Singular doubly nonlocal elliptic problems with Choquard type critical growth nonlinearities

Abstract

The theory of elliptic equations involving singular nonlinearities is well studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we study the very singular and doubly nonlocal singular problem (P λ )(See below). Firstly, we establish a very weak comparison principle and the optimal Sobolev regularity. Next using the critical point theory of non-smooth analysis and the geometry of the energy functional, we establish the global multiplicity of positive weak solutions.

Singular Fractional Choquard Equation with a Critical Nonlinearity and a Radon measure

“…Then Φ ǫ = 0 in R N \Ω. By Proposition 6.2 of Giacomoni et al [20], there exists a 1 , a 2 , a 3 , a 4 > 0 such that for 1 < q < min{2, N N −2s } we have the following four estimates.…”

“…Recently, Giacomoni et al in [20] dealt with fractional critical Choquard problem with singular nonlinearity, i.e. they considered problem (P β ) with α, λ = 0 and without the Radon measure µ.…”

“…they considered problem (P β ) with α, λ = 0 and without the Radon measure µ. In [20], the authors have explained a very weak comparison principle, established the existence of two positive weak solution and discussed about the Sobolev regularity of the solutions.…”

“…We show the existence of a SOLA to our problem with the method of approximations. We follow the approach closely related to the approaches used in [14], [20] and [28].…”