Brezis–Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem

By N. Hirano, C. Saccon, Naoki Shioji. 2008

108 citations recorded by the source corpus.

Abstract We study Brezis–Nirenberg type theorems for the equation − Δ u + g ( x , u ) = f ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain in R N , g ( x , ⋅ ) is increasing and f is a dissipative nonlinearity. We apply such theorems for studying existence and multiplicity of positive solutions for the equation − Δ u = u − q + λ u p in Ω , u = 0 on ∂ Ω , where q > 0 , p > 1 and λ > 0 .

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/a3e79e0c03107fa4805023bfa520888b097a95b4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity (2015, background): “4) has at least two solutions for μ > 0 small enough, such as [2],[6],[12],[26]-[29].”
Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient (2019, background): “[23] proved Brezis–Nirenberg type theorems for a singular elliptic problem.”
A bifurcation-type theorem for singular nonlinear elliptic equations (2015, unknown): “For singular equations, such problems were studied by Coclite-Palmieri [4], Crandall-Rabinowitz-Tartar [5], Hirano-Saccon-Shioji [11], LairShaker [14], Lazer-MacKenna [15], Sun-Wu-Long [21] for p = 2 (semilinear equations) and Giacomoni-Schindler-Takac [8], Perera-Zhang [19] for p = 2 (nonlinear equ”
Semilinear problems for the fractional laplacian with a singular nonlinearity (2015, background): “L2/ D 8̂̂<̂ :̂ u D u C u in ; u > 0 in ; u D 0 in @; where , p and are positive numbers (see, among other papers, [3, 4, 8, 12, 14, 21, 35] for an extensive analysis of this kind of problems).”
Positive solutions for nonlinear parametric singular Dirichlet problems (2018, methodology): “We mention the works of Coclite & Palmieri [2], Ghergu & Rădulescu [5], Hirano, Saccon & Shioji [10], Lair & Shaker [11], Sun, Wu & Long [21].”
On Dirichlet problem for fractional p-Laplacian with singular non-linearity (2016, background): “y 4 Lemma 2.2 For each w∈ X+, there exists a sequence {wk} in X0 such that,wk → wstrongly in X0, where 0 ≤ w1 ≤ w2 ≤ ...and wk has compact support in Ω, for each k . Proof. Proof here is adopted from [22]. Let w∈ X+ and {ψk} be sequence in Cc∞(Ω) such that ψk is non negative and converges strongly t”