Solutions in Lebesgue spaces to nonlinear elliptic equations with subnatural growth terms

Abstract:

The paper is devoted to the existence problem for positive solutions ${u \in L^{r}(\mathbb {R}^{n})}$, $0<r<\infty$, to the quasilinear elliptic equation \begin{equation*} -\Delta _{p} u = \sigma u^{q} \ \text { in } \ \mathbb {R}^n \end{equation*} in the subnatural growth case $0<q< p-1$, where $\Delta _{p}u = \mathrm {div}( |\nabla u|^{p-2} \nabla u )$ is the $p$-Laplacian with $1<p<\infty$, and $\sigma$ is a nonnegative measurable function (or measure) on $\mathbb {R}^n$.

The techniques rely on a study of general integral equations involving nonlinear potentials and related weighted norm inequalities. They are applicable to more general quasilinear elliptic operators in place of $\Delta _{p}$ such as the $\mathcal {A}$-Laplacian $\mathrm {div} \mathcal {A}(x,\nabla u)$, or the fractional Laplacian $(-\Delta )^{\alpha }$ on $\mathbb {R}^n$, as well as linear uniformly elliptic operators with bounded measurable coefficients $\mathrm {div} (\mathcal {A} \nabla u)$ on an arbitrary domain $\Omega \subseteq \mathbb {R}^n$ with a positive Green function.