Least energy sign-changing solutions for a class of fractional Kirchhoff–Poisson system

Abstract

Using the constraint variational method and a quantitative deformation lemma, we establish the existence of the least energy sign-changing solutions for the fractional Kirchhoff–Poisson system, a+b∫R3|(−Δ)s2u|2dx(−Δ)su+V(x)u+ϕ(x)u=f(x,u),(−Δ)tϕ=u2,x∈R3, where a > 0 is a constant, b∈R+ is a parameter, s, t ∈ (0, 1) and 4s + 2t > 3, (−Δ)s stands for the fractional Laplacian, V is a continuous, positive function, and f is nonlinear function satisfying suitable growth assumptions. Moreover, for any b > 0, we prove that the energy of the least energy sign-changing solution is strictly larger than twice the ground state energy. Furthermore, we show a convergence property of the least energy sign-changing solutions as the parameter b goes to zero. Our results complement an in-depth study of Wang, Radulescu, and Zhang [J. Math. Phys. 60, 011506 (2019)] in the sense that we are concerned with the nodal characteristics of the ground states.