Title: Nonlinear spectral analysis and the Fredholm alternative for the p-Laplacian
Speaker: Peter Takac, Institut fur Mathematik, Universitat Rostock, Germany
Seminar Type: Analysis and Operator Theory
Abstract: We briefly describe some basic mathematical challenges in the (nonlinear) spectral theory for the \(p\)-Laplace operator. It has been a long-standing open problem if all eigenvalues of the \(p\)-Laplace operator mapping the Sobolev space \(W_0^{1,p}(\Omega)\) into its dual space \(W^{-1,p'}(\Omega)\), \(\frac{1}{p} + \frac{1}{p'} = 1\), \(1 < p < \infty\), are variational in some reasonable sense, e.g., given by the Ljusternik-Schnirelmann formula. For a closely related nonlinear operator, we will provide a counterexample, although for the genuine \(p\)-Laplacian this problem still remains open.
The first eigenvalue \(\lambda_1\) and the corresponding eigenfunction \(\varphi_1\) enjoy analogous properties as in the (classical) linear case \(p=2\). We are concerned with the existence of a weak solution \(u\in W_0^{1,p}(\Omega)\) to the degenerate quasilinear Dirichlet boundary value problem \( \begin{equation*} \tag*{\rm (P)} - \Delta_p u = \lambda |u|^{p-2} u + f(x) \;\mbox{ in } \Omega ;\qquad u = 0 \;\mbox{ on } \partial\Omega . \end{equation*} \)
It is assumed that \(1 0\) small enough). More precisely, we obtain at least three distinct solutions if either \(p<2\) and \(\lambda_1 - \delta < \lambda < \lambda_1\), or else \(p>2\) and \(\lambda_1 < \lambda < \lambda_1 + \delta\). Naturally, the (linear selfadjoint) Fredholm alternative for the linearization of problem (P) about \(\varphi_1\) (with \(\lambda = \lambda_1\)) appears in the proofs.
