arXiv subjects

Geilson F. Germano

Publications and source records attributed to Geilson F. Germano.

3 recordsLinked to original sources

arXiv2019

Existence of ground state solution and concentration of maxima for a class of indefinite variational problems

In this paper we study the existence of ground state solution and concentration of maxima for a class of strongly indefinite problem like $$ \left\{\begin{array}{l} -Δu+V(x)u=A(εx)f(u) \quad \mbox{in} \quad \R^{N}, \\ u\in H^{1}(\R^{N}), \end{array}\right. \eqno{(P)_ε} $$ where $N \geq 1$, $ε$ is a positive parameter, $f: \mathbb{R} \to \mathbb{R}$ is a continuous function with subcritical growth and $V,A: \mathbb{R}^{N} \to \mathbb{R}$ are continuous functions verifying some technical conditions. Here $V$ is a $\mathbb{Z}^N$-periodic function, $0 \not\in σ(-Δ+ V)$, the spectrum of $-Δ+V$, and $$ 0 < \inf_{x \in \R^{N}}A(x)\leq \displaystyle\lim_{|x|\rightarrow+\infty}A(x)<\sup_{x \in \R^{N}}A(x). $$

math.AP↗

arXiv2018

Existence and concentration phenomena for a class of indefinite variational problems with critical growth

In this paper we are interested to prove the existence and concentration of ground state solution for the following class of problems $$ -Δu+V(x)u=A(εx)f(u), \quad x \in \R^{N}, \eqno{(P)_ε} $$ where $N \geq 2$, $ε>0$, $A:\R^{N}\rightarrow\R$ is a continuous function that satisfies $$ 0<\inf_{x\in\R^{N}}A(x)\leq\lim_{|x|\rightarrow+\infty}A(x)<\sup_{x\in\R^{N}}A(x)=A(0),\eqno{(A)} $$ $f:\R\rightarrow\R$ is a continuous function having critical growth, $V:\R^{N}\rightarrow\R$ is a continuous and $\Z^{N}$--periodic function with $0\notinσ(Δ+V)$. By using variational methods, we prove the existence of solution for $ε$ small enough. After that, we show that the maximum points of the solutions concentrate around of a maximum point of $A$.

math.AP↗

arXiv2017

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 $$ \left\{\begin{array}{l} -Δu+(V(x)-W(x))u=f(x,u) \quad \mbox{in} \quad \R^{N}, u\in H^{1}(\R^{N}), \end{array}\right. \eqno{(P)} $$ 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.

math.AP↗