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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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