Nonlinear Eigenvalue Problems for the Dirichlet (p,2)-Laplacian

Yunru Bai,Nikolaos S Papageorgiou,Leszek Gasiński

doi:10.3390/axioms11020058

Abstract

We consider a nonlinear eigenvalue problem driven by the Dirichlet (p,2)-Laplacian. The parametric reaction is a Carathéodory function which exhibits (p−1)-sublinear growth as x→+∞ and as x→0+. Using variational tools and truncation and comparison techniques, we prove a bifurcation-type theorem describing the “spectrum” as λ>0 varies. We also prove the existence of a smallest positive eigenfunction for every eigenvalue. Finally, we indicate how the result can be extended to (p,q)-equations (q≠2).

Highlights

We study the following nonlinear eigenvalue problem for the Dirichlet ( p, 2)-Laplacian
We are looking for positive solutions as the parameter λ > 0 varies
In the aforementioned works, the equation is driven by the p-Laplacian differential operator which is homogeneous, a property used by the authors in the proof of their results

Summary

Introduction

For every r ∈ (1, ∞) by ∆r we denote the r-Laplacian differential operator defined by. Λ > 0 is a parameter and f (z, x ) is a Carathéodory function. We are looking for positive solutions as the parameter λ > 0 varies. Our work complements those by Gasiński and Papageorgiou [1] and Papageorgiou, Rădulescu and Repovš [2] where the reaction is ( p − 1)-superlinear in x ∈ R. In the aforementioned works, the equation is driven by the p-Laplacian differential operator which is homogeneous, a property used by the authors in the proof of their results.

Mathematical Background—Hypotheses

Positive Solutions