Existence of nontrivial solutions for some elliptic equations with supercritical nonlinearity in exterior domains

The existence of positive solutions is discussed for some nonlinear elliptic equations involving the nonlinear terms with the growth order of super-critical exponents in exterior domains of balls such as $ -\Delta u = u^\beta $ in $\Omega$, ($(N+2)/(N-2) < \beta $), $u = 0 $ on $\partial B$, with $\Omega = \mathbb{R}^N \setminus\overline\Omega_0$ where $\Omega_0$ is the open ball. To recover the compactness of the embedding $L^{\beta+1}(\Omega) \subset H^1_0(\Omega)$, we work in the class of radially symmetric functions and introduce a new transformation, which reduces our problems to some nonlinear elliptic equations in annuli but with coefficients which have some singularity on the boundary. The difficulty caused by the singularity on the boundary will be managed by the arguments developed in our previous work.