Existence of solutions for strongly degenerate differential systems (Q1364185)

This paper is concerned with the existence of solutions of the following strongly degenerate problem \[ (g(t){\mathcal L}_v(u,u'))'= g(t)[{\mathcal L}_u(u,u')+ Q(t,u,u')],\;u:[T,\infty)\to \mathbb{R}^N\backslash \{0\},\;N\geq 1,\;\lim_{t\to \infty} u(t)=0, \] where \({\mathcal L}(u,v)= G(u,v)- F(u)\) with \(G(u,\cdot)\) strictly convex in \(\mathbb{R}^N\) for \(u\neq 0\), \(g\) is a non-decreasing function, \(F\) represents a restoring potential, and \(Q\) is a damping term. In applications, \(Q\) can have the form \(Q(t,u,v)= -\sigma(t)|u|^\alpha|v|^{\beta-1}v\), where \(\sigma\) is a nonnegative function, which can oscillate between zero and infinity, \(\alpha\) is a (possibly negative) real exponent, and \(\beta>0\). The cases when \(\alpha<0\) and when \(G(0,v)= 0\) have not been previously treated, and their discussion is a principal purpose of this paper. The approach is based on the construction of two Lyapunov functions, an idea suggested by Pucci and Serrin. Systems of this type arise naturally from degenerate elliptic systems, with or without a gradient term, as well as from degenerate parabolic equations (e.g. the porous media equation).