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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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