Geodesic
Existence of Multiple Solutions for a p-Laplacian System in RN with Sign-changing Weight Functions
Canadian mathematical bulletin, Tome 59 (2016) no. 2, pp. 417-434

Voir la notice de l'article provenant de la source Cambridge University Press

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 $\text{ }\lambda \text{ ,}\mu >\text{0,}\text{1}<\text{p}<\text{N,}\text{1}<\text{q}<\text{p}<\text{p}\left( \tau +1 \right)<\alpha +\beta <{{p}^{*}}=\frac{{{N}_{p}}}{N-p},0\le a<\frac{N-p}{p},a\le b0,M\left( s \right)=k+l{{s}^{\tau }},k>0,l,\tau \ge 0$ 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.
DOI : 10.4153/CMB-2015-035-4
Mots-clés : 35J66, Nehari manifold, quasilinear elliptic system, p-Laplacian operator, concave and convex nonlinearities 
Song, Hongxue; Chen, Caisheng; Yan, Qinglun. Existence of Multiple Solutions for a p-Laplacian System in RN with Sign-changing Weight Functions. Canadian mathematical bulletin, Tome 59 (2016) no. 2, pp. 417-434. doi: 10.4153/CMB-2015-035-4
@article{10_4153_CMB_2015_035_4,
     author = {Song, Hongxue and Chen, Caisheng and Yan, Qinglun},
     title = {Existence of {Multiple} {Solutions} for a {p-Laplacian} {System} in {RN} with {Sign-changing} {Weight} {Functions}},
     journal = {Canadian mathematical bulletin},
     pages = {417--434},
     year = {2016},
     volume = {59},
     number = {2},
     doi = {10.4153/CMB-2015-035-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2015-035-4/}
}
TY  - JOUR
AU  - Song, Hongxue
AU  - Chen, Caisheng
AU  - Yan, Qinglun
TI  - Existence of Multiple Solutions for a p-Laplacian System in RN with Sign-changing Weight Functions
JO  - Canadian mathematical bulletin
PY  - 2016
SP  - 417
EP  - 434
VL  - 59
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2015-035-4/
DO  - 10.4153/CMB-2015-035-4
ID  - 10_4153_CMB_2015_035_4
ER  - 
%0 Journal Article
%A Song, Hongxue
%A Chen, Caisheng
%A Yan, Qinglun
%T Existence of Multiple Solutions for a p-Laplacian System in RN with Sign-changing Weight Functions
%J Canadian mathematical bulletin
%D 2016
%P 417-434
%V 59
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2015-035-4/
%R 10.4153/CMB-2015-035-4
%F 10_4153_CMB_2015_035_4

[1] [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, 519–543. Google Scholar | DOI

[2] [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. Google Scholar

[3] [3] Bozhkov, Y. and Mitidieri, E., Existence of multiple solutions for quasilinear systems viafibering method. J. Differential Equations 190(2003), no. 1, 239–267. Google Scholar | DOI

[4] [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, 941–952. Google Scholar | DOI

[5] [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, 481–499. Google Scholar | DOI

[6] [6] Caffarelli, L., Kohn, R., Nirenberg, L., First order interpolation inequalities with weights. Compositio Math. 53(1984), no. 3, 259–275. Google Scholar

[7] [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, 1876–1908. Google Scholar | DOI

[8] [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, 1477–1486. Google Scholar | DOI

[9] [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, 4619–4627. Google Scholar | DOI

[10] [10] Corrêa, F. J. S. A., On positive solutions of nonlocal and nonvariational elliptic problems. Nonlinear Anal. 59(2004), no. 7, 1147–1155. Google Scholar | DOI

[11] [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, 153–194. Google Scholar

[12] [12] DiBenedetto, E., Degenerate parabolic equations. Universitext, Springer-Verlag, New York, 1993. Google Scholar | DOI

[13] [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, 703–726. Google Scholar | DOI

[14] [14] Ekeland, I., On the variationalprinciple. J. Math. Anal. Appl. 47(1974,) 324–353. Google Scholar | DOI

[15] [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, 139–148. Google Scholar | DOI

[16] [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, 1331–1354. Google Scholar

[17] [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, 818–833. Google Scholar | DOI

[18] [18] Nehari, Z., On a class of nonlinear second-order differential equations. Trans. Amer. Math. Soc. 95(1960), 101–123. Google Scholar | DOI

[19] [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, 819–851. Google Scholar | DOI

[20] [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. Google Scholar | DOI

[21] [21] Wu, T.-F., Onsemilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J. Math. Anal. Appl. 318(2006), no. 1, 253–270. Google Scholar | DOI

[22] [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. Google Scholar | DOI

[23] [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, 531–541. Google Scholar | DOI

[24] [24] Xuan, B., The solvability of quasilinearBrezis-Nirenberg-typeproblems with singular weights. Nonlinear Anal. 62(2005), no. 4, 703–725. Google Scholar | DOI

Cité par Sources :