Concentration of solutions for a class of biharmonic and p-Laplacian equations

Abstract

This article is concerned with the existence of a ground state solution for the class of elliptic Kirchhoff–Boussinesq type problems ${\mathrm{\Delta }}^{2}u\pm {\mathrm{\Delta }}_{p}u+(1+\mathrm{\lambda V}(x))u=f(u)+\gamma |u{|}^{2_{\ast \ast }-2}u\mathrm{in}{\mathbb{R}}^N,$ where $2<p<2^{\ast }=\frac{2N}{N-2}$ for $N\ge 3$ and $2_{\ast \ast }=\mathrm{\infty }$ for N = 3, N = 4, $2_{\ast \ast }=\frac{2N}{N-4}$ for $N\ge 5$. Here f is a continuous function and the term $1+\mathrm{\lambda V}(x)$ is the steep potential well introduced by Bartsch and Wang in [Bartsch T, Wang ZQ. Existence and multiplicity results for superlinear elliptic problems on ${\mathbb{R}}^N$. Commun Partial Differ Equ 1995;20:1725–1741]. The function f has subcritical growth and behaves like $|u{|}^{q-2}u$ with $p<q<2_{\ast \ast }$. We show the existence of a ground state solution using variational methods considering the subcritical case, i.e. $\gamma =0$ and the critical case, i.e. $\gamma =1$.

Keywords:

• Kirchhoff Boussinesq
• Nehari manifold
• Biharmonic operator
• concentration of solutions

AMS Subject Classifications:

• Primary 35B38
• 35J35
• Secondary 35J92

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Romulo D. Carlos and Giovany M. Figuereido were partially supported by CNPq, supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico-Brazil[163054/2020-7] Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Fundação de Apoio à Pesquisa do Distrito Federal.