Related papers: Existence and Concentration of Multiple Positive S…
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 }…
In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic…
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…
We prove the existence of a ground state positive solution of Schr\"odinger-Poisson systems in the plane of the form $$ -\Delta u + V(x)u + \frac{\gamma}{2\pi} \left(\log|\cdot| \ast u^2 \right)u = b |u|^{p-2}u \qquad\text{in}\…
In the paper we consider the following quasilinear Schr\"odinger--Poisson system in the whole space $\mathbb R^{3}$ $$ \begin{cases} - \varepsilon^2 \Delta u + (V + \phi) u = u |u|^{p - 1} \newline - \Delta \phi - \beta \Delta_4 \phi = u^2,…
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}…
In this manuscript, we consider the logarithmic Schr\"{o}dinger equation \begin{eqnarray*} -\varepsilon^2\Delta u+V(x)u=u\log u^{2},\,\,\,u>0, & \text{in}\,\,\,\mathbb{R}^{N}, \end{eqnarray*} where $N\geq3$, $\varepsilon>0$ is a small…
We consider a class of parametric Schr\"odinger equations driven by the fractional $p$-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. By using variational methods and…
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}\\…
In this paper, we study critical semilinear nonlocal elliptic equations involving the logarithmic Schr\"odinger operator and its fractional pseudo-relativistic counterpart, both arising in quantum models with nonlocal and relativistic…
We study the existence of positive solutions for nonlocal systems in gradient form and set in the whole $\mathbb R^N$. A quasilinear fractional Schr\"odinger equation, where the leading operator is the $\frac Ns$-fractional Laplacian, is…
We consider singularly perturbed nonlinear Schr\"odinger equations \be \label{eq:0.1} - \varepsilon^2 \Delta u + V(x)u = f(u), \ \ u > 0, \ \ v \in H^1(\R^N) \ee where $V \in C(\R^N, \R)$ and $f$ is a nonlinear term which satisfies the…
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},…
This paper is devoted to the study of the existence of positive and bounded solutions for a Schr\"odinger type equation defined on the entire Euclidean space, involving a general integro-differential operator. We consider the case where the…
In this article we study the existence and concentration behavior of bound states for a nonlinear Schr\"odinger-Poisson system with a parameter $\varepsilon>0$. Under some suitable conditions on the potential functions, we prove that for…
We study the existence of positive solutions of a particular elliptic system in $\mathbb{R}^3$ composed of two coupled non linear stationary Schr\"odinger equations (NLSEs), that is $-\epsilon^2 \Delta u + V(x) u= h_v(u,v), - \epsilon^2…
In this paper we deal with the following fractional Choquard equation \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{sp}(-\Delta)^{s}_{p} u + V(x)|u|^{p-2}u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u) \mbox{ 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…
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…
We study the multiplicity and concentration of complex valued solutions for a fractional magnetic Schr\"odinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with…