Energy bounds for entire nodal solutions of autonomous superlinear equations

Abstract

We prove new lower bounds for the energy of sign changing solutions of two different types of autonomous superlinear equations on R N . These bounds increase the energy range for which Palais-Smale sequences of related variational problems have a nontrivial weak limit. The results are obtained by transforming sign changing solutions of low energy into a convenient form and then applying different variants of the moving plane method.

Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains

“…Therefore, (1.16) implies that, the energy of any sign-changing solution of Eq. (1.7) is larger than two times the least energy, this property is called energy doubling by Weth in [21]. However, if b > 0 in (1.1), the property (1.16) is still unknown for the functional I b .…”

Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems inR3

“…The property (1.10) is called energy doubling in [20] for k = 1. However, the properties (1.9) and (1.10) are still unknown for the functional I b with b > 0.…”

Least energy sign-changing solutions for a class of nonlocal Kirchhoff-type problems

“…He obtained the energy of any sign-changing solutions is larger than that of the ground state solutions of (1), and claimed whether the energy of any sign-changing solutions is larger than twice that of the ground state solutions of (1) or not was unknown. In the present paper, we will give an affirmative answer that (1) has the property of the energy of any sign-changing solutions is larger than twice that of the ground state solutions of (1), which is called energy doubling property by Weth (2006). Precisely, we establish the second main result as follows.…”