Nonlinear potential estimates for sublinear problems with applications to elliptic semilinear and quasilinear equations

Abstract

We give a survey of nonlinear potential estimates and their applications obtained recently for positive solutions to sublinear problems of the type \[ u = \mathbf{G}(σu^q) + f \quad \textrm{in} \,\, Ω, \] where $0<q<1$, $\sigma \ge 0$ is a Radon measure in $\Omega$, $f\ge 0$ is a measurable function, and $\mathbf{G}$ is a linear integral operator with positive kernel $G$ on $\Omega \times \Omega$. For quasi-metric (or quasi-metrically modifiable) kernels $G$, these bilateral pointwise estimates yield existence criteria and uniqueness of solutions $u\in L_{\mathrm{loc}}^q(\Omega ,\sigma )$. Applications are considered to semilinear elliptic equations involving the (fractional) Laplacian, \[ (-Δ)^{\fracα{2}} u = σu^q + μ\quad \textrm{in} \,\, Ω, \qquad u=0 \, \, \textrm{in} \,\, Ω^c. \] Here $0<q<1$, $\mu ,\sigma \ge 0$ are Radon measures, and $\Omega$ is a bounded uniform domain in ${\mathbb{R}}^n$, if $0<\alpha \le 2$, or the entire space ${\mathbb{R}}^n$, a ball or half-space, if $0<\alpha <n$. Analogues of these results are presented for elliptic equations involving the $p$-Laplace operator on the entire space ${\mathbb{R}}^n$, \[ -Δ_p u = σu^q + μ\quad \textrm{in} \,\, {\mathbb R}^n, \qquad \liminf_{x\to \infty} u(x)=0, \] where $0<q<p-1$, and $\mu ,\sigma \ge 0$ are Radon measures. More general quasilinear equations with $\mathcal{A}$-Laplace operators $\mathrm{div}\mathcal{A}(x,\nabla u)$ in place of ${\Delta }_p$ are covered as well.

30 pages. arXiv admin note: text overlap with arXiv:2203.02531