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On the existence of solutions to a fourth-order quasilinear resonant problem. (English) Zbl 1065.31007
The authors consider the \(p\)-harmonic problem \(\Delta(| \Delta u|^{p-2}\Delta u)=g(x,u)\) in a smooth bounded domain \(\Omega\subset {\mathbb R}^ n\) with \(n\geq 2p+1\), for \(p>1\) with Navier Boundary conditions \(u= \Delta u=0\) on \(\partial \Omega\).
They suppose first a resonance condition around the origin and a “superlinearity” condition of mountain pass type: there exists \(\theta>p\) and \(M>0\) such that \(\big(| s| \geq M\) implies \(0<\theta G(x,s) < s(gx,s) \big)\). Using Morse theory and exploiting the analogy with the \(p\)-Laplacian, they prove the existence of a nontrivial solution. In the case the superlinearity condition is replaced by a nonresonance condition at \(+\infty\), the problem is shown to admit two nontrivial solutions in addition to the trivial one.

MSC:
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
35G30 Boundary value problems for nonlinear higher-order PDEs
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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