A uniform estimate for positive solutions of semilinear elliptic equations

Fusco, G.; Leonetti, F.; Pignotti, C.

doi:10.1090/S0002-9947-2011-05356-0

A uniform estimate for positive solutions of semilinear elliptic equations
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by G. Fusco, F. Leonetti and C. Pignotti

Trans. Amer. Math. Soc. 363 (2011), 4285-4307

DOI: https://doi.org/10.1090/S0002-9947-2011-05356-0

Published electronically: March 22, 2011

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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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