A concave–convex problem with a variable operator
We study the following elliptic problem - A(u) = λuq with Dirichlet boundary conditions, where A(u)(x)=Δu(x)χD1(x)+Δpu(x)χD2(x) is the Laplacian in one part of the domain, D1, and the p-Laplacian (with p> 2) in the rest of the domain, D2. We show that this problem exhibits a concave–convex na...
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todo:paper_09442669_v57_n1_p_Molino2023-10-03T15:49:13Z A concave–convex problem with a variable operator Molino, A. Rossi, J.D. 35J20 35J62 35J92 We study the following elliptic problem - A(u) = λuq with Dirichlet boundary conditions, where A(u)(x)=Δu(x)χD1(x)+Δpu(x)χD2(x) is the Laplacian in one part of the domain, D1, and the p-Laplacian (with p> 2) in the rest of the domain, D2. We show that this problem exhibits a concave–convex nature for 1 < q< p- 1. In fact, we prove that there exists a positive value λ∗ such that the problem has no positive solution for λ> λ∗ and a minimal positive solution for 0 < λ< λ∗. If in addition we assume that p is subcritical, that is, p< 2 N/ (N- 2) then there are at least two positive solutions for almost every 0 < λ< λ∗, the first one (that exists for all 0 < λ< λ∗) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every 0 < λ< λ∗) comes from an appropriate (and delicate) mountain pass argument. © 2017, Springer-Verlag GmbH Germany, part of Springer Nature. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_09442669_v57_n1_p_Molino |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
35J20 35J62 35J92 |
| spellingShingle |
35J20 35J62 35J92 Molino, A. Rossi, J.D. A concave–convex problem with a variable operator |
| topic_facet |
35J20 35J62 35J92 |
| description |
We study the following elliptic problem - A(u) = λuq with Dirichlet boundary conditions, where A(u)(x)=Δu(x)χD1(x)+Δpu(x)χD2(x) is the Laplacian in one part of the domain, D1, and the p-Laplacian (with p> 2) in the rest of the domain, D2. We show that this problem exhibits a concave–convex nature for 1 < q< p- 1. In fact, we prove that there exists a positive value λ∗ such that the problem has no positive solution for λ> λ∗ and a minimal positive solution for 0 < λ< λ∗. If in addition we assume that p is subcritical, that is, p< 2 N/ (N- 2) then there are at least two positive solutions for almost every 0 < λ< λ∗, the first one (that exists for all 0 < λ< λ∗) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every 0 < λ< λ∗) comes from an appropriate (and delicate) mountain pass argument. © 2017, Springer-Verlag GmbH Germany, part of Springer Nature. |
| format |
JOUR |
| author |
Molino, A. Rossi, J.D. |
| author_facet |
Molino, A. Rossi, J.D. |
| author_sort |
Molino, A. |
| title |
A concave–convex problem with a variable operator |
| title_short |
A concave–convex problem with a variable operator |
| title_full |
A concave–convex problem with a variable operator |
| title_fullStr |
A concave–convex problem with a variable operator |
| title_full_unstemmed |
A concave–convex problem with a variable operator |
| title_sort |
concave–convex problem with a variable operator |
| url |
http://hdl.handle.net/20.500.12110/paper_09442669_v57_n1_p_Molino |
| work_keys_str_mv |
AT molinoa aconcaveconvexproblemwithavariableoperator AT rossijd aconcaveconvexproblemwithavariableoperator AT molinoa concaveconvexproblemwithavariableoperator AT rossijd concaveconvexproblemwithavariableoperator |
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1782031135247695872 |