The index of a critical point for nonlinear elliptic operators with strong coefficient growth

Abstract

This paper is devoted to the computation of the index of a critical point for nonlinear operators with strong coefficient growth. These operators are associated with boundary value problems of the type \begin{center} $\displaystyle \sum_{|\alpha|=1}\mathscr{D}^{\alpha}\{\rho^{2}(u)\mathscr{D}^{\alpha}u+a_{\alpha}(x,\mathscr{D}^{1}u)\}=\lambda a_{0}(x,u,\mathscr{D}^{1}u),\ x\in\Omega$, $u(x)=0,\ x\in\partial\Omega$, \end{center} where $\Omega\subset\mathbf{R}^{n}$ is open, bounded and such that $\partial\Omega\in\mathbf{C}^{2}$, while $\rho: \mathbf{R} \to \mathbf{R}_{+}$ can have exponential growth. An index formula is given for such densely defined operators acting from the Sobolev space $W_0^{1,m}(\Omega)$ into its dual space. We consider different sets of assumptions for $m>2$ (the case of a real Banach space) and $m=2$ (the case of a real Hilbert space). The computation of the index is important for various problems concerning nonlinear equations: solvability, estimates for the number of solutions, branching of solutions, etc. The result of this paper are based upon recent results of the authors involving the computation of index of a critical point for densely defined abstract operators of type $(S_{+})$. The latter are based in turn upon a new degree theory for densely defined $(S_{+})$-mappings, which has also been developed by the authors in a recent paper. Applications of the index formula to the relevant bifurcation problems are also included.