Ground state sign-changing solutions for semilinear Dirichlet problems

Abstract

Abstract In the present paper, we consider the existence of ground state sign-changing solutions for the semilinear Dirichlet problem 0.1 {−△u+λu=f(x,u),x∈Ω;u=0,x∈∂Ω, $$ \left \{ \textstyle\begin{array}{l@{\quad}l} -\triangle u+\lambda u=f(x, u), & \hbox{$x\in\Omega$;} \\ u=0, & \hbox{$x\in\partial\Omega$,} \end{array}\displaystyle \right . $$ where Ω⊂RN $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with a smooth boundary ∂Ω, λ>−λ1 $\lambda>-\lambda_{1}$ is a constant, λ1 $\lambda_{1}$ is the first eigenvalue of (−△,H01(Ω)) $(-\triangle, H_{0}^{1}(\Omega))$, and f∈C(Ω×R,R) $f\in C(\Omega\times\mathbb{R}, \mathbb{R})$. Under some standard growth assumptions on f and a weak version of Nehari type monotonicity condition that the function t↦f(x,t)/|t| $t\mapsto f(x, t)/|t|$ is non-decreasing on (−∞,0)∪(0,∞) $(-\infty, 0)\cup(0, \infty)$ for every x∈Ω $x\in\Omega$, 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.

Keywords

• Semilinear Dirichlet problem
• Ground state sign-changing solutions
• Perturbation method