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Proceedings of the American Mathematical Society

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Uniform bounds for solutions to elliptic problems on simply connected planar domains


Author: Luca Battaglia
Journal: Proc. Amer. Math. Soc. 147 (2019), 4289-4299
MSC (2010): Primary 35J61, 35J47, 35B45
DOI: https://doi.org/10.1090/proc/14482
Published electronically: July 9, 2019
MathSciNet review: 4002542
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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.


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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

Keywords: Liouville equation, Lane-Emden equation, PDEs on simply connected domains, a priori estimates
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
Article copyright: © Copyright 2019 American Mathematical Society