The Existence of Nodal Solutions for the Half-Quasilinear $p$-Laplacian Problems
Received:January 05, 2016  Revised:November 23, 2016
Key Word: Bifurcation   Half-Quasilinear problems   Nodal solutions   $p$-Laplacian
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11561038) and the Natural Science Foundation of Gausu Province (Grant No.145RJZA087).
 Author Name Affiliation Wenguo SHEN Department of Basic Courses, Lanzhou Institute of Technology, Gansu 730050, P. R. China
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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), wheref_{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.