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

**Series:** 2006-07, Preprints

- MSC:
- 35J65 Nonlinear boundary value problems for linear elliptic equations

**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

**This paper was published in:**

Nonl. Differ. Equ. Appl. NoDEA 15, 69-90 (2008).