Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2020-03-26 , DOI: 10.1007/s00033-020-01296-7 Gelson G. dos Santos , Giovany M. Figueiredo , Rúbia G. Nascimento
In this paper, we are concerned with existence and behavior of positive solution to the class of nonlinear elliptic problems with discontinuous nonlinearities of the type
$$\begin{aligned} \left\{ \begin{array}{l} -\Delta u+V(x)u = H(u-\beta )f(x,u) \quad \text {in}\,\, {\mathbb {R}}^{N},\\ u\in D^{1,2}({\mathbb {R}}^{N})\cap W_{\hbox {loc}}^{2,2}({\mathbb {R}}^{N}), \end{array} \right. \qquad {(P)_{\beta }} \end{aligned}$$where \(\beta \ge 0\) is a positive parameter, \(N \ge 3\), \(V:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}\) is a nonnegative potential, which can vanish at infinity, \(f:{\mathbb {R}}^{N}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function with subcritical growth and H is the Heaviside function, i.e.,
$$\begin{aligned} H(t)=\left\{ \begin{array}{ll} 0 \quad \text{ if } \;\;t\le 0,\\ 1 \quad \text{ if }\;\;t>0. \end{array} \right. \end{aligned}$$We use a suitable nonsmooth truncation 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 \(u_{\beta }\) of \((P)_{\beta },\) for all \(\beta >0.\) In addition, we show that \(u_{\beta }\rightarrow u_{0}\) in \(D^{1,2}({\mathbb {R}}^{N})\) as \(\beta \rightarrow 0\), where \(u_{0}\) is a positive solution of \((P)_{0}\).
中文翻译：
通过不光滑惩罚在$$ {\ mathbb {R}} ^ {N} $$ RN中具有不连续非线性问题的正解的存在性和行为
在本文中，我们关注类型为非连续非线性的非线性椭圆问题的正解的存在性和行为
$$ \ begin {aligned} \ left \ {\ begin {array} {l}-\ Delta u + V（x）u = H（u- \ beta）f（x，u）\ quad \ text {in} \，\，{\ mathbb {R}} ^ {N}，\\ u \ in D ^ {1,2}（{\ mathbb {R}} ^ {N}）\ cap W _ {\ hbox {loc} } ^ {2,2}（{\ mathbb {R}} ^ {N}），\ end {array} \ right。\ qquad {（P）_ {\ beta}} \ end {aligned} $$其中\（\ beta \ ge 0 \）是一个正参数，\（N \ ge 3 \），\（V：{\ mathbb {R}} ^ {N} \ rightarrow {\ mathbb {R}} \）是一个非负势，它可以在无穷大处消失，\（f：{\ mathbb {R}} ^ {N} \ times {\ mathbb {R}} \ rightarrow {\ mathbb {R}} \}是Carathéodory函数具有亚临界生长，H是Heaviside函数，即
$$ \ begin {aligned} H（t）= \ left \ {\ begin {array} {ll} 0 \ quad \ text {if} \; \; t \ le 0，\\ 1 \ quad \ text {if } \; \; t> 0。\ end {array} \ right。\ end {aligned} $$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，以获得局部Lipschitz函数对于所有\（\ beta> 0. \）的\（（P）_ {\ beta}，\）的正解\（u _ {\ beta} \）此外，我们证明了\（u _ {\ beta} \ RIGHTARROW U_ {0} \）在\（d ^ {1,2}（{\ mathbb {R}} ^ {N}）\）作为\（\测试\ RIGHTARROW 0 \），其中\（U_ {0 } \）是\（（P）_ {0} \）的正解。
京公网安备 11010802027423号