Finite energy solutions to inhomogeneous nonlinear elliptic equations with sub-natural growth terms

TL;DR

This paper establishes necessary and sufficient conditions for the existence and uniqueness of finite energy solutions to certain inhomogeneous nonlinear elliptic equations with sub-natural growth terms, covering both p-Laplacian and fractional Laplacian cases.

Contribution

It provides a comprehensive analysis of existence and uniqueness criteria for finite energy solutions in inhomogeneous nonlinear elliptic equations with sub-natural growth, extending to fractional Laplacian scenarios.

Findings

Derived necessary and sufficient conditions for solutions.

Proved uniqueness of solutions under given conditions.

Extended results to fractional Laplacian and various domains.

Abstract

We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} + \mu \quad \text{on} \;\; \mathbb{R}^n \] in the sub-natural growth case $0<q<p-1$, where $\Delta_{p}$ ($1<p<\infty$) is the $p$-Laplacian, and $\sigma$, $\mu$ are positive Borel measures on $\mathbb{R}^n$. Uniqueness of such a solution is established as well. Similar inhomogeneous problems in the sublinear case $0<q<1$ are treated for the fractional Laplace operator $(-\Delta)^{\alpha}$ in place of $-\Delta_{p}$, on $\mathbb{R}^n$ for $0<\alpha<\frac{n}{2}$, and on an arbitrary domain $\Omega \subset \mathbb{R}^n$ with positive Green's function in the classical case $\alpha = 1$.