Nontrivial solutions of mountain pass type of quasilinear equations with slowly growing principal parts. (Q1850209)

The author studies the existence of nontrivial solutions of the quasilinear elliptic boundary value problem \[ -\text{ div}(a(| \nabla u| )\nabla u) = g(x,u)\text{ in } \Omega,\;u=0\text{ on }\partial\Omega,\tag{*} \] where \(\Omega\) is a bounded domain in \({\mathbb R}^{N}\). Here \(\phi(t):=a(t)t\) is increasing, odd and continuous, and the primitive \(\Phi\) of \(\phi\) is assumed to grow very slowly in the sense that \(\Phi(t) = o(t^p)\) as \(t\to\infty\) for any \(p>1\) (note that \(\Phi(t) = | t| ^p\) corresponds to the \(p\)-Laplacian). Then (*) is (formally) the Euler-Lagrange equation for the functional \[ I(u) \equiv J(u) + P(u) = \int_{\Omega} \Phi(| \nabla u| )\,dx - \int_{\Omega} G(x,u)\,dx \] (\(G\) is the primitive of \(g\)). It is shown that under appropriate conditions on \(\Phi\) and \(G\), \(J\) is convex and lower semicontinuous, \(P\) is of class \(C^1\) and \(I\) has the mountain pass geometry in the Orlicz-Sobolev space \(W^1_{0}L_{\Phi}\). For \(I\in C^1\), an appropriate theory in \(W^1_{0}L_{\Phi}\) has been developed in [\textit{P. Clément} et al., Mountain pass type solutions for quasilinear elliptic equations, Calc. Var., Partial Differ. Equ. 11, 33--62 (2000; Zbl 0959.35057)]. However, since here \(J\) may not be of class \(C^1\), the above theory is not applicable. Instead the author reformulates the problem in terms of a variational inequality and modifies a nonsmooth mountain pass theorem in [\textit{A. Szulkin}, Ann. Inst. H. Poincaré, Anal. Non Linéaire 3, 77--109 (1986; Zbl 0612.58011)] in order to obtain a Palais-Smale sequence \((u_{n})\) for \(I\). Passing to a subsequence, \(u_{n}\rightharpoonup u\neq 0\), and this \(u\) is shown to be a solution of (*). In a final section the author discusses a nonlinear eigenvalue problem (with \(\lambda\) in front of \(g\) in (*)).