Multiple positive solutions for a Schr\"odinger-Poisson-Slater system

In this paper we investigate the existence of positive solutions to the following Schr\"odinger-Poisson-Slater system [c]{ll} - \Delta u+ u + \lambda\phi u=|u|^{p-2}u & \text{in} \Omega -\Delta\phi= u^{2} & \text{in} \Omega u=\phi=0 & \text{on} \partial\Omega. where $\Omega$ is a bounded domain in $\mathbf{R}^{3},\lambda$ is a fixed positive parameter and $p<2^{*}=\frac{2N}{N-2}$. We prove that if $p$ is "near" the critical Sobolev exponent $2^*$, then the number of positive solutions is greater then the Ljusternik-Schnirelmann category of $\Omega$.Comment: added references and improved the resul

Nonlinear Schr\"odinger equation in the Bopp-Podolsky electrodynamics: solutions in the electrostatic case

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 prove existence and nonexistence results depending on the parameters $q,p$. Moreover we also show that, in the radial case, the solutions we find tend to solutions of the classical Schr\"odinger-Poisson system as $a\to0$.Comment: 30 pages, the nonexistence result has been improve

Existence of ground states for a modified nonlinear Schrodinger equation

In this paper we prove existence of ground state solutions of the modified nonlinear Schrodinger equation: $$-\Delta u+V(x)u-{1/2}u \Delta u^{2}=|u|^{p-1}u, x \in \R^N, N \geq 3,$$ under some hypotheses on $V(x)$. This model has been proposed in the theory of superfluid films in plasma physics. As a main novelty with respect to some previous results, we are able to deal with exponents $p\in(1,3)$. The proof is accomplished by minimization under a convenient constraint

A minimization problem for the Nonlinear Schr¨odinger-Poisson type Equation

In this paper we consider the stationary solutions of the Schr¨odinger-Poisson equation:it + − (|x|−1 | |2) + | |p−2 = 0 in R3. We are interested in the existence of standing waves, that is solutions of type (x, t) = u(x)e−i!t, where ! 2 R, with fixed L2 −norm. Then we are reduced to a constrained minimization problem. The main difficulty is the compactness of the minimizing sequences since the related functional is invariant  y translations. By using some abstract results, we give a positive answer, showing that the minimum of the functional is achieved on small L2 −spheres in the case 2 &lt; p &lt; 3 and large L2 − spheres in the case 3 &lt; p &lt; 10/3. The results exposed here can be found with more details in [6] and [7].  

On a generalized Kirchhoff equation with sublinear nonlinearities

In this paper we consider a generalized Kirchhoff? equation in a bounded domain under the effect of a sublinear nonlinearity. Under suitable assumptions on the data of the problem we show that, with a simple change of variable, the equation can be reduced to a classical semilinear equation and then studied with standard tools.Comment: 13 page

Klein-Gordon-Maxwell System in a bounded domain

This paper is concerned with the Klein-Gordon-Maxwell system in a bounded spatial domain. We discuss the existence of standing waves $\psi=u(x)e^{-i\omega t}$ in equilibrium with a purely electrostatic field $\mathbf{E}=-\nabla\phi(x)$. We assume an homogeneous Dirichlet boundary condition on $u$ and an inhomogeneous Neumann boundary condition on $\phi$. In the "linear" case we characterize the existence of nontrivial solutions for small boundary data. With a suitable nonlinear perturbation in the matter equation, we get the existence of infinitely many solutions.Comment: 17 page