Existence of positive solution for a nonlinear elliptic equation with saddle-like potential and nonlinearity with exponential critical growth in R2\mathbb{R}^{2}

In this paper, we use variational methods to prove the existence of positive solution for the following class of elliptic equation -\epsilon^{2}\Delta{u}+V(z)u=f(u) \,\,\, \mbox{in} \,\,\, \mathbb{R}^{2}, where $\epsilon >0$ is a positive parameter, $V$ is a saddle-like potential and $f$ has an exponential critical growth

Existence of heteroclinic solution for a double well potential equation in an infinite cylinder of RN\mathbb{R}^N

This paper concernes with the existence of heteroclinic solutions for the following class of elliptic equations -\Delta{u}+A(\epsilon x, y)V'(u)=0, \quad \mbox{in} \quad \Omega, where $\epsilon >0$, \Omega=\R \times \D is an infinite cylinder of $\mathbb{R}^N$ with $N \geq 2$. Here, we have considered a large class of potential $V$ that includes the Ginzburg-Landau potential $V(t)=(t^{2}-1)^{2}$ and two geometric conditions on the function $A$. In the first condition we assume that $A$ is asymptotic at infinity to a periodic function, while in the second one $A$ satisfies 0Comment: In this revised version we have corrected some typos and changed the proof of some lemma

Existence of standing waves solution for a Nonlinear Schr\"odinger equations in RN\mathbb{R}^{N}

In this paper, we investigate the existence of positive solution for the following class of elliptic equation - \epsilon^{2}\Delta u +V(x)u= f(u) \,\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, where $\epsilon >0$ is a positive parameter, $f$ has a subcritical growth and $V$ is a positive potential verifying some conditions

Existence of solutions for a class of singular elliptic systems with convection term

We show the existence of positive solutions for a class of singular elliptic systems with convection term. The approach combines pseudomonotone operator theory, sub and supersolution method and perturbation arguments involving singular terms

Existence of a positive solution for a logarithmic Schr\"{o}dinger equation with saddle-like potential

In this article we use the variational method developed by Szulkin \cite{szulkin} to prove the existence of a positive solution for the following logarithmic Schr\"{o}dinger equation \left\{ \begin{array}{lc} -{\epsilon}^2\Delta u+ V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} \right. where $\epsilon >0, N \geq 1$ and $V$ is a saddle-like potential

Positive solutions for a class of quasilinear singular elliptic systems

In this paper we establish the existence of two positive solutions for a class of quasilinear singular elliptic systems. The main tools are sub and supersolution method and Leray-Schauder Topological degree.Comment: 19 page

Existence of positive multi-bump solutions for a Schr\"odinger-Poisson system in R3\mathbb{R}^{3}

In this paper we are going to study a class of Schr\"odinger-Poisson system \left\{ \begin{array}{ll} - \Delta u + (\lambda a(x)+1)u+ \phi u = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3},\\ -\Delta \phi=u^2 \mbox{ in } \,\,\, \mathbb{R}^{3}.\\ \end{array} \right. Assuming that the nonnegative function $a(x)$ has a potential well $int (a^{-1}(\{0\}))$ 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: text overlap with arXiv:1402.683

Existence of solutions for a nonlocal variational problem in R2\mathbb{R}^2 with exponential critical growth

We study the existence of solution for the following class of nonlocal problem, -\Delta u +V(x)u =\Big( I_\mu\ast F(x,u)\Big)f(x,u) \quad \mbox{in} \quad \mathbb{R}^2, where $V$ is a positive periodic potential, $I_\mu=\frac{1}{|x|^\mu}$, $0<\mu<2$ and $F(x,s)$ is the primitive function of $f(x,s)$ in the variable $s$. In this paper, by assuming that the nonlinearity $f(x,s)$ has an exponential critical growth at infinity, we prove the existence of solutions by using variational methods

Ground state solution for a class of indefinite variational problems with critical growth

In this paper we study the existence of ground state solution for an indefinite variational problem of the type \left\{\begin{array}{l} -\Delta u+(V(x)-W(x))u=f(x,u) \quad \mbox{in} \quad \R^{N}, u\in H^{1}(\R^{N}), \end{array}\right. \eqno{(P)} where $N \geq 2$, $V,W:\mathbb{R}^N \to \mathbb{R}$ and $f:\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$ are continuous functions verifying some technical conditions and $f$ possesses a critical growth. Here, we will consider the case where the problem is asymptotically periodic, that is, $V$ is $\mathbb{Z}^N$-periodic, $W$ goes to 0 at infinity and $f$ is asymptotically periodic

Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz-Sobolev spaces

This work is concerned with the existence and multiplicity of solutions for the following class of quasilinear problems -\Delta_{\Phi}u+\phi(|u|)u=f(u)~\text{in} ~\Omega_{\lambda}, u(x)>0 ~\text{in}~\Omega_{\lambda}, u=0~ \mbox{on} ~\partial\Omega_{\lambda}, where $\Phi(t)=\int_0^{|t|} \phi(s) s \, ds$ is an $N-$function, $\Delta_{\Phi}$ is the $\Phi-$Laplacian operator, \linebreak $\Omega_{\lambda}=\lambda \Omega,$ $\Omega$ is a smooth bounded domain in $\mathbb{R}^N,$ $N \geq 2$, $\lambda$ is a positive parameter and $f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. Here, we use variational methods to get multiplicity of solutions by using of Lusternik-Schnirelmann category of ${\Omega}$ in itself