Nonlinear Analysis problems
A talk in the SPDEvent series by
Abdellahi Soumare from AIMS Senegal
| Abstract: | We consider a nonlinear variational elliptic problem with critical nonlinearity on a bounded domain of $\mathbb{R}^n, n \geq 3$ and mixed Dirichlet-Neumann boundary conditions. $-\Delta u = u^{\frac{n+2}{n-2}}$ $u > 0 \quad \text{ dans } \Omega,$ $u = 0 \quad \text{ sur } \Gamma_0,$ $\frac{\partial u}{\partial \nu} = 0 \quad \text{ sur }\Gamma_1.$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain with smooth boundary $\partial \Omega = \Gamma_0 \cup \Gamma_1$ and $\partial \nu$ denotes the derivation with respect to the outward unit normal $\nu$ on $\Gamma_1$. We study the effect of the domain's topology on the existence of solutions as Bahri-Coron did in their famous work on the homogeneous Dirichlet problem. However, due to the influence of the part of the boundary where the Neumann condition is prescribed, the blow-up picture in the present setting is more complicated and makes the mixed boundary problems different with respect to the homogeneous ones. Such complexity imposes modification of the argument of Bahri-Coron and demands new constructions and extra ideas. |