Solutions of elliptic problems with nonlinearities of linear growth

Abstract

In this paper, we study existence of nontrivial solutions to the elliptic equationNontrivial solutions are obtained in the case in which the nonlinearities have linear growth. That is, for some c > 0, | f (x, u)| ≤ c|u| for x ∈ and u ∈ R, and −cI m ≤ ∇ 2 u V (x, u) ≤ cI m for x ∈ and u ∈ R m , where I m is the m × m identity matrix. In sharp contrast to the existing results in the literature, we do not make any assumptions at infinity on the asymptotic behaviors of the nonlinearity f and ∇ u V . Mathematics Subject Classification (2000)35J55 · 35J65 · 58E05 Z.

A twist condition and periodic solutions of Hamiltonian systems

“…In spirit the idea was already used in [1,6,28] in the setting of asymptotically linear problems, though technically our constructions of modifications are different from those in [1,6,28]. Our approach follows closely to that in [23] for nonlinear elliptic problems. Due to the strongly indefinite nature of Hamiltonian systems, we need to overcome some more involved issues.…”

Some abstract critical point theorems for self-adjoint operator equations and applications

“…for all x ∈ X, with z ≥ K. The proof is the same as that of Lemma 3.4 in [19]. The only difference is that in [19,Lemma 3.4], it deals with the finite dimensional case, but here we deal with the infinite dimensional case.…”

“…The only difference is that in [19,Lemma 3.4], it deals with the finite dimensional case, but here we deal with the infinite dimensional case. Since where involved only formal computations, all the estimates are still valid, the proof carries over verbatim.…”

“…We can use the abstract critical point Theorem1.2 and Theorem1.3 to deal with the existence and multiplicity problems of solutions of nonlinear elliptic equations as in [19], the periodic solutions of asymptotically linear Hamiltonian systems as in [18] and the Lagrangian boundary value problems of asymptotically linear Hamiltonian systems as in [14,17]. To avoid tedious, in the following, we only show an application of the abstract critical point Theorem 1.2 and Theorem 1.3 to the problem of nonlinear Hamiltonian systems with P -periodic boundary conditions.…”

Homoclinic orbits for first order Hamiltonian systems with some twist conditions

“…In order to prove Theorem 1.4, inspired by [19] and [20], consider a sequence of modified problems −(Jż + L(t)z) = H m (t, z).…”