Superlinear quasilinearity and the second eigenvalue (Q5931375)

The paper gives (a long list of) conditions under which the \(m\)th-order superlinear elliptic differential equation in generalized divergence form NEWLINE\[NEWLINE\sum_{1\leq|\alpha|\leq m}(- 1)^{|\alpha|} D^\alpha A_\alpha(x, \xi_m'(u))= \lambda|u|^{q- 2} u+ b(x, u)|u|^{p- 2} u+ g(x,u)NEWLINE\]NEWLINE defined on a bounded open connected subset of \(\mathbb{R}^n\) has multiple nontrivial weak solutions. Here \(\xi_m'(u)\) is the vector of all partial derivatives of \(u\) of orders \(1\) through \(m\) and \(D^\alpha\) is standard multi-index notatio for derivatives. The proof is variational and uses the generalized mountain pass theorem.