Pointwise estimates of solutions to nonlinear equations for nonlocal operators

Abstract

We study pointwise behavior of positive solutions to nonlinear integral equations, and related inequalities, of the type \begin{equation*} u(x) - \int_ΩG(x, y) \, g(u(y)) d σ(y) = h, \end{equation*} where $(\Omega ,\sigma )$ is a locally compact measure space, $G(x,y):\Omega \times \Omega \to [0,+\infty ]$ is a kernel, $h\ge 0$ is a measurable function, and $g:[0,\infty )\to [0,\infty )$ is a monotone function. This problem is motivated by the semilinear fractional Laplace equation \begin{equation*} (-Δ)^{\fracα{2}} u - g(u) σ= μ\quad \text{in} \, \, Ω, \quad u=0 \, \, \, \text{in} \, \, Ω^c, \end{equation*} with measure coefficients $\sigma$, $\mu$, where $g(u)=u^q$, $q\in \mathbb{R}\setminus \{0\}$, and $0<\alpha <n$, in domains $\Omega \subseteq {\mathbb{R}}^n$, or Riemannian manifolds, with positive Green's function $G$.

25 pages

References in corpus (7)

• Elliptic Equations Involving Meausres
• Global estimates for kernels of Neumann series and Green's functions
• Pointwise estimates of solutions to semilinear elliptic equations and inequalities
• Pointwise estimates of solutions to nonlinear equations for nonlocal operators
• A sublinear version of Schur's lemma and elliptic PDE
• Pointwise estimates of Brezis-Kamin type for solutions of sublinear elliptic equations
• Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms

Cited by in corpus (5)

• Pointwise estimates of solutions to nonlinear equations for nonlocal operators
• Pointwise estimates of solutions to semilinear elliptic equations and inequalities
• Positive solutions to Schrödinger's equation and the exponential integrability of the balayage
• Quasilinear elliptic equations with sub-natural growth terms in bounded domains
• Positive solutions and harmonic measure for Schrödinger operators in uniform domains