A Nehari manifold for non-local elliptic operator with concave-convex non-linearities and sign-changing weight function

2013/07/19 by goyal, Sarika, Sreenadh, K. · 1 citation
#35J35 #35J60 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1307.5149

Abstract

In this article, we study the existence and multiplicity of non-negative solutions of following p-fractional equation: \\ds - 2∫_{\mb Rⁿ}\frac|u(y)-u(x)|^p-2(u(y)-u(x))|x-y|^n+p\al dxdy = \la h(x)|u|^q-1u+ b(x)|u|^r-1 u in \Om u ≥ 0 in \Om, u∈ W^\al,p(\mb Rⁿ), u =0 on \mb Rⁿ∖ \Om . where \Om is a bounded domain in \mb Rⁿ, p≥ 2, n> p\al, \al∈(0,1), 0< q0 and h, b are sign changing smooth functions. We show the existence of solutions by minimization on the suitable subset of Nehari manifold using the fibering maps. We find that there exists \la₀ such that for \la∈ (0,\la₀), it has at least two solutions.

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