Sublinear equations and Schur's test for integral operators

Abstract

We study weighted norm inequalities of $(p,r)$-type, $\Vert \mathbf{G}(fd\sigma ){\Vert }_{L^r(\Omega ,d\sigma )}\le C\Vert f{\Vert }_{L^p(\Omega ,\sigma )},\forall f\in L^p(\sigma ),$ for $0<r<p$ and $p>1$, where $\mathbf{G}(fd\sigma )(x)={\int }_{\Omega }G(x,y)f(y)d\sigma (y)$ is an integral operator associated with a nonnegative kernel $G$ on $\Omega \times \Omega$, and $\sigma$ is a locally finite positive measure in $\Omega$. We show that this embedding holds if and only if ${\int }_{\Omega }(\mathbf{G}\sigma {)}^{\frac{pr}{p-r}}d\sigma <+\infty ,$ provided $G$ is a quasi-symmetric kernel which satisfies the weak maximum principle. In the case $p=\frac{r}{q}$, where $0<q<1$, we prove that this condition characterizes the existence of a non-trivial solution (or supersolution) $u\in L^r(\Omega ,\sigma )$, for $r>q$, to the the sublinear integral equation $u-\mathbf{G}(u^{q}d\sigma )=0,u\ge 0.$ We also give some counterexamples in the end-point case $p=1$, which corresponds to solutions $u\in L^q(\Omega ,\sigma )$ of this integral equation. These problems appear in the investigation of weak solutions to the sublinear equation involving the (fractional) Laplacian, $(-\Delta {)}^{\alpha }u-\sigma u^q=0,u\ge 0,$ for $0<q<1$ and $0<\alpha <\frac{n}{2}$ in domains $\Omega \subseteq {\mathbb{R}}^n$ with a positive Green function.

Cited by in corpus (1)

• Minimal L^p-Solutions to Singular Sublinear Elliptic Problems