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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.

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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