Existence and Multiplicity of Nodal Solutions for Nonlinear Elliptic Equations with Critical Sobolev Growth

Let Ω be a smooth bounded domain of Rn, n ≥ 3, and let a(x) and f(x) be two smooth functions defined on a neighbourhood of Ω. First we study the existence of nodal solutions for the equation Δu + a(x)u = f(x) |u|4/(n − 2)u on Ω, u = 0 on ∂Ω. In particular, when Ω is a solid torus of R3, we describe three infinity (of pairs) of solutions of the equation Δu = |u|4/(n − 2)u on Ω, u = 0 on ∂Ω. Afterwards, we study the equation when the zero Dirichlet condition on the boundary is replaced by a non-zero Dirichlet condition.