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Nonsmooth critical point theory and quasilinear elliptic equations. (English) Zbl 0851.35038
Granas, Andrzej (ed.) et al., Topological methods in differential equations and inclusions. Proceedings of the NATO Advanced Study Institute and séminaire de mathématiques supérieures on topological methods in differential equations and inclusions, Montréal, Canada, July 11-22, 1994. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 472, 1-50 (1995).
The authors develop variational methods for the study of quasilinear elliptic equations of the type \[ \partial_j(a_{ij} \partial_i u)+ \textstyle{{1\over 2}} a_{ij}' \partial_i u\partial_j u= g(x, u)\tag{1} \] in a bounded domain \(\Omega\subset \mathbb{R}^n\). Here the coefficients \(a_{ij}(x, u)\) depend also on the unknown function \(u\) and \(a_{ij}'\) denotes differentiation with respect to the second variable. It is assumed that \(a_{ij}\) satisfy certain ellipticity conditions and are bounded. Equation (1), supplemented with a homogeneous Dirichlet boundary condition, is the Euler equation of the functional \[ F(u)= \textstyle{{1\over 2}} \displaystyle{\int_\Omega} a_{ij}(x, u) \partial_i u(x) \partial_j u(x)- G(x, u)dx\quad \text{in} \quad H^1_0(\Omega),\tag{2} \] where \(G\) is the primitive function of \(g\). This functional is continuous, but in general not differentiable (even not locally Lipschitz continuous). In the first part of the paper, the authors introduce a notion of generalized derivative, \(|dF|(u)\), of such functionals on metric spaces which they call the weak slope of \(F\) at \(u\). A critical point of \(F\) is then by definition a point \(u_0\) such that \(|dF|(u_0)= 0\). They show that classical tools from critical point theory, such as Palais-Smale condition, deformation lemma and its equivariant form, and Mountain Pass type theorems can be generalized to these functionals. Also Lyusternik-Schnirelman theory is considered and general results for the number of critical points are proved by means of the relative category.
As an application of this nonsmooth critical point theory, the authors prove in the second part of the paper the following multiplicity result for (1): If \(a_{ij}(x, u)\) are even in \(u\) and \(g(x, u)\) is superlinear, subcritical and odd in \(u\), then there exists a sequence \(u_k\) of weak solutions of (1), such that \(F(u_k)\to \infty\) as \(k\to \infty\). In addition, the solutions belong to \(L^\infty(\Omega)\).
For the entire collection see [Zbl 0829.00024].

35J65 Nonlinear boundary value problems for linear elliptic equations
49J52 Nonsmooth analysis
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49J35 Existence of solutions for minimax problems