Existence Theorems for Nonselfadjoint Semilinear Elliptic Boundary Value Problems

where E is a real elliptic linear differential operator in a bounded domain G of R” with a given system of linear homogeneous conditions, say, BX = 0 on the boundary aG of G and where N is a Nemitsky type nonnecessarily linear operator. We shall make use here of the alternative method, and particularly we shall make use for the elliptic case of new remarks. These remarks suggest that both the auxiliary and bifurcation equations can be analyzed under different topologies, and by a more specific construction of the operator S: Y0 -+ X0. Actually some of these remarks have been already used implicitly in previous papers on the semilinear wave equation in R2 (Cesari and Kannan [5], Cesari and Pucci [6]). For selfadjoint elliptic problems, Landesman and Lazer [9] proved, also by the alternative method, a remarkable theorem which was then extended by Williams [14] by the same method, and by others by different arguments. Later, Shaw [12] proved, again by the alternative method, that Landesman’s and Lazer’s theorem extends even to nonselfadjoint problems with equal Fredholm indices and whose eigenfunctions share regions of positivity and negativity with their corresponding adjoint eigenfunctions. In the present paper we definitely aim at elliptic problems which are not necessarily selfadjoint and do not necessarily satisfy Shaw’s requirements. The sufficient conditions we obtain are more quantitative in character and concern the cases Nx = f(t) + g(t, D”x) and NX = f(f) + g(t, x(t)), t E G. However, as we show by examples for the case Nx = f(t) + g(t, x(t)), our sufficient conditions for existence allow a great freedom on g, on which no monotonicity is required.