A variational approach to quasilinear elliptic systems with critical Hardy-Sobolev and sign-changing function exponents
محورهای موضوعی : Calculus of variations and optimal control; optimization
1 - Department of Mathematics, Shahid Beheshti Higher Education Center of Tehran, Tehran, Iran|Education System of Shahriar, Shahriar, Iran
کلید واژه: Nehari manifold, Multiple positive solutions, critical Hardy-Sobolev exponent, sign-changing function exponent,
چکیده مقاله :
The main aim of the present work is to review and study a variational method in existence and multiplicity of positive solutions for quasilinear elliptic systems with critical Hardy-Sobolev and sign-changing function exponents.
[1] C. O. Alves, D. C. De Morais Filho, M. A. S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal. 42 (2000), 771-787.
[2] P. A. Binding, P. Drabek, Y. X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electron. J. Differential Equations. 5 (1997), 1-11.
[3] H. Brèzis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486-490.
[4] K. J. Brown, The Nehari manifold for a semilinear elliptic equation involving a sublinear term, Calc. Var. 22 (2005), 483-494.
[5] K. J. Brown, T. F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign changing weight function, J. Math. Anal. Appl. 337 (2008), 1326-1336.
[6] K. J. Brown, T. F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl. 318 (2006), 253-270.
[7] K. J. Brown, Y. Zhang, The Nehari manifold for a semilinear elliptic problem with a sign changing weight function, J. Differential Equations. 193 (2003), 481-499.
[8] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weights, Compos. Math. 53 (1984), 259-275.
[9] P. Drabek, S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A. 127 (1997), 721-747.
[10] T. Horiuchi, Best constant in weighted Sobolev inequality with weights being powers of distance from origin, J. Inequal. Appl. 1 (1997), 275-292.
[11] T. S. Hsu, Multiple positive solutions for a critical quasilinear elliptic system with concave-convex nonlinearities, Nonlinear Anal. 71 (2009), 2688-2698.
[12] D. Kang, Nontrivial solutions to semilinear elliptic problems involving two critical Hardy-Sobolev exponents, Nonlinear Anal. 72 (2010), 4230-4243.
[13] O. H. Miyagaki, R. S. Rodrigues, On multiple solutions for a singular quasilinear elliptic system involving critical Hardy-Sobolev exponents, Houston J. Math. 34 (2008), 1271-1293.
[14] Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960), 101-123.
[15] V. Todorcevic, Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics, Springer Cham, 2019.
[16] T. F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign changing weight function, J. Math. Anal. Appl. 318 (2006), 253-270.
[17] B. Xuan, The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights, Nonlinear Anal. 62 (2005), 703-725.
