Multiplicity results for asymptotically homogeneous semilinear boundary value problems

Abstract

This paper treats nonlinear elliptic boundary value problems of the form $$\Delta u + f(x,u) = 0in\Omega ,u = 0on\partial \Omega $$ in the spaceL2(Ω) by degree theoretic methods. Emphasis is placed on existence of multiplesolutions in the case, where the nonlinearity f crosses several eigenvalues of the correspondingeigenvalue problem Δθ+λθ = 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 forsolutions (Theorem 1). The method is not confined to the selfadjoint case. It applies alsoto some time-periodic parabolic and hyperbolic problems.