Superlinear equations and a uniform anti-maximum principle for the multi-Laplacian operator

Abstract

In the first part of this paper, we study a nonlinear equation with the multi-Laplacian operator, where the nonlinearity intersects all but the first eigenvalue. It is proved that under certain conditions, involving in particular a relation between the spatial dimension and the order of the problem, this equation is solvable for arbitrary forcing terms. The proof uses a generalized Mountain Pass theorem. In the second part, we analyze the relationship between the validity of the above result, the first nontrivial curve of the Fucik spectrum, and a uniform anti-maximum principle for the considered operator.