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

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
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