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- Title
Infinitely Many Solutions for Critical Degenerate Kirchhoff Type Equations Involving the Fractional p–Laplacian.
- Authors
Binlin, Zhang; Fiscella, Alessio; Liang, Sihua
- Abstract
In this paper we study a class of critical Kirchhoff type equations involving the fractional p–Laplacian operator, that is M ∫ ∫ R 2 N | u (x) - u (y) | p | x - y | N + p s d x d y (- Δ) p s u = λ w (x) | u | q - 2 u + | u | p s ∗ - 2 u , x ∈ R N , where (- Δ) p s is the fractional p–Laplacian operator with 0 < s < 1 < p < ∞ , dimension N > p s , 1 < q < p s ∗ , p s ∗ is the critical exponent of the fractional Sobolev space W s , p (R N) , λ is a positive parameter, M is a non-negative function while w is a positive weight. By exploiting Kajikiya's new version of the symmetric mountain pass lemma, we establish the existence of infinitely many solutions which tend to zero under a suitable value of λ . The main feature and difficulty of our equations is the fact that the Kirchhoff term M is zero at zero, that is the equation is degenerate. To our best knowledge, our results are new even in the Laplacian and p–Laplacian cases.
- Subjects
DEGENERATE differential equations; SOBOLEV spaces; CRITICAL exponents; OPERATOR equations; BLOWING up (Algebraic geometry)
- Publication
Applied Mathematics & Optimization, 2019, Vol 80, Issue 1, p63
- ISSN
0095-4616
- Publication type
Article
- DOI
10.1007/s00245-017-9458-5