A perturbation approach to studying sign-changing solutions of Kirchhoff equations with a general nonlinearity

By employing a nonlocal perturbation approach and the method of invariant sets of descending flow, this manuscript investigates the existence and multiplicity of sign-changing solutions to a class of semilinear Kirchhoff equations in the following form -(a=b integral(R3) vertical bar del u vertical bar(2))Delta u+V(x)u = f(u), x is an element of R-3, where a, b > 0 are constants, V is an element of C(R-3, R), f is an element of C(R, R). The methodology proposed in the current paper is robust, in the sense that, neither the monotonicity condition on f nor the coercivity condition on V is required. Our result improves the study made by Deng et al. (J Funct Anal 269:3500-3527, 2015), in the sense that, in the present paper, the nonlinearities include the power-type case f (u) = vertical bar u vertical bar(p-2)u for p is an element of (2, 4), in which case, it remains open in the existing literature whether there exist infinitely many sign-changing solutions to the problem above. Moreover, energy doubling is established, namely, the energy of sign-changing solutions is strictly larger than two times that of the ground state solutions for small b > 0.

Automated summary

This paper investigates the existence and multiplicity of sign-changing solutions to a class of semilinear Kirchhoff equations using a novel methodology that does not require restrictions on the properties of the nonlinear term and V. The study improves existing research and establishes the conclusion of energy doubling for sign-changing solutions.