In this paper, we study the existence and behavior of positive solutions of the following quasilinear elliptic problems with discontinuous nonlinearities:

where \(\delta ,~\kappa >0\), \(N\ge 3\), \(V:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}\) is a nonnegative continuous function, which can vanish at infinity, that is, \(V(x)\rightarrow 0\) as \(|x|\rightarrow \infty \), \(f:{\mathbb {R}}^{N}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function and H is the Heaviside function. Via a suitable nonsmooth truncation, we apply the penalization method combined with the Mountain Pass Theorem for locally Lipschitz functional to obtain a positive solution \(u_{\delta }\) of \((P_{\delta })\) for all \(\delta >0\). Besides, we establish the convergent behavior of positive solution sequence \(\{u_{\delta }\}\), that is, \(u_{\delta }\rightarrow u_{0}\) in \(D^{1,2}({\mathbb {R}}^{N})\) as \(\delta \rightarrow 0^{+}\), where \(u_{0}\) is a positive solution of \((P_{0})\).

在本文中，我们研究了以下具有不连续非线性的拟线性椭圆问题的正解的存在性和行为：

其中\(\delta ,~\kappa >0\) , \(N\ge 3\) , \(V:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}\)是一个非负连续函数，可以在无穷远处消失，即\(V(x)\rightarrow 0\)为\(|x|\rightarrow \infty \)，\(f:{\mathbb {R}}^ {N}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\)是 Carathéodory 函数，H是 Heaviside 函数。经由合适的非光滑截断，我们应用惩罚方法与山路引理治疗局部李普希茨功能，得到正解\（U _ {\增量} \）的\（（P _ {\增量}）\）对于所有\ (\delta >0\). 此外，我们建立了正解序列\(\{u_{\delta }\}\)的收敛行为，即\(u_{\delta }\rightarrow u_{0}\)在\(D^{1 ,2}({\mathbb {R}}^{N})\)为\(\delta \rightarrow 0^{+}\)，其中\(u_{0}\)是\(( P_{0})\)。