GROUND STATE SIGN–CHANGING SOLUTIONS FOR FRACTIONAL KIRCHHOFF TYPE EQUATIONS IN <inline-formula><tex-math id="M1">$ \mathbb{R}^{3} $</tex-math></inline-formula>

Abstract

<abstract> In this paper, we investigate the existence of ground state sign–changing solutions for the following fractional Kirchhoff equation <p class="disp_formula">$ \left(a+b\int_{\mathbb{R}^{3}}|\left(-\triangle\right)^{\frac{\alpha}{2}}u|^{2}\mathrm{d}x\right) (-\triangle)^{\alpha}u+V(x)u = K(x)f(u) \rm{ \ \ in }\mathbb{R}^{3}, $ where $ \alpha\in (0, 1) $, $ a, b $ are positive parameters, $ V(x), \; K(x) $ are nonnegative continuous functions and $ f $ is a continuous function with quasicritical growth. By establishing a new inequality, we prove the above system possesses a ground state sign–changing solutions $ u_{b} $ with precisely two nodal domains, and its energy is strictly larger than twice that of the ground state solutions of Nehari–type. Moreover, we obtain the convergence property of $ u_{b} $ as the parameter $ b\rightarrow0 $. Our conditions weaken the usual increasing condition on $ f(t)/|t|^{3} $. </abstract>