paper, we consider the strongly coupled nonlinear Kirchhoff-type system with vanshing potentials: \begin{equation*}\begin{cases} -\left(a_1+b_1\int_{\mathbb{R}^3}|\nabla u|^2\dx\right)\Delta u+\lambda…

We consider the Schr\"odinger-Poisson system \begin{eqnarray}\left\{\begin{array} [c]{ll} -\Delta u+V(x) u+|u|^{p-2}u=\lambda \phi u, & \mbox{in}\mathbb{R}^{3},\\ -\Delta\phi= u^{2}, & \mbox{in}\mathbb{R}^{3}. \end{array} \right.\nonumber…

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…

We consider the following Schr\"odinger-Bopp-Podolsky system with critical and sublinear terms \begin{equation*} \begin{cases} - \Delta u+ u+Q(x)\phi u= \vert u\vert^4 u+ \lambda K(x)\vert u \vert^{p-1}u&\mbox{ in }\ \mathbb{R}^3 \smallskip…

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schr\"odinger equation $$ -\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\qquad u\in H^1_0(\Omega),\quad\int_\Omega u^2dx=\rho^2,\quad\lambda\in\mathbb{R}, $$…

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 Schr\"{o}dinger-Poisson-Slater equation \begin{equation*}\left\{\begin{array}{lll} -\Delta u + \lambda u + \big(|x|^{-1} \ast |u|^{2}\big)u = V(x) u^{ p_{\varepsilon}-1 }, \, \text{ in } \mathbb{R}^{3},\\[2mm]…

In this paper we consider the following quasilinear Schr\"odinger-Poisson system in a bounded domain in $\mathbb{R}^{2}$: $$ \left\{ \begin{array}[c]{ll} - \Delta u +\phi u = f(u) &\ \mbox{in } \Omega, -\Delta \phi - \varepsilon^{4}\Delta_4…

We are concerned with the study of positive solutions to the Gierer-Meinhardt system $$ \begin{cases} \displaystyle -\Delta u+\lambda u=\frac{u^p}{v^q}+\rho(x) &\quad\mbox{ in }\mathbb{R}^N\, , N\geq 3,\\[0.1in] \displaystyle -\Delta v+\mu…

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}\…

We study positive bound states for the equation $$- \epsilon^2 \Delta u + Vu = u^p, \qquad \text{in $\mathbf{R}^N$}, $$ where $\epsilon > 0$ is a real parameter, $\frac{N}{N-2} < p < \frac{N+2}{N-2}$ and $V$ is a nonnegative potential.…

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 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…

In this paper, we study the problem: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+u+\lambda K\left( x\right) \phi u=a\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{3}, \\ -\Delta \phi =K\left( x\right)…

We prove the existence of infinitely many non-radial positive solutions for the Schr\"{o}dinger-Newton system $$ \left\{\begin{array}{ll} \Delta u- V(|x|)u + \Psi u=0, &x\in\mathbb{R}^3,\newline \Delta \Psi+\frac12 u^2=0, &x\in\mathbb{R}^3,…

We consider the standing-wave problem for a nonlinear Schr\"{o}dinger equation, corresponding to the semilinear elliptic problem \begin{equation*} -\Delta u+V(x)u=|u|^{p-1}u,\ u\in H^1(\mathbb{R}^2), \end{equation*} where $V(x)$ is a…

We study a logarithmic fractional Schr\"odinger--Poisson system in \(\R^{3}\): \begin{equation*} \begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u+V(x)u+\phi u=u\log u^{2}+|u|^{p-2}u, & \text{in }\R^{3},\\…

We study the following class of linearly coupled Schr\"{o}dinger elliptic systems $$\left\{ \begin{array}{lr} -\Delta u+V_{1}(x)u=\mu|u|^{p-2}u+\lambda(x)v, & \quad x\in\mathbb{R}^{N}, \\ -\Delta v+V_{2}(x)v=|v|^{q-2}v+\lambda(x)u, &…