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

by Arioli, Gianni; Gazzola, Filippo; Grunau, Hans-Christoph; Sassone, Edoardo


Preprint series: 06-07, Preprints

The paper is published: Nonl. Differ. Equ. Appl. NoDEA 15, 69-90 (2008).

35J65 Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE


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 $-\Delta u=\lambda u+u^3$ in the unit ball of $\R^4$ under Dirichlet boundary conditions. We study the bifurcation branch arising from the second radial eigenvalue of $-\Delta$. It is known that it tends asymptotically to the first eigenvalue as the $L^\infty$-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 $\lambda$ coincides with the first eigenvalue.

Keywords: Brezis-Nirenberg problem, resonant case, dimension four, nonexistence, radial solution

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