Abstract
By virtue of $\Gamma-$convergence arguments, we investigate the stability of variational eigenvalues associated with a given topological index for the fractional $p-$Laplacian operator, in the singular limit as the nonlocal operator converges to the $p-$Laplacian. We also obtain the convergence of the corresponding normalized eigenfunctions in a suitable fractional norm.
Highlights
Many efforts have been devoted to the study of problems involving the fractional p−Laplacian operator, among which we mention eigenvalue problems [7,20,24,29], regularity theory [14,20,25, 27,28] and existence of solutions within the framework of Morse theory [26]
Our main concern is the study of the singular limit of these variational eigenvalues as s 1, in which case the limiting problem of (1.2) is formally given by
For every s ∈ (0, 1), let us ∈ W0s,p(Ω) be an eigenfunction corresponding to the variational eigenvalue λsm,p(Ω), normalized by us Lp(Ω) = 1
Summary
RN \Ω |x − y|N+sp dy dx = 0, and the claim is proved for u ∈ C0∞(Ω). There exists a sequence {φj}j∈N ⊂ C0∞(Ω) such that φj − u W 1,p(RN ) → 0 as j → ∞. By inequality (2.13) below we have (1 − s) p [φj − u]W s,p(RN ) ≤ C ∇(φj − u) Lp(Ω), with C independent of s and j. For every ε > 0, there exists j0 ∈ N independent of s such that (2.3). ∇u Lp(Ω) − ∇φj Lp(Ω) ≤ ∇φj − ∇u Lp(Ω) ≤ ε, and (1 − s) p [φj ]W s,p(RN ) − (1 − s) p [u]W s,p(RN ) ≤ (1 − s) p [φj − u]W s,p(RN ) ≤ C ε, for every j ≥ j0.
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