Existence and multiplicity of positive solutions for a class of semilinear elliptic system with nonlinear boundary conditions
الموضوعات :F. M. Yaghoobi 1 , J. Shamshiri 2
1 - Department of Mathemetics, College of Science, Hamedan Branch, Islamic Azad University, Hamedan, Iran
2 - Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran
الکلمات المفتاحية: Critical point, Semilinear elliptic system, Nonlinear boundary value problem, Fibering map, Nehari manifold,
ملخص المقالة :
This study concerns the existence and multiplicity of positive weak solutions for a class ofsemilinear elliptic systems with nonlinear boundary conditions. Our results is depending onthe local minimization method on the Nehari manifold and some variational techniques. Also,by using Mountain Pass Lemma, we establish the existence of at least one solution withpositive energy.
[1] A. Aghajani and J. Shamshiri and F. M. Yaghoobi, Existence and multiplicity of positive solutions for a class of nonlinear elliptic problems, Turk. J. Math (2012) doi:10.3906/mat-1107-23.
[2] A. Aghajani and J. Shamshiri, Multilicity of positive solutions for quasilinear elliptic p-Laplacian systems, E. J. D. E. 111 (2012) 1-16.
[3] A. Aghajani and F.M. Yaghoobi and J. Shamshiri, Multiplicity of positive solutions for a class of quasilinear elliptic p-Laplacian problems with nonlinear boundary conditions, Journal of Information and computing Science 8 (2013) 173-182.
[4] Y. Bozhkov and E. Mitidieri, Existence of multiple solutions for quasilinear systems via fibering method, Journal of Differential Equations 190 (2003) 239-267.
[5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010.
[6] K. J. Brown and T.-F.Wu, A semilinear elliptic system involving nonlinear boundary condition and signchanging weight function, J. Math.Anal. Appl. 337 (2008) 1326-1336.
[7] K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic problem with a sign changing weight function, J. Differential Equations 193 (2003) 481-499.
[8] C. M. Chu and C. L. Tang, Existence and multiplicity of positive solutions for semilinear elliptic systems with Sobolev critical exponents, Nonlinear Anal 71 (2009) 5118-5130.
[9] P. Drabek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proc. Royal Soc. Edinburgh Sect. A 127 (1997) 703-726.
[10] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974) 324-353.
[11] H. Fan, Multiple positive solutions for a critical elliptic system with concave and convex nonlinearities, Nonlinear Anal. Real World Appl. 18 (2014) 1422.
[12] M. F. Furtado a and J. P. P. da Silva, Multiplicity of solutions for homogeneous elliptic systems with critical growth, J.Math. Anal. Appl. 385 (2012) 770-785.
[13] T. -s. Hsu, Multiple positive solutions for a quasilinear elliptic system involving concave-convex nonlinearities and sign changing weight functions, Internat. J. Math. Math. Sci. (2012) doi:10.1155 (2012) 109-214.
[14] Feng-Yun Lu, The Nehari manifold and application to a semilinear elliptic system, Nonlinear Analysis 71 (2009) 3425-3433
[15] Y. Shen and J. Zhang, Multiplicity of positive solutions for a semilinear p-Laplacian system with Sobolev critical exponent, Nonlinear Analysis 74 (2011) 1019-1030.
[16] M. Struwe, Variational methods, Springer, Berlin, 1990.
[17] T. F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlin. Analysis 68 (6) (2008), 1733-1745.
