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Multiple solutions for quasilinear elliptic equations. (English) Zbl 0723.35034
The author uses degree theory for mappings of class \((S)_+\) [see F. E. Browder, Bull. Am. Math. Soc., New Ser. 9, 1-39 (1983; Zbl 0533.47053)] to determine the existence of multiple weak solutions to boundary value problems of the form \(Au-g(u)=0\) in \(\Omega \subset {\mathbb{R}}^ N\), \(u=0\) on \(\partial \Omega\), where \(\Omega\) is a bounded domain with smooth boundary. Many authors have contributed to the theory when \(A=\Delta\). Here A is taken of the form \[ Au:=- \sum^{N}_{i=1}\partial \{a(| \nabla u|^ 2)\partial u/\partial x_ i\}/\partial x_ i\text{ or } Au:=\sum^{N}_{i=1}\partial \{d(u)\partial u/\partial x_ i\}/\partial x_ i \] with additional technical assumptions on \(a(\cdot)\) and \(d(\cdot)\) as well as on \(g(\cdot)\) in the differential equation.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
55M25 Degree, winding number
Citations:
Zbl 0533.47053
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