相关论文: The fibering method approach for a non-linear Schr…

We study the following nonlinear Schr\"odinger-Bopp-Podolsky system \[ \begin{cases} -\Delta u + \omega u + q^{2}\phi u = |u|^{p-2}u -\Delta \phi + a^2 \Delta^2 \phi = 4\pi u^2 \end{cases} \hbox{ in }\mathbb{R}^3 \] with $a,\omega>0$. We…

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…

Consider the following nonlinear Schr\"odinger--Bopp--Podolsky system in $\mathbb{R}^3$: \[ \begin{cases} - \Delta v + v + \phi v = v |v|^{p - 2}; \\ \beta^2 \Delta^2 \phi - \Delta \phi = 4 \pi v^2, \end{cases} \] where $\beta > 0$ and $3 <…

Consider the following Schr\"odinger-Bopp-Podolsky system in $\mathbb{R}^3$ under an $L^2$-norm constraint, \[ \begin{cases} -\Delta u + \omega u + \phi u = u|u|^{p-2},\newline -\Delta \phi + a^2\Delta^2\phi=4\pi u^2,\newline…

In this paper, we consider in $\mathbb{R}^3$ the following zero mass Schr\"odinger-Bopp-Podolsky system \[ \begin{cases} -\Delta u +q^2\phi u=|u|^{p-2}u\\ -\Delta \phi+a^2\Delta^2\phi=4\pi u^2 \end{cases} \] where $a>0$, $q\ne 0$ and $p\in…

We prove a multiplicity result for \begin{equation*} \begin{cases} -\varepsilon^{2}\Delta_g u+\omega u+q^{2}\phi u=|u|^{p-2}u\\[1mm] -\Delta_g \phi +a^{2}\Delta_g^{2} \phi + m^2 \phi =4\pi u^{2} \end{cases} \text{ in }M, \end{equation*}…

In this paper, we consider the following zero mass Schr\"{o}dinger-Bopp-Podolsky system \[ \begin{cases} -\Delta u +q^2\phi u=|u|^{p-2}u, -\Delta \phi+a^2\Delta^2\phi=4\pi u^2, \end{cases} \text{ in } \mathbb{R}^3, \] where $a>0$ and $q\ne…

In this paper, we study the following nonlinear Dirac-Bopp-Podolsky system \begin{equation*} \left\lbrace \begin{array}{rll} \displaystyle{ -i\sum_{k=1}^{3}\alpha_{k}\partial_{k}u+[V(x)+q]\beta u+wu-\phi u}&=f(x,u), \ \ &\text{in}\…

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…

In this paper, we study the following Schr\"odinger-Poisson system: $$ \left\{\aligned&-\Delta u+V_\lambda(x)u+K(x)\phi u=f(x,u)&\quad\text{in }\bbr^3,\\ &-\Delta\phi=K(x)u^2&\quad\text{in }\bbr^3,\\…

In this paper, we study the existence of ground state solutions for the nonlinear fractional Schr\"{o}dinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^su+V(x)u+\phi u=|u|^{p-1}u, & \hbox{in $\mathbb{R}^3$,}…

In the spirit of the classical work of P. H. Rabinowitz on nonlinear Schr\"odinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schr\"odinger-Poisson system \begin{equation}\nonumber…

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 }…

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 work we study the following class of systems of coupled nonlinear fractional nonlinear Schr\"odinger equations, \begin{equation*} \left \{ \begin{array}{l} (-\Delta)^s u_1+ \lambda_1 u_1= \mu_1 |u_1|^{2p-2}u_1+\beta |u_2|^{p}…

In this paper, by adapting the perturbation method, we study normalized standing wave solutions for the following nonlinear Schr\"odinger-Bopp-Podolsky system: - Delta u + q(x) phi u = omega u + f(u) in Omega, - Delta phi + a^2 Delta^2 phi…

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

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…

In this paper we consider the following quasilinear Schr\"odinger-Poisson system $$ \left\{ \begin{array}[c]{ll} - \Delta u +u+\phi u = \lambda f(x,u)+|u|^{2^{*}-2}u &\ \mbox{in } \mathbb{R}^{3} \\ -\Delta \phi -\varepsilon^{4} \Delta_4…

We study the nonlinear Schr\"odinger system \[ \begin{cases} \displaystyle iu_t+\Delta u-u+(\frac{1}{9}|u|^2+2|w|^2)u+\frac{1}{3}\overline{u}^2w=0,\\ i\displaystyle \sigma w_t+\Delta w-\mu w+(9|w|^2+2|u|^2)w+\frac{1}{9}u^3=0, \end{cases} \]…