Positive solutions for singular double phase problems

TL;DR

This paper investigates positive solutions for a class of singular double phase equations combining p-Laplacian and weighted q-Laplacian operators, demonstrating the existence of multiple solutions for small parameters using the Nehari method.

Contribution

It introduces new results on the existence of multiple positive solutions for singular double phase problems with discontinuous weights, employing the Nehari method.

Findings

At least two positive solutions exist for small parameter values.

The Nehari method effectively handles the singular and discontinuous aspects.

The results extend the understanding of double phase problems with singular terms.

Abstract

We study the existence of positive solutions for a class of double phase Dirichlet equations which have the combined effects of a singular term and of a parametric superlinear term. The differential operator of the equation is the sum of a $p$-Laplacian and of a weighted $q$-Laplacian ($q<p$) with discontinuous weight. Using the Nehari method, we show that for all small values of the parameter $\lambda>0$, the equation has at least two positive solutions.