Journal of Fixed Point Theory and Applications ( IF 1.8 ) Pub Date : 2021-03-08 , DOI: 10.1007/s11784-021-00858-0 José Carlos de Albuquerque , Gelson G. dos Santos , Giovany M. Figueiredo
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.
中文翻译:
一类具有不连续非线性的$$ {\ mathbb {R}} ^ N $$ RN中的线性耦合系统正解的存在性和行为
在本文中,我们关注具有不连续非线性的下一类线性耦合椭圆系统的正解的存在性和行为
$$ \ begin {aligned} \ left \ {\ begin {array} {ll}-\ Delta u + V_ {1}(x)u = H(u- \ beta)f_ {1}(u)+ a( x)v,&{} \ text {in} {\ mathbb {R}} ^ {N},\\-\ Delta v + V_ {2}(x)v = H(v- \ beta)f_ {2 }(v)+ a(x)u,&{} \ text {in} {\ mathbb {R}} ^ {N},\\ u,v \ in D ^ {1,2}({\ mathbb { R}} ^ {N})\ cap W_ \ mathrm {loc} ^ {2,2}({\ mathbb {R}} ^ {N}),\ end {array} \ right。\ quad {(S)_ {\ beta}} \ end {aligned} $$其中\(\ beta \ ge 0 \),\(N \ ge 3 \),\(V_ {1},V_ {2},\) \(a:{\ mathbb {R}} ^ {N} \ rightarrow {\ mathbb {R}} \)是正电位,可以在无限远处消失,\(f_ {1},f_ {2}:{\ mathbb {R}} \ rightarrow {\ mathbb {R}} \)是连续函数,而H是Heaviside函数,即,如果((t> 0 \),如果\(t \ le 0,\) \(H(t)= 1 \),则\(H(t)= 0 \)。对于系统,我们使用适当的非平滑截断方法来应用Del Pino和Felmer的惩罚方法(Calc Var Partial Differ Equ 4:121–137,1996)与Mountain Pass定理相结合,以实现局部Lipschitz函数以得到正解\((U _ {\测试},V _ {\测试})\)的\((S)_ {\测试} \)在多值的意义。此外,我们表明,\((U _ {\测试},V _ {\测试})\ RIGHTARROW(U,V)\)在\(d ^ {1,2}({\ mathbb {R}} ^ { N})\ D ^ {1,2}({\ mathbb {R}} ^ {N})\)作为\(\ beta \ rightarrow 0 ^ {+} \),其中(u, v)是一个强烈意义上的连续系统\((S)_ {0} \)的正解。