Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity

Abstract

In this paper, we prove the existence of multiple weak solutions for a nonlinear nonlocal elliptic partial differential equation involving a singularity and a power nonlinearity, which is given as (−Δp)su=λuγ+uq;u>0 in Ω with zero Dirichlet boundary condition. Here, Ω is an open bounded domain in RN with smooth boundary, N > ps, s ∈ (0, 1), λ > 0, 0 < γ < 1, 1 < p < ∞, and p−1<q≤ps*=NpN−ps. We employ variational techniques to show the existence of multiple positive weak solutions of the above problem. We also prove that for some α ∈ (0, s], the weak solution to the problem is in C1,α(Ω¯).