首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 共查询到10条相似文献,搜索用时 140 毫秒
1.
2.
This paper is dedicated to studying the following Schrödinger–Poisson system Δ u + V ( x ) u K ( x ) ϕ | u | 3 u = a ( x ) f ( u ) , x 3 , Δ ϕ = K ( x ) | u | 5 , x 3 . Under some different assumptions on functions V(x), K(x), a(x) and f(u), by using the variational approach, we establish the existence of positive ground state solutions.  相似文献   

3.
We prove continuity and Harnack's inequality for bounded solutions to elliptic equations of the type div | u | p 2 u + a ( x ) | u | q 2 u = 0 , a ( x ) 0 , | a ( x ) a ( y ) | A | x y | α μ ( | x y | ) , x y , div | u | p 2 u 1 + ln ( 1 + b ( x ) | u | ) = 0 , b ( x ) 0 , | b ( x ) b ( y ) | B | x y | μ ( | x y | ) , x y , div | u | p 2 u + c ( x ) | u | q 2 u 1 + ln ( 1 + | u | ) β = 0 , c ( x ) 0 , β 0 , | c ( x ) c ( y ) | C | x y | q p μ ( | x y | ) , x y , $$\begin{eqnarray*} \hspace*{13pc}&&{\rm div}{\left(|\nabla u|^{p-2}\,\nabla u+a(x)|\nabla u|^{q-2}\,\nabla u\right)}=0, \quad a(x)\ge 0,\\ &&\quad |a(x)-a(y)|\le A|x-y|^{\alpha }\mu (|x-y|), \quad x\ne y, \\ &&{\rm div}{\left(|\nabla u|^{p-2}\,\nabla u {\left[1+\ln (1+b(x)\, |\nabla u|) \right]} \right)}=0, \quad b(x)\ge 0, \\ &&\quad |b(x)-b(y)|\le B|x-y|\,\mu (|x-y|),\quad x\ne y,\\ &&{\rm div}{\left(|\nabla u|^{p-2}\,\nabla u+ c(x)|\nabla u|^{q-2}\,\nabla u {\left[1+\ln (1+|\nabla u|) \right]}^{\beta } \right)}=0,\\ &&c(x)\ge 0, \, \beta \ge 0, |c(x)-c(y)|\le C|x-y|^{q-p}\,\mu (|x-y|), \quad x\ne y, \end{eqnarray*}$$ under the precise choice of μ.   相似文献   

4.
The following kind of Klein–Gordon–Maxwell system is investigated Δ u + V ( x ) u ( 2 ω + ϕ ) ϕ u = K ( x ) f ( u ) , in R 3 , Δ ϕ = ( ω + ϕ ) u 2 , in R 3 , $$\begin{equation*} \hspace*{4pc}{\left\lbrace \begin{aligned} &{-\Delta u+ V(x) u-(2\omega +\phi ) \phi u=K(x)f(u)}, & & {\quad \text{ in } \mathbb {R}^{3}}, \\ &{\Delta \phi =(\omega +\phi ) u^{2}}, & & {\quad \text{ in } \mathbb {R}^{3}}, \end{aligned}\right.} \end{equation*}$$ where ω > 0 $\omega >0$ is a parameter, and V is vanishing potential. By using some suitable conditions on K and f, we obtain a Palais–Smale sequence by using Pohožaev equality and prove the ground-state solution for this system by employing variational methods. Our result improves the related one in the literature.  相似文献   

5.
6.
7.
8.
9.
10.
In this work, we study the existence of positive solutions for the following class of semipositone quasilinear problems: Δ Φ u = λ f ( x , u ) + b ( u ) a in Ω , u > 0 in Ω , u = 0 on Ω , $$\begin{equation*} {\left\lbrace \def\eqcellsep{&}\begin{array}{rclcl}-\Delta _{\Phi } u & = & \lambda f(x,u)+b(u)-a & \mbox{in} & \Omega , \\[3pt] u& > & 0 & \mbox{in} & \Omega , \\[3pt] u & = & 0 & \mbox{on} & \partial \Omega , \end{array} \right.} \end{equation*}$$ where Ω R N $\Omega \subset \mathbb {R}^N$ is a bounded domain, N 2 $N\ge 2$ , λ , a > 0 $\lambda ,a > 0$ are parameters, f ( x , u ) $ f(x,u)$ is a Caractheodory function, and b ( t ) $b(t)$ has a critical growth with relation to the Orlicz–Sobolev space W 0 1 , Φ ( Ω ) $W_0^{1,\Phi }(\Omega )$ . The main tools used are variational methods, a concentration compactness theorem for Orlicz–Sobolev space and some priori estimates.  相似文献   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号