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Pointwise estimates of Brezis-Kamin type for solutions of sublinear elliptic equations (1601.05496v1)

Published 21 Jan 2016 in math.AP

Abstract: We study quasilinear elliptic equations of the type $$-\Delta_pu=\sigma \, uq \quad \text{in} \, \, \, \mathbb{R}n,$$ where $\Delta_p u=\nabla \cdot(\nabla u |\nabla u|{p-2})$ is the $p$-Laplacian (or a more general $\mathcal{A}$-Laplace operator $\text{div} \, \mathcal{A}(x, \nabla u)$), $0<q < p-1$, and $\sigma \ge 0$ is an arbitrary locally integrable function or measure on $\mathbb{R}n$. We obtain necessary and sufficient conditions for the existence of positive solutions (not necessarily bounded) which satisfy global pointwise estimates of Brezis-Kamin type given in terms of Wolff potentials. Similar problems with the fractional Laplacian $(-\Delta ){\alpha}$ for $0<\alpha<\frac{n}{2}$ are treated as well, including explicit estimates for radially symmetric $\sigma$. Our results are new even in the classical case $p=2$ and $\alpha=1$.

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