Folds and cusps in Banach spaces with applications to nonlinear partial differential equations. II

Berger, M. S.; Church, P. T.; Timourian, J. G.

doi:10.1090/S0002-9947-1988-0936814-8

Folds and cusps in Banach spaces with applications to nonlinear partial differential equations. II
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by M. S. Berger, P. T. Church and J. G. Timourian PDF

Trans. Amer. Math. Soc. 307 (1988), 225-244 Request permission

Abstract:

Earlier the authors have given abstract properties characterizing the fold and cusp maps on Banach spaces, and these results are applied here to the study of specific nonlinear elliptic boundary value problems. Functional analysis methods are used, specifically, weak solutions in Sobolev spaces. One problem studied is the inhomogeneous nonlinear Dirichlet problem \[ \Delta u + \lambda u - {u^3} = g\quad {\text {on}}\;\Omega ,\qquad u|\partial \Omega = 0,\] where $\Omega \subset {{\mathbf {R}}^n}(n \leqslant 4)$ is a bounded domain. Another is a nonlinear elliptic system, the von Kármán equations for the buckling of a thin planar elastic plate when compressive forces are applied to its edge.

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