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.
Submitted May 25, 2004. Published August 7, 2004.
Math Subject Classifications: 35G30, 49J35.
Key Words: Higher order elliptic boundary value problem; superlinear equation; mountain pass theorem; anti-maximum principle.
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| Eugenio Massa |
Dip. di Matematica, Universita degli Studi
Via Saldini 50
20133 Milano, Italy
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