A Singular Semilinear Elliptic Equation with a Variable Exponent

Abstract

In this paper we consider singular semilinear elliptic equations with a variable exponent whose model problem isHere Ω is an open bounded set of ℝ N , γ(x) is a positive continuous function and f(x) is a positive function that belongs to a certain Lebesgue space. We prove that there exists a solution to this problem in the natural energy space H (Ω) when γ(x) ≤ in a strip around the boundary. For another case, we prove that the solution belongs to H loc (Ω) and that it is zero on the boundary in a suitable sense.

Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

“…Without the aim to be complete, we refer to various works treating different aspects of problems as in (1.1) and in (1.2). The literature concerning the case of linear operators is [1,2,3,10,11,12,16,17,18,24,25]. For more general operators we refer to [21,22,29,32,36].…”

The Dirichlet problem for possibly singular elliptic equations with degenerate coercivity

“…Finally we quote [15] where the authors show uniqueness of solutions in presence of the p-Laplace operator and for a sufficiently regular function f . For more and different aspects concerning singular problems we refer to [3,14,16,22,23,24,26,28,32,42].…”

Quasilinear nonlocal elliptic problems with variable singular exponent

“…existence of at least one weak solution was proved in [12]. Motivated by this and the current interest in the study of nonlocal problems involving fractional Laplacian, we study the singular problem (1.1) where the singular exponent is a function of x.…”