Combined effects of singular and superlinear nonlinearities in singular double phase problems in $\mathbb{R}^N$

TL;DR

This paper investigates multiple solutions for a class of singular double phase problems in Euclidean space, using the Nehari manifold approach to establish the existence of at least two solutions when the parameter is small.

Contribution

It introduces a novel application of the Nehari manifold method to singular double phase problems, demonstrating the existence of multiple solutions under certain conditions.

Findings

At least two nontrivial solutions exist for small parameters.

The Nehari manifold can be split into three parts, with the third being empty for small parameters.

The energy functional attains minima on two parts of the Nehari manifold.

Abstract

This paper is concerned with multiplicity results for parametric singular double phase problems in $\mathbb{R}^N$ via the Nehari manifold approach. It is shown that the problem under consideration has at least two nontrivial weak solutions provided the parameter is sufficiently small. The idea is to split the Nehari manifold into three disjoint parts minimizing the energy functional on two of them. The third set turns out to be the empty set for small values of the parameter.