In this paper we consider the quasilinear critical problem $$(P_λ) ⎨ −Δ_p u = λu^{q-1} + u^{p*-1} in Ω, u>0 in Ω, u =0 on ∂Ω}$$, where Ω is a regular bounded domain in R^N, N≥p^2, 1 <2, p≤q0 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.

The Brezis–Nirenberg type problem for the p-laplacian (1 < p < 2): Multiple positive solutions / Cingolani, Silvia; Vannella, Giuseppina. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - STAMPA. - 266:8(2019), pp. 4510-4532. [10.1016/j.jde.2018.10.004]

The Brezis–Nirenberg type problem for the p-laplacian (1 < p < 2): Multiple positive solutions

Vannella, Giuseppina
2019-01-01

Abstract

In this paper we consider the quasilinear critical problem $$(P_λ) ⎨ −Δ_p u = λu^{q-1} + u^{p*-1} in Ω, u>0 in Ω, u =0 on ∂Ω}$$, where Ω is a regular bounded domain in R^N, N≥p^2, 1 <2, p≤q0 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.
2019
The Brezis–Nirenberg type problem for the p-laplacian (1 &lt; p &lt; 2): Multiple positive solutions / Cingolani, Silvia; Vannella, Giuseppina. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - STAMPA. - 266:8(2019), pp. 4510-4532. [10.1016/j.jde.2018.10.004]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11589/160096
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