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Existence of positive solutions for some polyharmonic nonlinear boundary-value problems. (English) Zbl 1055.35039

Summary: We present existence results for the polyharmonic nonlinear elliptic boundary-value problem \[ \begin{aligned} (-\Delta )^m u=\quad & f(\cdot,u) \quad \text{in }B \\ (\frac{\partial }{\partial \nu })^j u=\quad & 0 \quad \text{on }\partial B, \quad 0\leq j\leq m-1. \end{aligned} \] (in the sense of distributions), where \(B\) is the unit ball in \(\mathbb R^n\) and \(n\geq 2\). The nonlinearity \(f(x,t)\) satisfies appropriate conditions related to a Kato class of functions \(K_{m,n}\). Our approach is based on estimates for the polyharmonic Green function with zero Dirichlet boundary conditions and on the Schauder fixed point theorem.

MSC:

35J40 Boundary value problems for higher-order elliptic equations
35J60 Nonlinear elliptic equations
47N20 Applications of operator theory to differential and integral equations
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