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Existence of solutions to nonlinear, subcritical higher order elliptic Dirichlet problems. (English) Zbl 1185.35066
Summary: We consider the \(2m\)-th order elliptic boundary value problem \(Lu = f(x,u)\) on a bounded smooth domain \(\Omega \subset \mathbb R^N\) with Dirichlet boundary conditions on \(\partial \Omega \). The operator \(L\) is a uniformly elliptic linear operator of order \(2m\) whose principle part is of the form \((-\sum ^N_{i, j=1}a_{ij}(\chi )\frac{\partial ^2}{\partial \chi _{i}\partial \chi _{j}})^m\). We assume that \(f\) is superlinear at the origin and satisfies \(\lim _{s\to \infty } \frac{f(\chi ,s)}{s^q} = h(\chi ), \lim _{s\to -\infty }\frac{f(\chi ,s)}{|s|^q} = k(\chi )\), where \(h, k \in C(\overline {\Omega })\) are positive functions and \(q > 1\) is subcritical. By combining degree theory with new and recently established a priori estimates, we prove the existence of a nontrivial solution.

MSC:
35J40 Boundary value problems for higher-order elliptic equations
35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35J35 Variational methods for higher-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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