Ground state sign-changing solutions for semilinear Dirichlet problems

In the present paper, we consider the existence of ground state sign-changing solutions for the semilinear Dirichlet problem 0.1 $$ \left \{ \textstyle\begin{array}{l@{\quad}l} -\triangle u+\lambda u=f(x, u), & \hbox{$x\in\Omega$;} \\ u=0, & \hbox{$x\in\partial\Omega$,} \end{array}\displaystyle \right . $$ where $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with a smooth boundary ∂Ω, $\lambda>-\lambda_{1}$ is a constant, $\lambda_{1}$ is the first eigenvalue of $(-\triangle, H_{0}^{1}(\Omega))$ , and $f\in C(\Omega\times\mathbb{R}, \mathbb{R})$ . Under some standard growth assumptions on f and a weak version of Nehari type monotonicity condition that the function $t\mapsto f(x, t)/|t|$ is non-decreasing on $(-\infty, 0)\cup(0, \infty)$ for every $x\in\Omega$ , we prove that (0.1) possesses one ground state sign-changing solution, which has precisely two nodal domains. Our results improve and generalize some existing ones.