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

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.