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

Abstract:

      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.