A Variational Approach to the Fredholm Alternative for the p-Laplacian near the First Eigenvalue

Abstract

We are concerned with the existence of a weak solution \(u \in W_0^{1,p}(\Omega)\) to the degenerate quasi-linear Dirichlet boundary value problem $$- \Delta_p u = \lambda |u|^{p-2} u + f(x) \quad \hbox{in}\ \Omega ;\qquad u = 0 \quad \hbox{on}\, \partial \Omega.\qquad\quad \hbox{(P)}$$ It is assumed that 1 0 small enough). Moreover, we obtain at least three distinct solutions if either p 2 and λ1 < λ ≤ λ1 + δ. The proofs use a minimax principle for the corresponding energy functional performed in the orthogonal decomposition \(W_0^{1,p}(\Omega) = {\rm lin} \{\varphi_1\} \oplus W_0^{1,p}(\Omega)^\perp\) induced by the inner product in L2(Ω). First, the global minimum is taken over \(W_0^{1,p}(\Omega)^\perp\), and then either a local minimum or a local maximum over lin {φ1}. If the latter is a local minimum, the local minimizer in \(W_0^{1,p}(\Omega)\) thus obtained provides a solution to problem (P). On the other hand, if it is a local maximum, one gets only a pair of sub- and supersolutions to problem (P), which is then used to obtain a solution by a topological degree argument.