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Transactions of the American Mathematical Society

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A uniform estimate for positive solutions of semilinear elliptic equations


Authors: G. Fusco, F. Leonetti and C. Pignotti
Journal: Trans. Amer. Math. Soc. 363 (2011), 4285-4307
MSC (2010): Primary 35J61, 35B09
DOI: https://doi.org/10.1090/S0002-9947-2011-05356-0
Published electronically: March 22, 2011
MathSciNet review: 2792988
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Abstract: We consider the semilinear elliptic equation $ \Delta u=W'(u)$ with Dirichlet boundary condition in a Lipschitz, possibly unbounded, domain $ \Omega\subset\mathbb{R}^n.$ Under suitable assumptions on the potential $ W$, we deduce a condition on the size of the domain that implies the existence of a positive solution satisfying a uniform pointwise estimate. Here uniform means that the estimate is independent of $ \Omega$.

Under some geometric restrictions on the domain, we extend the analysis to the case of mixed Dirichlet-Neumann boundary conditions.

As an application of our estimate we give a proof of the existence of potentials such that, independent of the choice of $ \Omega$ and of the value of $ \lambda>0$, the equation $ \Delta u=\lambda W'(u)$ has infinitely many positive solutions.


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Additional Information

G. Fusco
Affiliation: Dipartimento di Matematica Pura e Applicata, Università degli Studi di L’Aquila, Via Vetoio, Loc. Coppito, 67010 L’Aquila Italy

F. Leonetti
Affiliation: Dipartimento di Matematica Pura e Applicata, Università degli Studi di L’Aquila, Via Vetoio, Loc. Coppito, 67010 L’Aquila Italy

C. Pignotti
Affiliation: Dipartimento di Matematica Pura e Applicata, Università degli Studi di L’Aquila, Via Vetoio, Loc. Coppito, 67010 L’Aquila Italy

DOI: https://doi.org/10.1090/S0002-9947-2011-05356-0
Keywords: Semilinear elliptic equations, positive solutions
Received by editor(s): November 2, 2009
Published electronically: March 22, 2011
Article copyright: © Copyright 2011 American Mathematical Society

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