Multiple periodic solutions for Hamiltonian systems with not coercive potential
The authors consider the existence of \(T\)-periodic solutions for the system of ordinary differential equations \(-u''+A(t)u = \lambda b(t)\nabla G(u)\), where \(A\) is an \(N\times N\) symmetric positive definite matrix with entries \(a_{ij}: [0,T]\to\mathbb{R}\), \(b\) is a nonnegative function on \([0,T]\) (\(b\not\equiv 0\)) and \(\lambda\) is a positive parameter. The first main result asserts that if \(G\) satisfies a certain comparison condition, then, for \(\lambda\) in an explicitly given interval, there exist at least two nontrivial periodic solutions. The second main result asserts the existence of infinitely many periodic solutions provided \(\lambda\) is in an explicitly given interval and \(G(\xi)/|\xi|^2\) has oscillatory behaviour as \(|\xi|\to\infty\) (roughly, \(\liminf_{|\xi|\to\infty}G(\xi)/|\xi|^2 < L\limsup_{|\xi|\to\infty}G(\xi)/|\xi|^2\), \(L\) a suitable constant). The proofs use some abstract critical point theorems due to the first author and his co-authors.
scientific article; zbMATH DE number 5651218