On a Class of Weighted p-Laplace Equation with Singular Nonlinearity

Abstract

We consider a class of singular weighted anisotropic p-Laplace equations. We provide sufficient condition on the weight function that may vanish or blow up near the origin to ensure the existence of at least one weak solution in the purely singular case and at least two different weak solutions in the purturbed singular case.

On an anisotropic p-Laplace equation with variable singular exponent

“…In the perturbed singular case, see Arcoya-Boccardo [4] for the semilinear case and Giacomoni-Schindler-Takáč [32], Bal-Garain [5] for the quasilinear case respectively. See also Garain-Mukherjee [27,29] and Leggat-Miri [36], dos Santos-Figueiredo-Tavares [22], Bal-Garain [6,28] for the study of singular problems in the context of weighted and anisotropic p-Laplace operator respectively. Furthermore, we refer to the nice survey by Ghergu-Rȃdulescu [31] for an extensive study of singular elliptic problems.…”

Weighted singular problem with variable singular exponent and $p$-admissible weights

“…In this article, we provide sufficient conditions on the weight function w, the singular exponent γ and on the nonlinearity f to ensure existence of weak solutions of (1.1). We remark that the weights w and the operator −div(A(x, ∇u)) that we considered in (1.1) is more general than considered in [7,8,17,26,28] and in contrast to [9], we complement on the existence and uniqueness results for 1 < p < 2 and γ > 1 in the weighted setting.…”

“…In such situation, equation (1.4) becomes degenerate that is captured by the weight function w, one can refer to Drábek-Kufner-Nicolosi [23], Heinonen-Kilpeläinen-Martio [34], Gol'dshtein-Motreanu-Motreanu-Ukhlov [31,32] for a wide range of investigation of degenerate problems with non-singular nonlinearities. Recently, authors in [7,8,26,28] studied the following type of weighted singular problems (1.7) −∆ p,w u = g(x, u) in Ω, u > 0 in Ω, u = 0 on ∂Ω to deal with the question of existence for various type of singular nonlinearity g, where the weight w belong to a subclass of Muckenhoupt weights. Recently, Garain-Kinnunen [27] settled the question of nonexistence for singular p-Laplace equations in a general metric measure space with doubling weights supporting a weak Poincaré inequality.…”

Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

“…De Cave [31], Canino-Sciunzi-Trombetta [25] and Garain [45] considered the quasilinear case to obtain existence, uniqueness and regularity results. In the perturbed singular case, multiplicity results for the problem (P) have been obtained by Arcoya-Mérida [5], Arcoya-Boccardo [4] in the semilinear case; Giacomoni-Schindler-Takáč [51], Bal-Garain-Mukherjee [7,48] in the quasilinear case and the references therein.…”