Infinitely many nodal solutions of Kirchhoff-type equations with asymptotically cubic nonlinearity without oddness hypothesis

Abstract

In this paper, we consider the existence and asymptotic behavior of infinitely many nodal solutions of Kirchhoff-type equations with an asymptotically cubic nonlinear term without oddness assumptions. Combining variational methods and convex analysis techniques, we show, for any positive integer k, the existence of a radial nodal solution that changes sign exactly k times. Meanwhile, we prove that the energy of such solution is an increasing function of k. Moreover, the asymptotic behavior of these solutions are also studied upon varying the parameters. By using different analytical approaches, the question of the existence of infinite solutions to some elliptic nonlinear equations is addressed without invoking oddness assumptions. At the same time, we propose a method to overcome the difficulties caused by the complicated competition between the nonlocal term and the asymptotically cubic nonlinearity.