Principal eigenvalues and eigenfunctions for fully nonlinear equations in punctured balls
This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions (λ¯...
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| Veröffentlicht in: | Journal de mathématiques pures et appliquées Jg. 186; S. 74 - 102 |
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| Abstract | This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions (λ¯γ,uγ) of the equationF(D2uγ)+λ¯γuγrγ=0inB(0,1)∖{0},uγ=0on∂B(0,1) where uγ>0 in B(0,1)∖{0} and γ>0. We prove existence of radial solutions which are continuous on B(0,1)‾ in the case γ<2, existence of unbounded solutions in the case γ=2 and a non existence result for γ>2. We also give, in the case of Pucci's operators, the explicit value of λ¯2, which generalizes the Hardy–Sobolev constant for the Laplacian.
Dans cet article nous nous intéressons à l'existence de valeur propre principale et de fonctions propres associées pour des opérateurs complètement non linéaires dans un domaine épointé, en présence d'un potentiel singulier.
Plus pécisément, nous analysons l'existence, l'unicité et la régularité de solutions de l'équationF(D2uγ)+λ¯γuγrγ=0dansB(0,1)∖{0},uγ=0sur∂B(0,1) où uγ>0 est définie sur B(0,1)∖{0} et γ>0. Nous montrons l'existence de solutions radiales qui sont continues sur B(0,1)‾ dans le cas γ<2, l' existence de solutions non bornées dans le cas γ=2 et un résultat de non existence dans le cas γ>2. Nous donnons aussi, dans le cas des opérateurs de Pucci, la valeur explicite de λ¯2, ce qui généralise la constante de Hardy–Sobolev dans le cas du Laplacien. |
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| AbstractList | This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and ¯λγ regularity of solutions ( , u γ ) of the equation¯λ γ uγ F(D 2 u γ ) + = 0 in B(0, 1) 0 , uγ = 0 on ∂B(0, 1) r γ \ { }where u γ > 0 in B(0, 1) \ { 0 } and γ > 0. We prove existence of radial solutions which are continuous on B(0, 1) in the case γ < 2, existence of unbounded solutions in the case γ = 2 and a non existence result for γ > 2. We also give, in the case of Pucci’s operators, the ¯λ2 explicit value of , which generalizes the Hardy–Sobolev constant for the Laplacian.
Dans cet article nous nous intéressons à l'existence de valeur propre principale et de fonctions propres associées pour des opérateurs complètement non linéaires dans un domaine épointé, en présence d'un potentiel singulier. This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions (λ¯γ,uγ) of the equationF(D2uγ)+λ¯γuγrγ=0inB(0,1)∖{0},uγ=0on∂B(0,1) where uγ>0 in B(0,1)∖{0} and γ>0. We prove existence of radial solutions which are continuous on B(0,1)‾ in the case γ<2, existence of unbounded solutions in the case γ=2 and a non existence result for γ>2. We also give, in the case of Pucci's operators, the explicit value of λ¯2, which generalizes the Hardy–Sobolev constant for the Laplacian. Dans cet article nous nous intéressons à l'existence de valeur propre principale et de fonctions propres associées pour des opérateurs complètement non linéaires dans un domaine épointé, en présence d'un potentiel singulier. Plus pécisément, nous analysons l'existence, l'unicité et la régularité de solutions de l'équationF(D2uγ)+λ¯γuγrγ=0dansB(0,1)∖{0},uγ=0sur∂B(0,1) où uγ>0 est définie sur B(0,1)∖{0} et γ>0. Nous montrons l'existence de solutions radiales qui sont continues sur B(0,1)‾ dans le cas γ<2, l' existence de solutions non bornées dans le cas γ=2 et un résultat de non existence dans le cas γ>2. Nous donnons aussi, dans le cas des opérateurs de Pucci, la valeur explicite de λ¯2, ce qui généralise la constante de Hardy–Sobolev dans le cas du Laplacien. |
| Author | Demengel, Françoise Leoni, Fabiana Birindelli, Isabeau |
| Author_xml | – sequence: 1 givenname: Isabeau surname: Birindelli fullname: Birindelli, Isabeau email: isabeau@mat.uniroma1.it organization: Sapienza Università di Roma, P.le Aldo Moro 2, 00185 Roma, Italy – sequence: 2 givenname: Françoise surname: Demengel fullname: Demengel, Françoise email: francoise.demengel@cyu.fr organization: CY Paris University, 33, Boulevard du Port, 95011 Cergy-Pontoise Cedex, France – sequence: 3 givenname: Fabiana surname: Leoni fullname: Leoni, Fabiana email: leoni@mat.uniroma1.it organization: Sapienza Università di Roma, P.le Aldo Moro 2, 00185 Roma, Italy |
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| Keywords | 35J75 Fully nonlinear elliptic equations 35J60 35P15 Regularity of eigenfunctions 35P30 Singular potential Principal eigenvalues singular potential regularity of eigenfunctions principal eigenvalues |
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| SubjectTerms | Analysis of PDEs Fully nonlinear elliptic equations Mathematics Principal eigenvalues Regularity of eigenfunctions Singular potential |
| Title | Principal eigenvalues and eigenfunctions for fully nonlinear equations in punctured balls |
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