The Brezis–Nirenberg type problem for the p-laplacian (1 < p < 2): Multiple positive solutions
| Autor: | Giuseppina Vannella, Silvia Cingolani |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
p-Laplace operator
Applied Mathematics Perturbation results 010102 general mathematics Critical Sobolev exponent Multiplicity (mathematics) Type (model theory) 01 natural sciences 010101 applied mathematics Combinatorics Bounded function Domain (ring theory) p-Laplacian Morse theory 0101 mathematics Analysis Mathematics |
| Zdroj: | Journal of Differential Equations. 266:4510-4532 |
| ISSN: | 0022-0396 |
| DOI: | 10.1016/j.jde.2018.10.004 |
| Popis: | In this paper we consider the quasilinear critical problem ( P λ ) { − Δ p u = λ u q − 1 + u p ⁎ − 1 in Ω u > 0 in Ω u = 0 on ∂ Ω where Ω is a regular bounded domain in R N , N ≥ p 2 , 1 p 2 , p ≤ q p ⁎ , p ⁎ = N p / ( N − p ) , λ > 0 is a parameter. In spite of the lack of C 2 regularity of the energy functional associated to ( P λ ) , we employ new Morse techniques to derive a multiplicity result of solutions. We show that there exists λ ⁎ > 0 such that, for each λ ∈ ( 0 , λ ⁎ ) , either ( P λ ) has P 1 ( Ω ) distinct solutions or there exists a sequence of quasilinear problems approximating ( P λ ) , each of them having at least P 1 ( Ω ) distinct solutions. These results complete those obtained in [23] for the case p ≥ 2 . |
| Databáze: | OpenAIRE |
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