Document Type : Research Paper


Department of Basic Sciences, Babol Noushirvani University of Technology, 47148-71167, Babol, Iran.


In this work, we study the existence of non-trivial multiple solutions for a class of quasilinear elliptic systems equipped with concave-convex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and Palais-Smale condition, we prove the existence and multiplicity results of positive solutions.


Main Subjects

[1] G.A. Afrouzi and S.H. Rasouli, A remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system, Nonlin. Anal., 71 (2009), pp. 445-455.

[2] A. Ambrosetti, H. Brezis, and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), pp.519-543.

[3] A. Ambrosetti, J. Garcia-Azorero, and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996), pp. 219-242.

[4] C.O. Alves, D.C. de Morais Filho, and M.A.S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlin. Anal., 42 (2000), pp. 771-787.

[5] G. Azorero and I. Peral, Some results about the existence of a second positive solution in a quasilinear critical problem, Indiana Univ. Math. J., 43 (1994), pp. 941-957.

[6] T. Barstch and M. Willem, On a elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123 (1995), pp. 3555-3561.

[7] P.A. Binding, P. Drabek, and Y.X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electron J. Differ. Eqs, 5 (1997), pp. 1-11.
[8] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), pp. 437-477.

[9] H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), pp. 486-490.

[10] K.J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differ. Eq.s, 193 (2003), pp. 481-499.

[11] K.J. Brown and T.F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function, J. Math. Anal. Appl., 337 (2008), pp. 1326-1336.

[12] P. Han, The effect of the domain topology on the number of positive solutions of elliptic systems involving critical Sobolev exponents, Houston J. Math., 32 (2006), pp. 1241-1257.

[13] T.S. Hsu, Multiple positive solutions for a critical quasilinear elliptic system with concave-convex nonlinearities, Nonlin. Anal., 71 (2009), pp. 2688-2698.

[14] T.S. Hsu, Multiplicity results for P-Laplacian with critical nonlinearity of concave-convex type and sign-changing weight functions, Abs. and Appl Anal. Article ID 652109, 24 pages, 2009.

[15] G. Tarantello, On nonhomogeneous elliptic involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), pp. 281-304.

[16] T.F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlin. Anal., 68 (2008), pp. 1733-1745.

[17] T.F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318 (2006), pp. 253-270.