   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
MathSciNet review: 4002542
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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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