Existence and behavior of positive solutions for a class of linearly coupled systems with discontinuous nonlinearities in ${\mathbb {R}}^N$

Abstract

In this paper we are concerned with existence and behavior of positive solutions to the following class of linearly coupled elliptic systems with discontinuous nonlinearities

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V_{1}(x)u = H(u-\beta )f_{1}(u)+ a(x)v, &{} \text {in } {\mathbb {R}}^{N},\\ -\Delta v+V_{2}(x)v = H(v-\beta )f_{2}(v)+ a(x)u, &{} \text {in } {\mathbb {R}}^{N},\\ u,v\in D^{1,2}({\mathbb {R}}^{N})\cap W_\mathrm{loc}^{2,2}({\mathbb {R}}^{N}), \end{array} \right. \quad {(S)_{\beta }} \end{aligned}$$

where $\beta \ge 0$, $N \ge 3$, $V_{1},V_{2},$ $a:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}$ are positive potentials, which can vanish at infinity, $f_{1},f_{2}:{\mathbb {R}}\rightarrow {\mathbb {R}}$ are continuous functions and H is the Heaviside function, i.e, $H(t)=0$ if $t\le 0,$ $H(t)=1$ if $t>0$. We use a suitable nonsmooth truncation, for systems, to apply a version of the penalization method of Del Pino and Felmer (Calc Var Partial Differ Equ 4:121–137, 1996) combined with the Mountain Pass Theorem for locally Lipschitz functional to obtain a positive solution $(u_{\beta },v_{\beta })$ of $(S)_{\beta }$ in multivalued sense. In addition, we show that $(u_{\beta },v_{\beta })\rightarrow (u,v)$ in $D^{1,2}({\mathbb {R}}^{N})\times D^{1,2}({\mathbb {R}}^{N})$ as $\beta \rightarrow 0^{+}$, where (u, v) is a positive solution of the continuous system $(S)_{0}$ in strong sense.