On a class of infinite semipositone problems for (p,q) Laplace operator

TL;DR

This paper establishes the existence of positive solutions for a class of nonlinear elliptic boundary value problems involving the $(p, q)$ Laplace operator, using fixed point theorems, and identifies maximal solutions for specific nonlinearities.

Contribution

It is the first to analyze positive solutions for $(p, q)$ Laplace problems in arbitrary bounded domains, introducing new existence and maximality results.

Findings

Proved existence of positive solutions using fixed point theorem.

Derived maximal solutions for specific nonlinearities.

First results of this kind for arbitrary bounded domains with $(p, q)$ Laplace operators.

Abstract

We analyze a non-linear elliptic boundary value problem, that involves $(p, q)$ Laplace operator, for the existence of its positive solution in an arbitrary smooth bounded domain. The non-linearity here is driven by a continuous function in $(0,\infty)$ which is singular, monotonically increasing and eventually positive. We prove the existence of a positive solution of this problem using a fixed point theorem due to Amann\cite{amann1976fixed}. In addition, for a specific nonlinearity we derive that the obtained solution is maximal in nature. The main results obtained here are first of its kind for a $(p, q)$ Laplace operator in an arbitrary bounded domain.