## A uniform estimate for positive solutions of semilinear elliptic equations

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- by G. Fusco, F. Leonetti and C. Pignotti PDF
- Trans. Amer. Math. Soc.
**363**(2011), 4285-4307 Request permission

## 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
- MR Author ID: 70195
**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
- Received by editor(s): November 2, 2009
- Published electronically: March 22, 2011
- © Copyright 2011 American Mathematical Society
- 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
- MathSciNet review: 2792988