Existence and asymptotic behaviour of solutions for a quasi-linear schrodinger-poisson system under a critical nonlinearity

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 \phi = u^{2} & \ \mbox{in } \mathbb{R}^{3}, \end{array} \right. depending on the two parameters $\lambda,\varepsilon>0$. We first prove that, for $\lambda$ larger then a certain $\lambda^{*}>0$, there exists a solution for every $\varepsilon>0$. Later, we study the asymptotic behaviour of these solutions whenever $\varepsilon$ tends to zero, and we prove that they converge to the solution of the Schr\"odinger-Poisson system associated

A multiplicity result via Ljusternick-Schnirelmann category and morse theory for a fractional schr\"odinger equation in RN\mathbb R^{N}

In this work we study the following class of problems in $\mathbb R^{N}, N>2s$ $$\varepsilon^{2s} (-\Delta)^{s}u + V(z)u=f(u), \,\,\, u(z) > 0$$ where $0<s<1$, $(-\Delta)^{s}$ is the fractional Laplacian, $\varepsilon$ is a positive parameter, the potential $V:\mathbb{R}^N \to\mathbb{R}$ %is a continuous functions and the nonlinearity $f:\mathbb R \to \mathbb R$ satisfy suitable assumptions; in particular it is assumed that $V$ achieves its positive minimum on some set $M.$ By using variational methods we prove existence, multiplicity and concentration of maxima of positive solutions when $\varepsilon\to 0^{+}$. In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the "topological complexity" of the set $M$

Some remarks on the comparison principle in Kirchhoff equations

In this paper we study the validity of the comparison principle and the sub-supersolution method for Kirchhoff type equations. We show that these principles do not work when the Kirchhoff function is increasing, contradicting some previous results. We give an alternative sub-supersolution method and apply it to some models

Ground state solution for a Kirchhoff problem with exponential critical growth

We establish the existence of a positive ground state solution for a Kirchhoff problem in $\mathbb{R}^2$ involving critical exponential growth, that is, the nonlinearity behaves like $\exp(\alpha_{0}s^{2})$ as $|s| \to \infty$, for some $\alpha_{0}>0$. In order to obtain our existence result we used minimax techniques combined with the Trudinger-Moser inequality

Multi-bump solutions for a Kirchhoff problem type

In this paper, we are going to study the existence of solution for the following Kirchhoff problem \left\{ \begin{array}{l} M\biggl(\displaystyle\int_{\mathbb{R}^{3}}|\nabla u|^{2} dx +\displaystyle\int_{\mathbb{R}^{3}} \lambda a(x)+1)u^{2} dx\biggl) \biggl(- \Delta u + (\lambda a(x)+1)u\biggl) = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3}, \\ \mbox{}\\ u \in H^{1}(\mathbb{R}^{3}). \end{array} \right. Assuming that the nonnegative function $a(x)$ has a potential well with $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint components $\Omega_1, \Omega_2, ....., \Omega_k$ and the nonlinearity $f(t)$ has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by variational methods.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0293

Multiplicity of solutions for a NLS equations with magnetic fields in RN\mathbb{R}^{N}}

We investigate the multiplicity of nontrivial weak solutions for a class of complex equations. This class of problems are related with the existence of solitary waves for a nonlinear Sch\"{o}dinger equation. The main result is established by using minimax methods and Lusternik-Schnirelman theory of critical points

Existence of least energy nodal solution with two nodal domains for a generalized Kirchhoff problem in an Orlicz Sobolev space

We show the existence of a nodal solution with two nodal domains for a generalized Kirchhoff equation of the type -M\left(\displaystyle\int_\Omega \Phi(|\nabla u|)dx\right)\Delta_\Phi u = f(u) \ \ \mbox{in} \ \ \Omega, \ \ u=0 \ \ \mbox{on} \ \ \partial\Omega, where $\Omega$ is a bounded domain in $\mathbf{R}^N$, $M$ is a general $C^{1}$ class function, $f$ is a superlinear $C^{1}$ class function with subcritical growth, $\Phi$ is defined for $t\in \mathbf{R}$ by setting $\Phi(t)=\int_0^{|t|}\phi(s)sds$, $\Delta_\Phi$ is the operator $\Delta_\Phi u:=div(\phi(|\nabla u|)\nabla u)$. The proof is based on a minimization argument and a quantitative deformation lemma

On a nonlocal multivalued problem in an Orlicz-Sobolev space via Krasnoselskii's genus

This paper is concerned with the multiplicity of nontrivial solutions in an Orlicz-Sobolev space for a nonlocal problem involving N-functions and theory of locally Lispchitz continuous functionals.Comment: arXiv admin note: substantial text overlap with arXiv:1411.375

Nodal solutions of a NLS equation concentrating on lower dimensional spheres

In this work we deal with a following nonlinear Schrodinger equation in dimension greater or equal to 3, with a subcritical power-type nonlinearity and a positive potential satisfying a local condition. We prove the existence and concentration of nodal solutions which concentrate around a k - dimensional sphere of RN, where k is between 1 and N-1, as a parameter goes to 0. The radius of such sphere is related with the local minimum of a function which takes into account the potential. Variational methods are used together with the penalization technique in order to overcome the lack of compactness.Comment: 25 page

Multiplicity and concentration behavior of positive solutions for a Schrodinger-Kirchhoff type problem via penalization method

In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem \left\{\begin{array}{rcl} \mathcal{L}_{\varepsilon}u = f(u) \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u \in H^1 (\mathbb{R}^3), \end{array} \right. where $\varepsilon$ is a small positive parameter, $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function, $\mathcal{L}_{\varepsilon}$ is a nonlocal operator defined by $$\mathcal{L}_{\varepsilon} u = M \left(\frac{1}{\varepsilon} \int_{\mathbb{R}^3} |\nabla u|^2 + \frac{1}{\varepsilon^3} \int_{\mathbb{R}^3} V(x) u^{2}\right) \left[-\varepsilon^2\Delta u + V(x)u \right],$$ $M:\mathbb{R}_+\to \mathbb{R}_+$ and $V:\mathbb{R}^3 \to \mathbb{R}$ are continuous functions which verify some hypotheses.Comment: 32 page