The second bifurcation branch for radial solutions of the Brezis-Nirenberg problem in dimension four

Abstract

Existence results available for the semilinear Brezis-Nirenberg eigenvalue problem suggest that the compactness problems for the corresponding action functionals are more serious in small dimensions. In space dimension n = 3, one can even prove nonexistence of positive solutions in a certain range of the eigenvalue parameter. In the present paper we study a nonexistence phenomenon manifesting such compactness problems also in dimension n = 4. We consider the equation -Δu = λu + u 3 in the unit ball of ℝ4 under Dirichlet boundary conditions. We study the bifurcation branch arising from the second radial eigenvalue of -Δ. It is known that it tends asymptotically to the first eigenvalue as the L ∞-norm of the solution tends to blow up. Contrary to what happens in space dimension n = 5, we show that it does not cross the first eigenvalue. In particular, the mentioned Dirichlet problem in n = 4 does not admit a nontrivial radial solution when λ coincides with the first eigenvalue. © 2008 Birkhaueser.