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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Multiple solutions for an indefinite elliptic problem with critical growth in the gradient
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by Louis Jeanjean and Humberto Ramos Quoirin PDF
Proc. Amer. Math. Soc. 144 (2016), 575-586 Request permission

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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Additional 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