Sub and supersolutions, invariant cones and multiplicity results for p-Laplace equations

TL;DR

This paper develops a new critical point theory for p-Laplace equations using sub-supersolutions and invariant cones, leading to existence and multiplicity results including a sign-changing solution.

Contribution

It introduces an abstract framework based on invariant cones in the gradient flow for quasilinear elliptic equations, extending multiplicity results.

Findings

Proves existence of multiple solutions for p-Laplace equations.

Establishes a sign-changing solution among the solutions.

Develops a novel invariance approach in the $W^{1,p}_0$ topology.

Abstract

For a class of quasilinear elliptic equations involving the p-Laplace operator, we develop an abstract critical point theory in the presence of sub-supersolutions. Our approach is based upon the proof of the invariance under the gradient flow of enlarged cones in the $W^{1,p}_0$ topology. With this, we prove abstract existence and multiplicity theorems in the presence of variously ordered pairs of sub-supersolutions. As an application, we provide a four solutions theorem, one of the solutions being sign-changing.