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Existence results for perturbations of the \(p\)-Laplacian. (English) Zbl 0818.35029

The existence of solutions to the problem \[ - \Delta_ pu \equiv - \text{div} \bigl( | \nabla u |^{p-2} \nabla u \bigr) = f(x,u)\;\text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega \] is studied. Here \(\Omega\) is a bounded domain in \(\mathbb{R}^ N\), \(f\) is a Carathéodory function with a subcritical growth. A class of abstract functionals \(I(u) = J(u) - N(u)\) in a real Banach space is considered, where \(J\) is \(p\)-homogeneous and the perturbation \(N\) is not \(p\)- homogeneous at infinity in a certain sense. It is proved that a Palais- Smale type condition is fulfilled and therefore minimax techniques can be applied to the proof of the existence of critical points of such functionals. Assumptions about the potential \(F\) of \(f\) are given under which the boundary value problem mentioned can be considered in this general setting. Particularly, Ambrosetti-Rabinowitz’ value mountain-pass theorem is applied under a condition on “crossing of the first eigenvalue \(\lambda_ p\) of \(- \Delta_ p\)”. In a situation when \(F\) interacts with \(\lambda_ p\), Rabinowitz’ saddle point theorem is used.
Reviewer: M.Kučera (Praha)

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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References:

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