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

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Abstract

In this paper we are interested to prove the existence and concentration of ground state solution for the following class of problems

where N ≥ 2, 𝜖 > 0, \(A:\mathbb {R}^{N}\rightarrow \mathbb {R}\) is a continuous function that satisfies

$$0< \inf\limits_{x\in\mathbb{R}^{N}}A(x)\leq \lim\limits_{|x|\rightarrow+\infty}A(x)< \sup\limits_{x\in\mathbb{R}^{N}}A(x)=A(0), $$
(A)

\(f:\mathbb {R}\rightarrow \mathbb {R}\) is a continuous function having critical growth, \(V:\mathbb {R}^{N}\rightarrow \mathbb {R}\) is a continuous and \(\mathbb {Z}^{N}\)–periodic function with 0∉σ(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.

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Acknowledgments

C. O. Alves was partially supported by CNPq/Brazil 304804/2017-7 and INCT-MAT. G. F. Germano was partially supported by CAPES.

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Authors and Affiliations

  1. Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, CEP: 58429-900, Campina Grande, Pb, Brazil

    Claudianor O. Alves & Geilson F. Germano

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  1. Claudianor O. Alves
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  2. Geilson F. Germano
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Correspondence to Claudianor O. Alves.

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Alves, C.O., Germano, G.F. Existence and Concentration Phenomena for a Class of Indefinite Variational Problems with Critical Growth. Potential Anal 52, 135–159 (2020). https://doi.org/10.1007/s11118-018-9734-2

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