Multiplicity results for asymptotically homogeneous semilinear boundary value problems (Q1085336)

This paper treats nonlinear elliptic boundary value problems of the form \[ \Delta u+f(x,u)=0\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega \] in the space \(L^ 2(\Omega)\) by degree theoretic methods. Emphasis is placed on existence of multiple solutions in the case, where the nonlinearity f crosses several eigenvalues of the corresponding eigenvalue problem \(\Delta \theta +\lambda \theta =0\) with zero boundary values. No differentiability conditions (but Lipschitz type conditions) on f are assumed. A main tool is a new a priori bound for solutions (Theorem 1). The method is not confined to the selfadjoint case. It applies also to some time-periodic parabolic and hyperbolic problems.