Nonlinear equations and weighted norm inequalities

Abstract:

We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem \[ \begin {aligned} - \Delta u & = v u^{q} + w, \quad u \ge 0 && \text {on $\Omega $},\\ u &= 0 & \text {on $\partial \Omega $}, \end {aligned} \] on a regular domain $\Omega$ in $\mathbf {R}^{n}$ in the “superlinear case” $q > 1$. The coefficients $v, w$ are arbitrary positive measurable functions (or measures) on $\Omega$. We also consider more general nonlinear differential and integral equations, and study the spaces of coefficients and solutions naturally associated with these problems, as well as the corresponding capacities. Our characterizations of the existence of positive solutions take into account the interplay between $v$, $w$, and the corresponding Green’s kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on $v$ and $w$; we also obtain sharp two-sided estimates of solutions up to the boundary. Some of our results are new even if $v \equiv 1$ and $\Omega$ is a ball or half-space. The corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so-called $3 G$-inequality by an elementary “integration by parts” argument. This also gives a new unified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.