Existence and Concentration Phenomena for a Class of Indefinite Variational Problems with Critical Growth

Abstract

In this paper we are interested to prove the existence and concentration of ground state solution for the following class of problemsf : R → R is a continuous function having critical growth, V : R N → R is a continuous and Z N -periodic function with 0 / ∈ σ(∆ + V ). By using variational methods, we prove the existence of solution for ǫ small enough. After that, we show that the maximum points of the solutions concentrate around of a maximum point of A. (2010): 35B40, 35J2, 47A10 . However, in [16], Pankov has found a ground state solution by minimizing the energy functional J on the set Mathematics Subject ClassificationsThe reader is invited to see that if J is strongly definite, that is, when E − = {0}, the set O is exactly the Nehari manifold associated with J. Hereafter, we say that u 0 ∈ H 1 (R N ) is a ground state solution if J ′ (u 0 ) = 0, u 0 ∈ O and J(u 0 ) = inf w∈O J(w).

Existence of solution for a class of indefinite variational problems with discontinuous nonlinearity

“…F (x, t)/|t| 2 → +∞ uniformly in x as |t| → +∞, (h 6 ) and t → f (x, t)/|t| is strictly increasing on R \ {0}. (h 7 ) The same approach was used by Alves and Germano [4,5], and Zhang, Xu and Zhang [51,52].…”

A Berestycki–Lions type result for a class of degenerate elliptic problems involving the Grushin operator

“…Finally, in order to show that , we need only show that the above inequality is true on balls of , that is, there exist and such that The argument follows with some adjusts from the ideas found in [4, lemma 2.13].…”