On a class of parametric $(p,2)$-equations

TL;DR

This paper investigates the existence and multiplicity of solutions for a class of parametric $(p,2)$-equations involving the sum of a $p$-Laplacian and Laplace operator, especially near the principal eigenvalue.

Contribution

It provides new multiplicity results with sign information for solutions when the parameter is near the principal eigenvalue from above and below.

Findings

Multiple solutions are established near the principal eigenvalue.

Sign information about solutions is characterized.

Results apply to equations involving combined $p$-Laplacian and Laplace operators.

Abstract

We consider parametric equations driven by the sum of a $p$-Laplacian and a Laplace operator (the so-called $(p,2)$-equations). We study the existence and multiplicity of solutions when the parameter $\lambda>0$ is near the principal eigenvalue $\hat{\lambda}_1(p)>0$ of $(-\Delta_p,W^{1,p}_{0}(\Omega))$. We prove multiplicity results with precise sign information when the near resonance occurs from above and from below of $\hat{\lambda}_1(p)>0$.