Quasilinearity below the 1st eigenvalue (Q2718955)

The author establishes the existence of two nontrivial weak solutions for the \(2m\)th-order quasilinear Dirichlet problem NEWLINE\[NEWLINE\sum_{|\alpha|\leq m}(- 1)^{|\alpha|} D^\alpha A_\alpha(x, \xi_m(u))=|u|^{q- 2}u+\lambda|u|^{p- 2} u,NEWLINE\]NEWLINE where \(\xi_m= \{\xi_\alpha:\mid |\alpha\leq m|\}\) is a certain vector space, \(\lambda\) is a real number strictly less than the first eigenvalue in \(W^{m,p}_0(\Omega)\) and \(1< p< q\) where \(q\) satisfies certain Sobolev restrictions.