Mónica Clapp: Elliptic boundary value problems with critical and supercritical nonlinearities

We consider the problem

-\Delta u=\left\vert u\right\vert ^{p-2}u\text{ \ in }\Omega,\text{\qquad }u=0\text{ \ on }\partial\Omega,

where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq3,$ and $p\geq2^{\ast}$ with $2^{\ast}:=\frac{2N}{N-2}$, the critical Sobolev exponent.

The existence of solutions to this problem depends on the domain. It is well known that it does not have a nontrivial solution if $\Omega$ is strictly starshaped and that it has infinitely many solutions if $\Omega$ is an annulus. A remarkable existence result was obtained by Bahri and Coron for $p=2^{\ast}:$ they showed that, if the homology of $\Omega$ is nontrivial, the problem has a positive solution. However, this condition is not enough to guarantee existence in the supercritical case: for $p\geq\frac{2(N-1)}{N-3}$ Passaseo exhibited domains carrying one nontrivial homology class in which no nontrivial solution exists.

We shall give examples of domains whose homology becomes richer as $p$ increases, in which no nontrivial solution exists. We shall also present some new multiplicity results for this problem, both in the critical and the supercritical case. This is joint work with Jorge Faya (Universidad Nacional Aut\'{o}noma de M\'{e}xico) and Angela Pistoia (Universit\`{a} di Roma "La Sapienza").