Least Energy Sign-Changing Solution for N-Kirchhoff Problems with Logarithmic and Exponential Nonlinearities

Abstract

In this paper, we are concerned with the existence of least energy sign-changing solutions for the following N-Laplacian Kirchhoff-type problem with logarithmic and exponential nonlinearities:

$$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b \int _{\Omega }|\nabla u|^{N} d x\right) \Delta _{N} u=|u|^{p-2} u \ln |u|^{2}+\lambda f(u), &{} \text{ in } \Omega , \\ u=0, &{} \text{ on } \partial \Omega , \end{array}\right. \end{aligned}$$

where f(t) behaves like \(\ exp\left( {\alpha |t|^{{\frac{N}{{N - 1}}}} } \right) \). Combining constrained variational method, topological degree theory and quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution \(u_{b}\) with precisely two nodal domains. Moreover, we show that the energy of \(u_{b}\) is strictly larger than two times of the ground state energy and analyze the convergence property of \(u_{b}\) as \(b\searrow 0\).