Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition
The authors consider the Hamiltonian system \(\dot z = JH_z(t,z)\), \(z\in\mathbb{R}^{2N}\). The Hamiltonian \(H\) is of the form \(H(t,z) = \frac12B(t)z\cdot z + R(t,z)\), where \(B\) and \(R\) are 1-periodic in \(t\) and either 0 is not in the spectrum of \(-J\frac{d}{dt}+B\) or 0 is the left endpoint of a spectral gap (i.e., \(0\in \sigma(-J\frac{d}{dt}+B)\) and \(\sigma(-J\frac{d}{dt}+B)\cap(0,\alpha)=\emptyset\) for some \(\alpha>0\)). The main result asserts that this system has a homoclinic (to 0) solution under a superlinearity condition on \(R_z\) which is weaker than the usual condition of Ambrosetti and Rabinowitz. The proof uses a combination (and adaptation) of the methods which may be found in [\textit{Y.\,H.\thinspace Ding} and \textit{M.\,Willem}, ``Homoclinic orbits of a Hamiltonian system'', Z.\ Angew.\ Math.\ Phys.\ 50, No.\,5, 759--778 (1999; Zbl 0997.37041), \textit{A.\,Szulkin} and \textit{W.\,Zou}, ``Homoclinic orbits for asymptotically linear Hamiltonian systems'', J.~Funct.\ Anal.\ 187, No.\,1, 25--41 (2001; Zbl 0984.37072) and \textit{M.\,Willem} and \textit{W.\,M.\thinspace Zou}, ``On a Schrödinger equation with periodic potential and spectrum point zero'', Indiana Univ.\ Math.\ J.\ 52, No.\,1, 109--132 (2003; Zbl 1030.35068)].
scientific article; zbMATH DE number 5725434