Problem involving nonlocal operator

2017/07/12 by Giri, Ratan Kr., Choudhuri, D., Soni, Amita
#35J35 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1707.03636

Abstract

The aim of this paper is to deal with the elliptic pdes involving a nonlinear integrodifferential operator, which are possibly degenerate and covers the case of fractional p-Laplacian operator. We prove the existence of a solution in the weak sense to the problem \beginsplit -\mathscrL_Φu amp; = λ|u|^q-2u in Ω,
u amp; = 0 in ℝ^N∖ Ω\endsplit if and only if a weak solution to \beginsplit -\mathscrL_Φu amp; = λ|u|^q-2u +f, f∈ L^p'(Ω),
u amp; = 0 on ℝ^N∖ Ω\endsplit (p' being the conjugate of p), exists in a weak sense, for q∈(p, p_s^*) under certain condition on λ, where -\mathscrL_Φ is a general nonlocal integrodifferential operator of order s∈(0,1) and p_s^* is the fractional Sobolev conjugate of p. We further prove the existence of a measure μ^* corresponding to which a weak solution exists to the problem \beginsplit -\mathscrL_Φu amp; = λ|u|^q-2u +μ^* in Ω,
u amp; = 0 in ℝ^N∖ Ω\endsplit depending upon the capacity.

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