Skip to main content Accessibility help

Existence of Multiple Solutions for a p-Laplacian System in ℝ N with Sign-changing Weight Functions

  • Hongxue Song (a1), Caisheng Chen (a1) and Qinglun Yan (a2)


In this paper, we consider the quasi-linear elliptic problem

$$-M\left( {{\int }_{{{\mathbb{R}}^{N}}}}{{\left| x \right|}^{-ap}}{{\left| {{\nabla }_{u}} \right|}^{p}}dx \right)\,\text{div}\left( {{\left| x \right|}^{-ap}}{{\left| \nabla u \right|}^{p-2}}\nabla u \right)=\frac{\alpha }{\alpha +\beta }H\left( x \right){{\left| u \right|}^{\alpha -2}}u{{\left| v \right|}^{\beta }}+\text{ }\lambda \text{ }{{\text{h}}_{1}}\left( x \right){{\left| u \right|}^{q-2}}u,$$

$$-M\left( {{\int }_{{{\mathbb{R}}^{N}}}}{{\left| x \right|}^{-ap}}{{\left| \nabla v \right|}^{p}}dx \right)\,\text{div}\left( {{\left| x \right|}^{-ap}}{{\left| \nabla v \right|}^{p-2}}\nabla v \right)=\frac{\beta }{\alpha +\beta }H\left( x \right){{\left| v \right|}^{\beta -2}}v{{\left| u \right|}^{\alpha }}+\mu {{h}_{2}}\left( x \right){{\left| v \right|}^{q-2}}v,$$

$$u\left( x \right)>0,v\left( x \right)>0,x\in {{\mathbb{R}}^{N}},$$

where $\text{ }\lambda \text{ ,}\mu >\text{0,}\text{1}<\text{p}<\text{N,}\text{1}<\text{q}<\text{p}<\text{p}\left( \tau +1 \right)<\alpha +\beta <{{p}^{*}}=\frac{{{N}_{p}}}{N-p},0\le a<\frac{N-p}{p},a\le b<a+1,d=a+1-b>0,M\left( s \right)=k+l{{s}^{\tau }},k>0,l,\tau \ge 0$ and the weight $H\left( x \right),\,{{h}_{1}}\left( x \right),\,{{h}_{2}}\left( x \right)$ are continuous functions that 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.



Hide All
[1] Ambrosetti, A., Brezis, H., and Cerami, G., Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122(1994), no. 2, 519543.
[2] Binding, P. A., Drabek, P., and Huang, Y. X., On Neumann boundary problems for some quasilinear elliptic equations. Electron. J. Differential Equations 5(1997), no. 05.
[3] Bozhkov, Y. and Mitidieri, E., Existence of multiple solutions for quasilinear systems viafibering method. J. Differential Equations 190(2003), no. 1, 239267.
[4] Brock, E., Iturriaga, L., Sanchez, J. S., and Ubilla, P., Existence of positive solutions for p-Laplacian problems with weights. Commun. Pure and Appl. Anal. 5(2006), no. 4, 941952.
[5] Brown, K. J. and Zhang, Y., The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. J. Differential Equations 193(2003), no. 2, 481499. 6/S0022-0396(03)00121-9
[6] Caffarelli, L., Kohn, R., Nirenberg, L., First order interpolation inequalities with weights. Compositio Math. 53(1984), no. 3, 259275.
[7] Chen, C.-Y., Kuo, Y.-C., Wu, T.-F., TheNehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differential Equations. 250(2011), no. 4, 18761908. 6/j.jde.2O10.11.01 7
[8] Chen, S.-J. and Li, L., Multiple solutions for the nonhomogeneous Kirchhoff equation on HN. Nonlinear Anal. Real World Appl. 14(2013), no. 3, 14771486.
[9] Chipot, M. and Lovat, B., Some remarks on nonlocal elliptic and parabolic problems.Proceedings of the Second World Congress of Nonlinear Analysts, Part 7 (Athens, 1996).Nonlinear Anal. 30(1997),no. 7, 46194627.
[10] Corrêa, F. J. S. A., On positive solutions of nonlocal and nonvariational elliptic problems. Nonlinear Anal. 59(2004), no. 7, 11471155.
[11] de Thélinand, K. J. Vélin, Existence and non-existence of nontrivial solution for some nonlinear elliptic systems. Rev. Mat. Univ. Complutense Madrid 6(1993), no. 1, 153194.
[12] DiBenedetto, E., Degenerate parabolic equations. Universitext, Springer-Verlag, New York, 1993.
[13] Drabek, P. and Pohozaev, S. I., Positive solutions for the p-Laplacian: application of the fibering method. Proc. Roy. Soc. Edinburgh Sect. A 127(1997), no. 4, 703726.
[14] Ekeland, I., On the variationalprinciple. J. Math. Anal. Appl. 47(1974,) 324353.
[15] Kristalyand, A. Varga, C., Multiple solutions for a degenerate elliptic equation involving sublinear terms at infinity. J. Math. Anal. Appl. 352(2009), no. 1, 139148.
[16] Mitidieri, E., Sweers, G., R. van der Vorst, Non-existence theorems for systems of quasilinear partial defferential equations. Differential Integral Equations 8(1995), no. 6, 13311354.
[17] Miyagakiand, O. H. Rodrigues, R. S., On positive solutions for a class of singular quasilinear elliptic systems. J. Math. Anal. Appl. 334(2007), no. 2, 818833.
[18] Nehari, Z., On a class of nonlinear second-order differential equations. Trans. Amer. Math. Soc. 95(1960), 101123.
[19] Ni, W.-M. and Takagi, I., On the shape of least energy solution to a Neumann problem. Comm. Pure Appl. Math. 44(1991), no. 7, 819851.
[20] Rabinowitz, P. H., Minimax methods in criticalpoint theory with applications to different equations. CBMS Regional Conference Series in Mathematics, 65, American Mathematical Society, Providence, RI, 1986.
[21] Wu, T.-F., Onsemilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J. Math. Anal. Appl. 318(2006), no. 1, 253270.
[22] Wu, T.-F., Multiplicity results for a semilinear elliptic equation involving sign-changing weight function. Rocky Mountain J. Math.39(2009), no. 3,995-1011.
[23] Xiu, Z. H., Chen, C. S., and Huang, J. C., Existence of multiple solution for an elliptic system with sign-changing weight functions. J. Math. Anal. Appl. 395(2012), no. 2, 531541.
[24] Xuan, B., The solvability of quasilinearBrezis-Nirenberg-typeproblems with singular weights. Nonlinear Anal. 62(2005), no. 4, 703725.
MathJax is a JavaScript display engine for mathematics. For more information see


Existence of Multiple Solutions for a p-Laplacian System in ℝ N with Sign-changing Weight Functions

  • Hongxue Song (a1), Caisheng Chen (a1) and Qinglun Yan (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed