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Multiple solutions for an indefinite elliptic problem with critical growth in the gradient

Authors: Louis Jeanjean and Humberto Ramos Quoirin
Journal: Proc. Amer. Math. Soc. 144 (2016), 575-586
MSC (2010): Primary 35J20, 35J61, 35J91
Published electronically: May 28, 2015
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Abstract: We consider the problem

$\displaystyle -\Delta u =c(x)u+\mu \vert\nabla u\vert^2 +f(x), \quad u \in H^1_0(\Omega ) \cap L^{\infty }(\Omega ), \leqno {(P)}$    

where $ \Omega $ is a bounded domain of $ \mathbb{R}^N$, $ N \geq 3$, $ \mu >0$ and $ c, f \in L^q(\Omega )$$ \text { for some}$
$ q>\frac {N}{2}$ with $ f\gneqq 0. $ Here $ c$ is allowed to change sign and we assume that $ c^+ \not \equiv 0$. We show that when $ c^+$ and $ \mu f$ are suitably small this problem has at least two positive solutions. This result contrasts with the case $ c \leq 0$, where uniqueness holds. To show this multiplicity result we first transform $ (P)$ into a semilinear problem having a variational structure. Then we are led to the search of two critical points for a functional whose superquadratic part is indefinite in sign and has a so-called slow growth at infinity. The key point is to show that the Palais-Smale condition holds.

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Additional Information

Louis Jeanjean
Affiliation: Laboratoire de Mathématiques (UMR 6623), Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France

Humberto Ramos Quoirin
Affiliation: Departamento de Matematica y Computación Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

Keywords: Indefinite variational problem, critical growth in the gradient, superlinear term with slow growth, Cerami condition
Received by editor(s): July 15, 2014
Received by editor(s) in revised form: December 25, 2014
Published electronically: May 28, 2015
Additional Notes: The second author was supported by the FONDECYT project 11121567. This work has been carried out in the framework of the project NONLOCAL (ANR-14-CE25-0013) funded by the French National Research Agency (ANR)
Communicated by: Catherine Sulem
Article copyright: © Copyright 2015 American Mathematical Society