Abstract
\This article is devoted to the existence and multiplicity to the following Brezis–Nirenberg-type problems involving singular nonlinearities:−Δu=up−1u+λu−1−β/xαuin Ωu=0on ∂Ω, whereΩis a smooth bounded domain inℝNN≥3,0∈Ω,λ>0,p=2∗−1with2∗≔2N/N−2is the critical Sobolev exponent,0≤α<Np+β/p+1, and0<β<1. By using the Nehari manifold and maximum principle theorem, the existence of at least two distinct positive solutions is obtained.
Highlights
Introduction e main purpose of this article is to investigate the existence of positive solutions of the following problem:
We research the critical points as the minimizers of the energy functional associated to problem (1) on the constraint defined by the Nehari manifold, which are solutions of our system
It is well known that J is of class C1 in H, and the solutions of (1) are the critical points of J which is not bounded below on H
Summary
It is well known that J is of class C1 in H, and the solutions of (1) are the critical points of J which is not bounded below on H. In order to obtain the first positive solution, we give the following important lemmas. If u0 is a local minimizer for J on M, u0 is a Lemma 3. Ere exists a positive number λ∗ such that for all solution of the optimization problem:. (i) For all λ such that 0 < λ < λ∗, there exists a (PS)c+ sequence in M+. (ii) For all λ such that 0 < λ < λ∗∗, there exists a (PS)c− sequence in M− and for each u ∈ H\{0}, we write tM tmax(u). J t+u infJ(tu), for 0 ≤ t ≤ tM, J(t− u) supJ(tu), for t ≥ 0
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