Existence and concentration of nodal solutions for a subcritical p&amp;q equation

Gustavo S. Costa; Giovany M. Figueiredo

doi:10.3934/cpaa.2020227

Article Contents

2020, Volume 19, Issue 11: 5077-5095. Doi: 10.3934/cpaa.2020227

This issue Previous Article Quasilinear nonlocal elliptic problems with variable singular exponent Next Article A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains

Existence and concentration of nodal solutions for a subcritical p&q equation

Gustavo S. Costa and
Giovany M. Figueiredo^,

Departamento de Matemática - Universidade de Brasília, 70.910-900, Brasília - DF, Brazil

^* Corresponding author
^* Corresponding author

Received: January 31, 2020

Revised: April 30, 2020

Early access: July 2020

Published: November 2020

Gustavo S. Costa and Giovany M. Figueiredo were partially supported by CNPq, Capes and Fapesp - Brazil

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Abstract

In this paper we prove existence and concentration results for a family of nodal solutions for a some quasilinear equation with subcritical growth, whose prototype is

$ -\Delta_{p}u- \Delta_{q}u+V( x)(|u|^{p-2}u+|u|^{q-2}u) = f(u) \quad \mbox{in} \ \mathbb{R}^{N}. $

Each nodal solution changes sign exactly once in $ \mathbb{R}^{N} $ and has an exponential decay at infinity. Here we use variational methods and Del Pino and Felmer's technique [10] in order to overcome the lack of compactness.

Keywords:

p&q Laplacian operator,
Del Pino and Felmer's penalization method.

Mathematics Subject Classification: Primary: 35J60; Secondary: 35J10, 35J20.

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