Abstract
Under some local superquadratic conditions on $W(t,u)$ with respect to u, the existence of infinitely many homoclinic solutions is obtained for the nonperiodic second order Hamiltonian systems $\ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0$ , $\forall t\in\mathbb{R}$ , where $L(t)$ is unnecessarily coercive.
Highlights
Introduction and main resultsLet us consider the second order Hamiltonian systems u (t) – L(t)u + ∇W t, u(t) =, ∀t ∈ R, ( ) where L ∈ C(R, RN )is a symmetric matrix-valued function and ∇W (t, x) = ∂ ∂x W
L does not satisfy the coercive condition (A ) and W is superquadratic near the origin, Theorem is different from Theorem . in [ ]
As far as the authors know, there is little research concerning the multiplicity of homoclinic solutions for problem ( ) simultaneously under local conditions and noncoercive conditions, so our result is different from the previous results in the literature
Summary
L does not satisfy the coercive condition (A ) and W is superquadratic near the origin, Theorem is different from Theorem . in [ ]. L does not satisfy the coercive condition (A ) and W is superquadratic near the origin, Theorem is different from Theorem . As far as the authors know, there is little research concerning the multiplicity of homoclinic solutions for problem ( ) simultaneously under local conditions and noncoercive conditions, so our result is different from the previous results in the literature. The proof is motivated by the argument in [ ]. We will modify and extend W to an appropriate W and show for the associated modified functional I the existence of a sequence of homoclinic solutions converging to zero in L∞ norm, and we obtain infinitely many homoclinic solutions for the original problem. We introduce the Sobolev space that we study. Let A be the self-adjoint extension of the operator
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