Quasilinear nonlocal elliptic problems with variable singular exponent

Abstract

In this article, we provide existence results to the following nonlocal equation (−∆) s p u = g(x, u), u > 0 in Ω, u = 0 in R N \ Ω, (P λ) where (−∆) s p is the fractional p-Laplacian operator. Here Ω ⊂ R N is a smooth bounded domain, s ∈ (0, 1), p > 1 and N > sp. We establish existence of at least one weak solution for (P λ) when g(x, u) = f (x)u −q(x) and existence of at least two weak solutions when g(x, u) = λu −q(x) + u r for a suitable range of λ > 0. Here r ∈ (p − 1, p * s − 1) where p * s is the critical Sobolev exponent and 0 < q ∈ C 1 (Ω).

On an anisotropic p-Laplace equation with variable singular exponent

“…Papageorgiou-Scapellato [40] obtained existence results for a purely singular (p(x), q(x))-Laplace equation. In the nonlocal setting problem (1.11) has been investigated by Garain-Mukherjee in [30]. See also Rȃdulescu-Repovš [43] for an extensive literature of variable exponent problems.…”

On a class of mixed local and nonlocal semilinear elliptic equation with singular nonlinearity

“…In the second part of this article, we investigate the multiplicity result for the purturbed singularity (g 2 ) in Theorem 1.8. Here, we utilise the variational approach introduced in Arcoya-Boccardo [2] in combination with the technique from [26] to deal with the nonlocality. To this end, we obtain existence multiple solutions of the associated approximate problem (3.3).…”

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

“…has been studied by Fang [42] in the semilinear case p = 2; Canino-Montoro-Sciunzi-Squassina [23], Garain-Mukherjee [49] in the quasilinear case. The perturbed singular case is investigated by Barrios-De Bonis-Medina-Peral [10], Adimurthi-Giacomoni-Santra [1] for p = 2; Mukherjee-Sreenadh [57] in the quasilinear case and the references therein.…”