Superlinear equations and a uniform anti-maximum principle for the multi-Laplacian operator. (English) Zbl 1109.35353

Massa, Eugenio

Electron. J. Differ. Equ. 2004, Paper No. 97, 19 p. (2004).

Summary: 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 Fučík spectrum, and a uniform anti-maximum principle for the considered operator.

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35B50	Maximum principles in context of PDEs
35J50	Variational methods for elliptic systems
47J30	Variational methods involving nonlinear operators
49J35	Existence of solutions for minimax problems

Keywords:

Higher order elliptic boundary value problem; superlinear equation; mountain pass theorem; anti-maximum principle