Some nonlinear elliptic equations involving nonlinearities of critical growth

Abstract

We consider the following Neumann boundary value problem: ?Anu+??u?n?2u=f(x,u)?u?n?2in ?- A_n u + \lambda |u|^{n-2}u = f(x, u) |u|^{n-2} \quad \text{in } \Omega?An?u+??u?n?2u=f(x,u)?u?n?2in ? in a bounded domain ??Rn\Omega \subset \mathbb{R}^n??Rn with smooth boundary. Here, Anu=div(??u?n?2?u)A_n u = \text{div}(|\nabla u|^{n-2} \nabla u)An?u=div(??u?n?2?u) is the n-Laplacian, ?>0\lambda > 0?>0, and f(x,t)f(x, t)f(x,t) is a C1C^1C1-function of critical growth. We prove the existence of a nonnegative solution. Under some additional conditions on ?\Omega?, we show that the problem admits more than one solution. In the special case when ?\Omega? is a ball, we show that there are in fact infinitely many solutions. We also consider a semilinear Dirichlet problem in ?\Omega? for the Laplacian with nonlinearity of critical growth. A solution to this problem is known to exist under certain conditions which relate the size of the domain with the growth of the lower-order term in the nonlinearity. In this thesis, we show that the above condition is not optimal by producing a counterexample. Finally, we consider a problem of critical growth for the n-Laplacian in the whole of Rn\mathbb{R}^nRn and prove the existence of a solution.