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

Abstract

We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation \[ -Δ_{p} u = σu^{q} + μ\quad \text{on} \;\; \mathbb{R}^n \] 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 $\alpha =1$.

33 pages, published online in Advances in Calculus of Variations

References in corpus (3)

• Global estimates for kernels of Neumann series and Green's functions
• Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms
• Positive solutions to Schrödinger's equation and the exponential integrability of the balayage

Cited by in corpus (4)

• Solutions to sublinear elliptic equations with finite generalized energy
• Existence of minimal solutions to quasilinear elliptic equations with several sub-natural growth terms
• Minimal L^p-Solutions to Singular Sublinear Elliptic Problems
• Nonlocal Sublinear Elliptic Problems Involving Measures