$(p,2)$-equations asymmetric at both zero and infinity

TL;DR

This paper studies a nonlinear elliptic equation combining p-Laplacian and Laplacian operators with asymmetric reaction terms, proving multiple solution existence and sign properties using variational and Morse theory methods.

Contribution

It introduces new multiplicity theorems for asymmetric $(p,2)$-equations, including solutions with specific sign characteristics.

Findings

Proved existence of multiple solutions including positive, negative, and nodal.

Developed variational, truncation, and Morse theory techniques for asymmetric problems.

Established sign information for all solutions.

Abstract

We consider a $(p,2)$-equation, that is, a nonlinear nonhomogeneous elliptic equation driven by the sum of a $p$-Laplacian and a Laplacian with $p>2$. The reaction term is $(p-1)$-linear but exhibits asymmetric behaviour at $\pm\infty$ and at $0^{\pm}$. Using variational tools, together with truncation and comparison techniques and Morse theory, we prove two multiplicity theorems, one of them providing sign information for all the solutions (positive, negative, nodal).