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Transactions of the American Mathematical Society

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

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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