A Berestycki–Lions type result and applications

Abstract

In this paper we show an abstract theorem involving the existence of critical points for a functional I, which permit us to prove the existence of solutions for a large class of Berestycki-Lions type problems. In the proof of the abstract result we apply the deformation lemma on a special set associated with I, which we call of Pohozaev set.2010 Mathematics Subject Classification. 35J60; 35A15, 49J52.

A global minimization trick to solve some classes of Berestycki-Lions type problems

“…In the present paper, we show a new abstract theorems, see Theorems 2.1, 4.4 and 6.1, which can be used to prove the existence of solution for (1.1) and related problems. These theorems are totally different from that proved in [1] and their proofs are very simple, however Theorem 4.4 permits to show the existence of multiple solutions for a class of zero mass problem that is a new result for this class of problem. The present article completes the study made in [1], in the sense that we can consider other class of problems, for example, our approach can be applied to study a version of Berestycki-Lions type problems for elliptic system, which were not considered until moment in the literature, see Section 6.…”

“…For more details about this subject, we would like to cite the references found in the above mentioned papers. Recently, Alves, Duarte and Souto [1] have proved an abstract theorem that was used to solve a large class of Berestycki-Lions type problems, which includes Anisotropic operator, Discontinuous nonlinearity, etc.. In that paper, the authors have used the deformation lemma together with a notation of Pohozaev set to prove the abstract theorem.…”

A Berestycki–Lions type result for a class of problems involving the 1-Laplacian operator

“…showing that J (u 0 )w = 0 for all w ∈ D 1,2 γ,radx,y R N , that is, u 0 is a critical point of J in D 1,2 γ,radx,y R N . We claim that u 0 = 0, because if u 0 = 0, the Lemma 2.4 gives that…”

“…By Lemma 3.3, the sequence (u n ) is bounded in H 1,2 γ,radx,y (R N ), then for some subsequence, still denoted by itself, there is u 0 ∈ H 1,2 γ,radx,y (R N ) such that…”

“…Recently, Alves, Duarte and Souto [1] have proved an abstract theorem that was used to solve a large class of Berestycki-Lions type problems, which includes Anisotropic operator, Discontinuous nonlinearity, etc.. In that paper, the authors have used the deformation lemma together with a notation of Pohozaev set to prove the abstract theorem.…”