Abstract
In this paper, we consider the existence of nontrivial solutions for a fractional p-q Laplacian equation with critical nonlinearity in a bounded domain. Our approach is based on variational methods and some analytical techniques.
Highlights
1 Introduction In this paper, we are interested in the existence of nontrivial solutions for the following equation: (– )spu + (– )squ = λ|u|r–2u + |u|p∗s –2u, x ∈ Ω, u = 0, x ∈ RN \ Ω, (1.1)
Where Ω is a bounded domain in RN, 0 < s < 1, 1 < q < p < r < p∗s, λ is a positive constant, p∗s = pN/(N – sp) is the fractional critical exponent, and (– )sp is the fractional p-Laplacian operator defined on smooth functions as
The definition is consistent, up to a normalization constant depending on N and s, with the usual definition of the linear fractional Laplacian operator (– )s when p = 2
Summary
We are interested in the existence of nontrivial solutions for the following equation:. In a bounded domain with p ≥ 2, where g is a subcritical nonlinearity By variational methods they prove the existence of a local minimizer of the associated functional to (1.4), which turns to be a weak solution of problem (1.4), provided that the constant λ is sufficiently small. Where f satisfies some subcritical conditions; see Fiscella et al [14] By variational methods they obtain multiplicity and bifurcation results for (1.5), which generalized those given in [15] to the nonlocal framework of the fractional Laplacian. We mention that there is a local weak lower semicontinuity result for the corresponding energy functional of problem (1.1), which leads to the existence of a critical point under certain conditions. Taking if necessary a subsequence, we can assume that there exists u ∈ W0s,p(Ω) such that uk u in W0s,p(Ω), uk → u in Ls(Ω), 1 ≤ s < p∗, uk → u a.e. in Ω
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