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On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption



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Díaz Díaz, Jesús Ildefonso and Hernández, J. and Ilyasov, Y. (2015) On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption. Nonlinear Analysis: Theory, Methods & Applications, 119 . pp. 484-500. ISSN 0362-546X

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Official URL: http://www.sciencedirect.com/science/article/pii/S0362546X1500005X


We study a semilinear elliptic equation with a strong absorption term given by a non-Lipschitz function. The motivation is related with study of the linear Schrödinger equation with an infinite well potential. We start by proving a general existence result for non-negative solutions. We use also variational methods, more precisely Nehari manifolds, to prove that for any λ>λ1 (the first eigenvalue of the Laplacian operator) there exists (at least) a non-negative solution. These solutions bifurcate from infinity at λ1 and we obtain some interesting additional information. We sketch also an asymptotic bifurcation approach, in particular this shows that there exists an unbounded continuum of non-negative solutions bifurcating from infinity at λ=λ1. We prove that for some neighborhood of (λ1,+∞) the positive solutions are unique. Then Pohozaev’s identity is introduced and we study the existence (or not) of free boundary solutions and compact support solutions. We obtain several properties of the energy functional and associated quantities for the ground states, together with asymptotic estimates in λ, mostly for λ↗λ1. We also consider the existence of solutions with compact support in Ω for λ large enough.

Item Type:Article
Uncontrolled Keywords:Semilinear elliptic equation; Strong absorption term; Spectral problem; Positive solutions; Nehari manifolds; Bifurcation from infinity; Pohozaev identity; Solutions with compact support
Subjects:Sciences > Mathematics
ID Code:30297
Deposited On:22 May 2015 09:00
Last Modified:12 Dec 2018 15:06

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