## 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

### Keywords:

quasilinear; multiple weak solutions

Zbl 0533.47053
Full Text:

### References:

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