Radially symmetric solutions to a Dirichlet problem involving critical exponents

Abstract:

In this paper we answer, for $N = 3,4$, the question raised in [1] on the number of radially symmetric solutions to the boundary value problem $- \Delta u(x) = \lambda u(x) + u(x)|u(x){|^{4/(N - 2)}}$, $x \in B: = \{ x \in {R^N}:\left \| x \right \| < 1\}$, $u(x) = 0$, $x \in \partial B$, where $\Delta$ is the Laplacean operator and $\lambda > 0$. Indeed, we prove that if $N = 3,4$, then for any $\lambda > 0$ this problem has only finitely many radial solutions. For $N = 3,4,5$ we show that, for each $\lambda > 0$, the set of radially symmetric solutions is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.