Existence and multiplicity of solutions for resonant $(p,2)$-equations

TL;DR

This paper investigates the existence and multiplicity of solutions for a class of resonant elliptic equations involving the p-Laplacian and Laplacian, using variational methods and critical groups.

Contribution

It establishes new existence and multiplicity results for resonant $(p,2)$-equations, including five smooth solutions with specific sign properties.

Findings

Proves existence of one nontrivial smooth solution.

Establishes multiplicity of five solutions, including four with constant sign and one nodal.

Solutions are ordered and obtained via variational methods and critical groups.

Abstract

We consider Dirichlet elliptic equations driven by the sum of a $p$-Laplacian $(2<p)$ and a Laplacian. The conditions on the reaction term imply that the problem is resonant at both $\pm\infty$ and at zero. We prove an existence theorem (producing one nontrivial smooth solution) and a multiplicity theorem (producing five nontrivial smooth solutions, four of constant sign and the fifth nodal; the solutions are ordered). Our approach uses variational methods and critical groups.