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

  Published:2016-03-18
 Printed: Jun 2016
  • Hongxue Song,
    College of Science, Nanjing University of Posts and Telecommunications, , Nanjing 210023, P. R. China
  • Caisheng Chen,
    College of Science, Hohai University, Nanjing 210098, P. R. China
  • Qinglun Yan,
    College of Science, Nanjing University of Posts and Telecommunications, , Nanjing 210023, P. R. China
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Abstract

In this paper, we consider the quasi-linear elliptic problem \[ \left\{ \begin{aligned} & -M \left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla u|^{p}dx \right){\rm div} \left(|x|^{-ap}|\nabla u|^{p-2}\nabla u \right) \\ & \qquad=\frac{\alpha}{\alpha+\beta}H(x)|u|^{\alpha-2}u|v|^{\beta}+\lambda h_{1}(x)|u|^{q-2}u, \\ & -M \left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla v|^{p}dx \right){\rm div} \left(|x|^{-ap}|\nabla v|^{p-2}\nabla v \right) \\ & \qquad=\frac{\beta}{\alpha+\beta}H(x)|v|^{\beta-2}v|u|^{\alpha}+\mu h_{2}(x)|v|^{q-2}v, \\ &u(x)\gt 0,\quad v(x)\gt 0, \quad x\in \mathbb{R}^{N} \end{aligned} \right. \] where $\lambda, \mu\gt 0$, $1\lt p\lt N$, $1\lt q\lt p\lt p(\tau+1)\lt \alpha+\beta\lt p^{*}=\frac{Np}{N-p}$, $0\leq a\lt \frac{N-p}{p}$, $a\leq b\lt a+1$, $d=a+1-b\gt 0$, $M(s)=k+l s^{\tau}$, $k\gt 0$, $l, \tau\geq0$ and the weight $H(x), h_{1}(x), h_{2}(x)$ are continuous functions which change sign in $\mathbb{R}^{N}$. We will prove that the problem has at least two positive solutions by using the Nehari manifold and the fibering maps associated with the Euler functional for this problem.
Keywords: Nehari manifold, quasilinear elliptic system, $p$-Laplacian operator, concave and convex nonlinearities Nehari manifold, quasilinear elliptic system, $p$-Laplacian operator, concave and convex nonlinearities
MSC Classifications: 35J66 show english descriptions Nonlinear boundary value problems for nonlinear elliptic equations 35J66 - Nonlinear boundary value problems for nonlinear elliptic equations
 

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