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Multiplicity of solutions for quasilinear elliptic equations. (English) Zbl 0863.35038
The quasilinear elliptic variational equation $-\sum^n_{i,j=1}{\partial\over\partial x_j}\Biggl(a_{ij}(x,u){\partial u\over\partial x_i}\Biggr)+{1\over 2}\sum^n_{i,j=1}{\partial a_{ij}\over\partial u}(x,u){\partial u\over\partial x_i}{\partial u\over\partial x_j}=g(x,u)\quad\text{in }\Omega,\;u|_{\partial\Omega}=0$ is shown to have infinitely many distinct weak solutions, if one assumes among other things: $$\Omega\subset\mathbb{R}^n$$ is a bounded domain, the principal part is uniformly elliptic, even in $$u$$, and satisfies some sign condition with respect to $$u$$. The nonlinear lower order term is odd with respect to $$u$$, superlinear and grows subcritically: $$|g(x,u)|\leq a+b|u|^p$$ with $$1<p<(n+2)/(n-2)$$; $$(n\geq 3)$$. The semilinear case $$a_{ij}=\delta_{ij}$$ is a direct consequence of critical point theory, see e.g. M. Struwe [Variational Methods, 1st edition, Theorem 6.6 (1990; Zbl 0746.49010)], while in the present paper, the functional which has to be considered is not differentiable in $$H^1_0(\Omega)$$. This difficulty is overcome by a nonsmooth critical point theory, for which the author refers to the paper [J. N. Corvellec, M. Degiovanni and M. Morzocchi, Topol. Methods Nonlinear Anal. 1, No. 1, 151-171 (1993; Zbl 0789.58021)].

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 49J52 Nonsmooth analysis
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