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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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Uniform bounds for solutions to elliptic problems on simply connected planar domains
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by Luca Battaglia PDF
Proc. Amer. Math. Soc. 147 (2019), 4289-4299 Request permission

Abstract:

We consider the following elliptic problems on simply connected planar domains: \begin{equation*} \left \{\begin {array}{ll}-\Delta u=\lambda |x|^{2\alpha }K(x)e^u&\text {in }\Omega \\u=0&\text {on }\partial \Omega \end{array}\right . ;\quad \quad \left \{\begin {array}{ll}-\Delta u=|x|^{2\alpha }K(x)u^p&\text {in }\Omega \\u>0&\text {in }\Omega \\u=0&\text {on }\partial \Omega \end{array}\right . \end{equation*} with $\alpha >-1,\lambda >0,p>1,0<K(x)\in C^1\left (\overline \Omega \right )$.

We show that any solution to each problem must satisfy a uniform bound on the mass, which is given, respectively, by $\lambda \int _\Omega |x|^{2\alpha }K(x)e^u\mathrm dx$ and $p\int _\Omega |x|^{2\alpha }K(x)u^{p+1}\mathrm dx$. The same results apply to some systems and more general nonlinearities.

The proofs are based on the Riemann mapping theorem and a Pohožaev-type identity.

References
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Additional Information
  • Luca Battaglia
  • Affiliation: Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy
  • MR Author ID: 1059262
  • Email: lbattaglia@mat.uniroma3.it
  • Received by editor(s): September 15, 2018
  • Received by editor(s) in revised form: November 21, 2018
  • Published electronically: July 9, 2019
  • Communicated by: Joachim Krieger
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4289-4299
  • MSC (2010): Primary 35J61, 35J47, 35B45
  • DOI: https://doi.org/10.1090/proc/14482
  • MathSciNet review: 4002542