The homogeneous Dirichlet problem for non-elliptic partial differential equations with strong nonlinearities. (English) Zbl 0702.35061

The existence of weak solutions to certain nonelliptic partial differential equations is shown. The use of anisotropic Sobolev spaces makes it possible to apply methods of the elliptic theory. The functions, generating the Nemytskii operators are not necessarily of polynomial growth. The theory covers, for example, the following differential equations: \[ u_{yy}-u_{xxxx}-(g(u_ x))_ x=f, \]
\[ \partial_{xxxx}(u_{xxxx}+u_{xxyy}+u_{yyyy})+g(u)=f\quad or\quad u_{xxyy}+g(u)=f. \] Here g: \({\mathbb{R}}\to {\mathbb{R}}\) is a function that satisfies g(t)\(\cdot t\geq 0\) for all \(t\in {\mathbb{R}}.\)
The paper is connected with the author’s previous article [Math. Methods Appl. Sci. 9, 493-519 (1987; Zbl 0638.35024)]. The references contain 35 items.
Reviewer: O.John


35G30 Boundary value problems for nonlinear higher-order PDEs


Zbl 0638.35024
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