A Probabilistic Approach to the Existence of Solutions to Semilinear Elliptic Equations

TL;DR

This paper introduces a probabilistic method to establish the existence of positive solutions for semilinear elliptic equations with power nonlinearities, relaxing restrictions on the exponent and providing new insights into solution conditions.

Contribution

It presents a novel probabilistic approach to determine solution existence without upper bounds on the exponent, extending previous geometric and topological criteria.

Findings

Provided sufficient probabilistic conditions for solution existence

Established a multiplicity result for solutions

Discussed potential for characterizing domain properties probabilistically

Abstract

We study a semilinear elliptic equation with a pure power nonlinearity with exponent $p>1$, and provide sufficient conditions for the existence of positive solutions. These conditions involve expected exit times from the domain, $D$, where a solution is defined, and expected occupation times in suitable subdomains of $D$. They provide an alternative new approach to the geometric or topological sufficient conditions given in the literature for exponents close to the critical Sobolev exponent. Moreover, unlike standard results, in our probabilistic approach no \emph{a priori} upper bound restriction is imposed on $p$, which might be supercritical. The proof is based on a fixed point argument using a probabilistic representation formula. We also prove a multiplicity result and discuss possible extensions to the existence of sign changing solutions. Finally, we conjecture that necessary conditions for the existence of solutions might be obtained using a similar probabilistic approach. This motivates a series of natural questions related to the characterisation of topological and geometrical properties of a domain in probabilistic terms.