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Landesman-Lazer conditions for strongly nonlinear boundary value problems. (English) Zbl 0706.35051

The authors study equations of the type \[ div(| \nabla u|^{p- 2} \nabla u)+\lambda_ 1| u|^{p-2} u+f(x,u)=g, \] under Dirichlet or Neumann boundary conditions. Here \(\lambda_ 1\) is a smallest “eigenvalue”. They obtain Landesman-Lazer type results by using degree theory and the Borsuk Ulam theorem. Note that one does not have the linear structure of the classical case. A. Anane and J. P. Gossez [Commun. Partial Differ. Equations 15, No.8, 1141-1159 (1990)] have obtained some related results by variational methods.
Reviewer: E.Dancer

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J70 Degenerate elliptic equations
55M25 Degree, winding number
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