半拟线性p-Laplacian问题结点解的存在性

The Existence of Nodal Solutions for the Half-Quasilinear $p$-Laplacian Problems

中文关键词:  半拟线性问题  分岐  结点解  $p$-Laplacian

英文关键词:Bifurcation  Half-Quasilinear problems  Nodal solutions  $p$-Laplacian

中文摘要:

      本文将研究半拟线性 $p$--Laplacian 问题 $-(\varphi_{p}(x'))'=\alpha(t) \varphi_{p}(x^{+})+\beta(t)\varphi_{p}(x^{-}) +ra(t)f(x),\,\,00$成立, 且 $f_{0}, f_{\infty}\not\in (0,\infty)$,其中 $f_{0}=\lim_{|s|\rightarrow0} f(s)/\varphi_{p}(s), f_{\infty}=\lim_{|s|\rightarrow+\infty} f(s)/\varphi_{p}(s)$.我们用分歧技巧与集序列的逼近法来证明主要定理.

英文摘要:

      In this paper, we study the existence of nodal solutions for the following problem: $$\align &-(\varphi_{p}(x'))'=\alpha(t) \varphi_{p}(x^{+})+\beta(t)\varphi_{p}(x^{-}) +ra(t)f(x),\,\,00$ for $s\neq0$, and $f_{0}, f_{\infty}\not\in (0,\infty)$, where $$f_{0}= \lim_{|s|\rightarrow0} f(s)/\varphi_{p}(s),~~f_{\infty}=\lim_{|s|\rightarrow+\infty} f(s)/\varphi_{p}(s).$$ We use bifurcation techniques and the approximation of connected components to prove our main results.