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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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