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Ground state sign--changing solutions for fractional Kirchhoff type equations in $\mathbb{R}^{3}$
Guofeng Che,Haibo Chen
Keywords:Fractional Kirchhoff equations; Ground state energy sign--changing solutions; Non--Nehari manifold method; Variational methods
      In this paper, we investigate the existence of ground state sign--changing solutions for the following fractional Kirchhoff equations $$(a+b\int_{\mathbb{R}^{3}}|(-\triangle)^{\frac{\alpha}{2}}u|^{2}\mathrm{d}x) (-\triangle)^{\alpha}u+V(x)u=K(x)f(u) \mbox{ \ \ in }\mathbb{R}^{3},$$ 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 $f(t)/|t|^{3}$.