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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
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Abstract: We consider the following elliptic problems on simply connected planar domains:

$\displaystyle \left \{\begin {array}{ll}-\Delta u=\lambda \vert x\vert^{2\alpha... ... \\ u>0&\text {in }\Omega \\ u=0&\text {on }\partial \Omega \end{array}\right .$    

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 \vert x\vert^{2\alpha }K(x)e^u\mathrm dx$ and
$ p\int _\Omega \vert x\vert^{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
Email: lbattaglia@mat.uniroma3.it

DOI: https://doi.org/10.1090/proc/14482
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