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

2018/01/24 by Claudianor O. Alves, Alves, Claudianor O., Geilson F. Germano +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.1801.08138

Abstract

In this paper we are interested to prove the existence and concentration of\nground state solution for the following class of problems -
Deltaν+V(x)u=A(
epsilon x)f(u),
quad x
in
R^N,
eqno(P)
_epsilon where\nN \≥ 2, \ε>0, A: R^N\→ R is a continuous function that\nsatisfies \n0lt;
inf_x
in
R^NA(x)
leq
lim_{|x|
rightarrow+
infty}A(x)lt;
sup_x
in
R^NA(x)=A(0),
eqno(A)\n f: R\→ R is a continuous function having critical growth,\nV: R^N\→ R is a continuous and Z^N--periodic function with\n0\∉\σ(\Δ+V). By using variational methods, we prove the existence\nof solution for \ε small enough. After that, we show that the maximum\npoints of the solutions concentrate around of a maximum point of A.\n

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