Abstract
Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.
Highlights
Introduction and main resultsConsider the following class of mixed local–nonlocal elliptic equations (−∆)s u − ε∆u = f (x, u) in Ω, u= in RN \ Ω, (1.1)where Ω is a bounded smooth domain of RN, N >, s ∈ (, ), ε > is a real parameter and (−∆)s u(x) = cN,s PV u(x) |x −− u(y) y|N+ s d yRN is the fractional Laplacian operator
In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian
The issue of the existence of solutions to fractional Laplacian problems, as well as the study of qualitative properties of such solutions, has been addressed by many authors and the literature devoted to the eld grows up continuously
Summary
Consider the following class of mixed local–nonlocal elliptic equations (−∆)s u − ε∆u = f (x, u) in Ω, u=. Integro-di erential equations of the form (1.1) arise naturally in the study of stochastic processes with jumps. The generator of an N-dimensional Lévy process has the following general structure: Lu = aij ∂ij u + bj ∂j u + (u(x + y) − u(x) − y · ∇u(x))χB (y)dν(y),. I,j j where ν is the Lévy measure and satis es RN min{ , |y| }dν(y) < +∞, and χB is the usual characteristic function of the unit ball B of RN. The rst term of (1.2) on the right-hand side corresponds to the di usion, the second one to the drift, and the third one to the jump part
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
