On the existence of solutions to a fourth-order quasilinear resonant problem. (English) Zbl 1065.31007

Liu, Shibo; Squassina, Marco

Abstr. Appl. Anal. 7, No. 3, 125-133 (2002).

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.

Reviewer: Youssef Jabri (Oujda)

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

Keywords:

\(p\)-harmonic problem; Morse theory; Local linking; Resonance