Complex Analysis and Operator Theory ( IF 0.8 ) Pub Date : 2024-03-08 , DOI: 10.1007/s11785-024-01495-4 Ting Huang , Yan-Ying Shang
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 \(b\searrow 0\).
中文翻译：
具有对数和指数非线性的 N-基尔霍夫问题的最小能量变号解
在本文中，我们关注以下具有对数和指数非线性的N -拉普拉斯基尔霍夫型问题的最小能量符号变化解的存在性：
$$\begin{对齐} \left\{ \begin{array}{ll} -\left( a+b \int _{\Omega }|\nabla u|^{N} dx\right) \Delta _{ N} u=|u|^{p-2} u \ln |u|^{2}+\lambda f(u), &{} \text{ in } \Omega , \\ u=0, &{ } \text{ 上 } \partial \Omega 、 \end{array}\right。\end{对齐}$$其中f ( t ) 的行为类似于\(\ exp\left( {\alpha |t|^{{\frac{N}{{N - 1}}}} } \right) \)。结合约束变分法、拓扑度理论和定量变形引理，证明该问题具有一个具有精确两个节点域的最小能量变号解\(u_{b}\) 。此外，我们证明\(u_{b}\)的能量严格大于基态能量的两倍，并将\(u_{b}\)的收敛性分析为\(b\searrow 0\)。
京公网安备 11010802027423号