Author
Jacques Giacomoni
Other affiliations: University of Toulouse, Centre national de la recherche scientifique
Bio: Jacques Giacomoni is an academic researcher from University of Pau and Pays de l'Adour. The author has contributed to research in topics: Type (model theory) & Uniqueness. The author has an hindex of 18, co-authored 99 publications receiving 1015 citations. Previous affiliations of Jacques Giacomoni include University of Toulouse & Centre national de la recherche scientifique.
Papers published on a yearly basis
Papers
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TL;DR: In this paper, the authors employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 1,p 0 ( ).
Abstract: We investigate the following quasilinear and singular problem, { − pu = λ uδ + uq in ; u|∂ = 0, u > 0 in , (P) where is an open bounded domain with smooth boundary, 1 0, and 0 N . We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 1,p 0 ( ). While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions to problem (P) in C1,β ( ) with some β ∈ (0, 1). Furthermore, we show that δ < 1 is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) in C1( ). Mathematics Subject Classification (2000): 35J65 (primary); 35J20, 35J70 (secondary).
183 citations
TL;DR: In this paper, the existence of positive solutions for fractional elliptic equations of the type (-Δ)1/2u = h(u), u > 0 in (-1,1), u = 0 in ℝ∖(-1, 1) where h is a real valued function that behaves like eu2 as u → ∞.
Abstract: Abstract We study the existence of positive solutions for fractional elliptic equations of the type (-Δ)1/2u = h(u), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1) where h is a real valued function that behaves like eu2 as u → ∞ . Here (-Δ)1/2 is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t = 0. In case h is concave near t = 0, we show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold.
58 citations
TL;DR: In this article, the authors studied the singular, critical elliptic problem (P λ ) and showed the existence of multiple positive solutions by combining variational and sub super-solution methods.
Abstract: In this paper, we study the following singular, critical elliptic problem : ( P λ ) { − Δ u = λ ( u − δ + u q + ρ ( u ) ) in Ω u | ∂ Ω = 0 , u > 0 in Ω where Ω ⊂ R N ( N ≥ 3 ) is a bounded domain with smooth boundary ∂ Ω , λ > 0 , δ > 0 , 1 q ≤ 2 ∗ − 1 = N + 2 N − 2 and ρ a smooth function with subcritical asymptotic behaviour at ∞ . We show the existence of multiple positive solutions of the problem ( P λ ) by combining variational and sub super-solution methods.
55 citations
TL;DR: In this paper, the authors study the following fractional elliptic equation with critical growth and singular nonlinearity, and show the existence and multiplicity of positive solutions with respect to the parameter λ.
Abstract: Abstract In this article, we study the following fractional elliptic equation with critical growth and singular nonlinearity: ( - Δ ) s u = u - q + λ u 2 s * - 1 , u > 0 in Ω , u = 0 in ℝ n ∖ Ω , (-\\Delta)^{s}u=u^{-q}+\\lambda u^{{2^{*}_{s}}-1},\\qquad u>0\\quad\\text{in }% \\Omega,\\qquad u=0\\quad\\text{in }\\mathbb{R}^{n}\\setminus\\Omega, where Ω is a bounded domain in ℝ n {\\mathbb{R}^{n}} with smooth boundary ∂ Ω {\\partial\\Omega} , n > 2 s {n>2s} , s ∈ ( 0 , 1 ) {s\\in(0,1)} , λ > 0 {\\lambda>0} , q > 0 {q>0} and 2 s * = 2 n n - 2 s {2^{*}_{s}=\\frac{2n}{n-2s}} . We use variational methods to show the existence and multiplicity of positive solutions with respect to the parameter λ.
54 citations
TL;DR: In this article, it was shown that the condition at infinity for h is optimal for getting the existence of the second solution in a smooth bounded domain, where h = 0, 0 < δ < 3, Ω ≥ 3.
Abstract: In this paper, we are interested in the following singular problem: where 0 0, 0 < δ < 3, Ω a smooth bounded domain. We show on suitable conditions on h there exist two solutions in . Investigating the radial case, we are able to prove that the condition at infinity for h is optimal for getting the existence of the second solution.
49 citations
Cited by
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01 Jan 2003
2,810 citations
Book•
04 Oct 2007
TL;DR: In this article, the authors propose a model for solving the model elliptic problems and model parabolic problems. But their model is based on Equations with Gradient Terms (EGS).
Abstract: Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.
935 citations
374 citations
TL;DR: In this paper, the authors employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 1,p 0 ( ).
Abstract: We investigate the following quasilinear and singular problem, { − pu = λ uδ + uq in ; u|∂ = 0, u > 0 in , (P) where is an open bounded domain with smooth boundary, 1 0, and 0 N . We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 1,p 0 ( ). While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions to problem (P) in C1,β ( ) with some β ∈ (0, 1). Furthermore, we show that δ < 1 is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) in C1( ). Mathematics Subject Classification (2000): 35J65 (primary); 35J20, 35J70 (secondary).
183 citations