Continuation and limit behavior for damped quasi-variational systems

In this paper paper we study continuation and limit behavior as r→∞ of solutions of a system of the form (∇G(u′))′ +f(r,u) = Q(r,u,u′), r∈J=[R,∞). Here G is of class C1, strictly convex on Rn with ∇G(0)=0, f(r,u)=∇uF(r,u) for some F of class C1, and Q is continuous with (Q(r,u,p),p)≤0 for any r∈J, u,p∈Rn. A problem treated in this paper is that of finding conditions under which a solution u defined on an interval [R1,R2)⊂J can be extended to the entire [R1,∞). Let H denote the Legendre transform of G, namely H(p)=(∇G(p),p)−G(p). Theorem 3.1 states that extendibility indeed holds if H(p)→∞ as |p|→∞, F(r,u)≥c(r) for some c∈C(J) and Fr(r,u)≤ψ(r){1+|F(r,u)|} for some ψ∈L1loc(J). Theorems 3.2 shows that the same is still true if one allows polynomial growth on u for F and Fr, under an additional growth restriction on Q, while Theorem 3.3 does it by imposing polynomial growth on H. In addition, several results concerning boundedness and limit behavior of globally defined solutions are provided.