Existence of Multiple Solutions for a $p$-Laplacian System in $extbf{R}^{N}$ with Sign-changing Weight Functions

Canadian mathematical bulletin

Qinglun Yan¹  Caisheng Chen²  Hongxue Song¹ 

[1] College of Science, Nanjing University of Posts and Telecommunications, , Nanjing 210023, P. R. China;College of Science, Hohai University, Nanjing 210098, P. R. China

关键词: Nehari manifold;    quasilinear elliptic system;    $p$-Laplacian operator;    concave and convex nonlinearities;   

DOI  :  10.4153/CMB-2015-035-4

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

In this paper, we consider the quasi-linear ellipticproblem [ left{egin{aligned}&-M left(int_{mathbb{R}^{N}}|x|^{-ap}|abla u|^{p}dxight){mdiv} left(|x|^{-ap}|abla u|^{p-2}abla uight) \&qquad=frac{alpha}{alpha+eta}H(x)|u|^{alpha-2}u|v|^{eta}+lambdah_{1}(x)|u|^{q-2}u, \&-M left(int_{mathbb{R}^{N}}|x|^{-ap}|abla v|^{p}dx ight){mdiv} left(|x|^{-ap}|abla v|^{p-2}abla vight) \&qquad=frac{eta}{alpha+eta}H(x)|v|^{eta-2}v|u|^{alpha}+muh_{2}(x)|v|^{q-2}v, \&u(x)gt 0,quad v(x)gt 0, quad xin mathbb{R}^{N}end{aligned} ight.]where $lambda, mugt 0$, $1lt plt N$,$1lt qlt plt p(au+1)lt alpha+etalt p^{*}=frac{Np}{N-p}$, $0leqalt frac{N-p}{p}$, $aleq blt a+1$, $d=a+1-bgt 0$, $M(s)=k+l s^{au}$,$kgt 0$, $l, augeq0$ and the weight $H(x), h_{1}(x), h_{2}(x)$arecontinuous functions which change sign in $mathbb{R}^{N}$. Wewill prove that the problem has at least two positive solutionsbyusing the Nehari manifold and the fibering maps associated withthe Euler functional for this problem.

【 授权许可】

Unknown