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.


35J65 Nonlinear boundary value problems for linear elliptic equations
55M25 Degree, winding number


Zbl 0533.47053
Full Text: DOI


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