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

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- by Louis Jeanjean and Humberto Ramos Quoirin
- Proc. Amer. Math. Soc.
**144**(2016), 575-586 - DOI: https://doi.org/10.1090/proc12724
- Published electronically: May 28, 2015
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## Abstract:

We consider the problem \begin{equation} -\Delta u =c(x)u+\mu |\nabla u|^2 +f(x), \quad u \in H^1_0(\Omega ) \cap L^{\infty }(\Omega ), \tag {(P)} \end{equation} 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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## Bibliographic Information

**Louis Jeanjean**- Affiliation: Laboratoire de Mathématiques (UMR 6623), Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France
- MR Author ID: 318795
- Email: louis.jeanjean@univ-fcomte.fr
**Humberto Ramos Quoirin**- Affiliation: Departamento de Matematica y Computación Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
- MR Author ID: 876954
- Email: humberto.ramos@usach.cl
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 575-586 - MSC (2010): Primary 35J20, 35J61, 35J91
- DOI: https://doi.org/10.1090/proc12724
- MathSciNet review: 3430835