Quasilinear elliptic equations with sub-natural growth terms in bounded domains

Abstract

We consider the existence of positive solutions to weighted quasilinear elliptic differential equations of the type \[ \begin{cases} - Δ_{p, w} u = σu^{q} & \text{in $\Omega$}, \\ u = 0 & \text{on $\partial \Omega$} \end{cases} \] in the sub-natural growth case $0<q<p-1$, where $\Omega$ is a bounded domain in ${\mathbb{R}}^n$, ${\Delta }_{p,w}$ is a weighted $p$-Laplacian, and $\sigma$ is a nonnegative (locally finite) Radon measure on $\Omega$. We give criteria for the existence problem. For the proof, we investigate various properties of $p$-superharmonic functions, especially the solvability of Dirichlet problems with infinite measure data.

References in corpus (7)

• Nonlinear elliptic equations and intrinsic potentials of Wolff type
• Pointwise estimates of solutions to nonlinear equations for nonlocal operators
• Pointwise estimates of solutions to semilinear elliptic equations and inequalities
• A sublinear version of Schur's lemma and elliptic PDE
• Pointwise estimates of Brezis-Kamin type for solutions of sublinear elliptic equations
• Solutions to sublinear elliptic equations with finite generalized energy
• Existence of minimal solutions to quasilinear elliptic equations with several sub-natural growth terms

Cited by in corpus (2)

• Trace inequalities of the Sobolev type and nonlinear Dirichlet problems
• A stability result for elliptic equations with singular nonlinearity and its applications to homogenization problems