Positive Solutions for Singular Anisotropic (p,\xa0q)-Equations

Nikolaos S. Papageorgiou,Patrick Winkert

doi:10.1007/s12220-021-00703-3

Nikolaos S. Papageorgiou, Patrick Winkert

Open Access

https://doi.org/10.1007/s12220-021-00703-3

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Abstract

In this paper, we consider a Dirichlet problem driven by an anisotropic (p, q)-differential operator and a parametric reaction having the competing effects of a singular term and of a superlinear perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter moves. Moreover, we prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.

Highlights

Let ⊆ RN be a bounded domain with a C2-boundary ∂
There are very few works on this subject. We mention two such papers which are close to our problem (Pλ). These are the works of Byun–Ko [4] and Saoudi–Ghanmi [26] who examine equations driven by the anisotropic p-Laplacian and the parameter multiplies only the singular term
We recall some basic facts about Lebesgue and Sobolev spaces with variable exponents

Summary

Introduction

Let ⊆ RN be a bounded domain with a C2-boundary ∂. In problem (Pλ) we have the sum of two such anisotropic differential operators with distinct exponents. Even in the case of constant exponents, the differential operator in (Pλ) is not homogeneous This makes the study of problem (Pλ) more difficult. Boundary value problems driven by a combination of differential operators of different nature, such as ( p, q)-equations, arise in many mathematical models of physical processes. There are very few works on this subject We mention two such papers which are close to our problem (Pλ). These are the works of Byun–Ko [4] and Saoudi–Ghanmi [26] who examine equations driven by the anisotropic p-Laplacian and the parameter multiplies only the singular. Related works to the topic can be found in the papers of Ambrosio [1], Ambrosio–Radulescu [2], Liu–Motreanu– Zeng [15], Papageorgiou–Zhang [23], Ragusa–Tachikawa [25], Zeng–Bai–Gasinski– Winkert [28,29] and the references therein

Preliminaries and Hypotheses

An Auxiliary Purely Singular Problem

Positive Solutions

Minimal Positive Solutions