A sublinear version of Schur's lemma and elliptic PDE

Abstract

We study the weighted norm inequality of $(1,q)$-type, \[ \Vert \mathbf{G}ν\Vert_{L^q(Ω, dσ)} \le C \Vert ν\Vert, \quad \text{ for all } ν\in \mathcal{M}^+(Ω), \] along with its weak-type analogue, for $0<q<1$, where $\mathbf{G}$ is an integral operator associated with the nonnegative kernel $G(x,y)$. Here ${\mathcal{M}}^{+}(\Omega )$ denotes the class of positive Radon measures in $\Omega$; $\sigma ,\nu \in {\mathcal{M}}^{+}(\Omega )$, and $||\nu ||=\nu (\Omega )$. For both weak-type and strong-type inequalities, we provide conditions which characterize the measures $\sigma$ for which such an embedding holds. The strong-type $(1,q)$-inequality for $0<q<1$ is closely connected with existence of a positive function $u$ such that $u\ge \mathbf{G}(u^q\sigma )$, i.e., a supersolution to the integral equation \[ u - \mathbf{G}(u^q σ) = 0, \quad u \in L^q_{\rm loc} (Ω, σ). \] This study is motivated by solving sublinear equations involving the fractional Laplacian, \[ (-Δ)^{\fracα{2}} u - u^q σ= 0\] in domains $\Omega \subseteq {\mathbf{R}}^n$ which have a positive Green function $G$, for $0<\alpha <n$.

29 pages

References in corpus (4)

• Global estimates for kernels of Neumann series and Green's functions
• Pointwise estimates of Brezis-Kamin type for solutions of sublinear elliptic equations
• 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 (8)

• Pointwise estimates of solutions to nonlinear equations for nonlocal operators
• Pointwise estimates of solutions to semilinear elliptic equations and inequalities
• Solutions to sublinear elliptic equations with finite generalized energy
• Quasilinear elliptic equations with sub-natural growth terms in bounded domains
• Accretivity and form boundedness of second order differential operators
• Trace inequalities of the Sobolev type and nonlinear Dirichlet problems
• Minimal L^p-Solutions to Singular Sublinear Elliptic Problems
• Nonlocal Sublinear Elliptic Problems Involving Measures