The Leray-Lions existence theorem under general growth conditions

TL;DR

This paper extends the Leray-Lions existence theorem to elliptic equations with general growth conditions, allowing explicit dependence on the solution and spatial variables, thus broadening the class of solvable problems.

Contribution

It introduces a novel approach to handle general p,q-growth conditions with explicit dependence on (x,u), expanding the applicability of the Leray-Lions theorem beyond natural growth scenarios.

Findings

Existence of weak solutions under general growth conditions.

Extension of Leray-Lions theorem to broader elliptic equations.

Handling of explicit (x,u) dependence in the differential operator.

Abstract

We prove an existence result of weak solutions $u\in W_{0}^{1,p}\left( \Omega \right) \cap W_{\mathrm{loc}}^{1,q}\left( \Omega \right) $, to a Dirichlet problem for a second order elliptic equation in divergence form, under general and $p,q-$growth conditions of the differential operator. This is a first attempt to extend to general growth the well known Leray-Lions existence theorem, which holds under the so-called natural growth conditions with $q=p$. We found a way to treat the general context with explicit dependence on $\left( x,u\right) $, other than on the gradient variable $\xi =Du$; these aspects require particular attention due to the $p,q-$context, with some differences and new difficulties compared to the standard case $p=q$.