Elliptic equations with critical growth and Moser’s inequality

Abstract

Let Ω be a bounded domain in R² and let

$$M = \{ u \in H_0^1(\Omega ):\int_\Omega {{{\left| {\nabla u} \right|}^2}} \leqslant 1\} .$$

(1.1)

Then

$$ S\mathop = \limits^{{\text{def}}} \mathop {{\text{sup}}}\limits_M \int_\Omega {{e^{a{u^2}}}} \leqslant c\left| \Omega \right|,$$

(1.2)

where c does not depend on Ω, whenever

$$\alpha \leqslant 4\pi .$$

(1.3)