Abstract
In this paper, we study the existence of infinitely many weak solutions for nonlocal elliptic equations with critical exponent driven by the fractional p-Laplacian of order s. We show the above result when λ>0 is small enough. We achieve our goal by making use of variational methods, more specifically, the Nehari Manifold and Lusternik-Schnirelmann theory.
Highlights
This work is concerned with the existence of weak solutions of the following critical fractional p-Laplacian problem:(−Δ)sp u = |u|ps∗−2 u + λ |u|r−2 u in Ω, (1)u = 0 on ∂Ω, where Ω is a smoothly bounded domain of RN, N ≥ sp, 0 < s < 1, 1 < r < p < ps∗ fl Np/(N − sp) is the fractional critical Sobolev exponent, and λ is positive parameter.(−Δ)sp denotes the fractional p-Laplacian operator defined on smooth functions by (−Δ)sp u (x) = lim ε↘0 ∫ RN \Bε (x) u (x) −u (y)p−2 (u (x) x − yN+sp (y)) dy
We study the existence of infinitely many weak solutions for nonlocal elliptic equations with critical exponent driven by the fractional p-Laplacian of order s
This work is concerned with the existence of weak solutions of the following critical fractional p-Laplacian problem: (−Δ)sp u = |u|ps∗−2 u + λ |u|r−2 u in Ω, (1)
Summary
We study the existence of infinitely many weak solutions for nonlocal elliptic equations with critical exponent driven by the fractional p-Laplacian of order s. This work is concerned with the existence of weak solutions of the following critical fractional p-Laplacian problem: (−Δ)sp u = |u|ps∗−2 u + λ |u|r−2 u in Ω, (1) U = 0 on ∂Ω, where Ω is a smoothly bounded domain of RN, N ≥ sp, 0 < s < 1, 1 < r < p < ps∗ fl Np/(N − sp) is the fractional critical Sobolev exponent, and λ is positive parameter.
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