This paper is concerned with the forward and inverse problems for fractional semilinear elliptic equation (??)su+a(x,u)=0 0<s<1. For problem, we proved problem well-posed has a unique solution small exterior data. The consider here consists of two cases. First demonstrate that an unknown coefficient a(x,u) can be uniquely determined from knowledge measurements, known as Dirichlet-to-Neumann map...

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumption of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma devel...

An interesting field of modern mathematical research is the study of geometric properties of solutions to elliptic problems. Remarkably, this is often done without any explicit representation of the solution. This paper concentrates on the problem of convexity of level sets for solutions to some elliptic semilinear boundary-value problems in convex rings.More precisely, letΩ0,Ω1 be convex, boun...

We adopt the integral definition of fractional Laplace operator and analyze solution techniques for fractional, semilinear, elliptic optimal control problems posed on Lipschitz polytopes. consider two strategies discretization: a semidiscrete scheme where admissible set is not discretized fully discrete such with piecewise constant functions. As an instrumental step, we derive error estimates f...