The Nehari manifold for indefinite semilinear elliptic systems involving critical exponent

Abstract In this paper, we study the combined effect of concave and convex nonlinearities on the number of solutions for an indefinite semilinear elliptic system ( E λ , μ ) involving critical exponents and sign-changing weight functions. Using Nehari manifold, the system is proved to have at least two nontrivial nonnegative solutions when the pair of the parameters ( λ , μ ) belongs to a certain subset of R 2 .

[1]  G. Tarantello.  On nonhomogeneous elliptic equations involving critical Sobolev exponent , 1992 .

[2]  A. E. Hamidi,et al.  The Nehari manifold for systems of nonlinear elliptic equations , 2006 .

[3]  A. Cañada.  Nonlinear Ordinary Boundary Value Problems under a Combined Effect of Periodic and Attractive Nonlinearities , 2000 .

[4]  P. Felmer,et al.  On superquadratic elliptic systems , 1994 .

[5]  A. E. Hamidi.  Multiple solutions with changing sign energy to a nonlinear elliptic equation , 2004 .

[6]  Jianfu Yang,et al.  Positive solutions for critical inhomogeneous elliptic problems in non-contractible domains , 2009 .

[7]  M. Squassina.  An Eigenvalue Problem for Elliptic Systems , 2000 .

[8]  A. Ahammou.  Positive Radial Solutions of Nonlinear Elliptic Systems , 2001 .

[9]  Tsing-San Hsu,et al.  Multiple positive solutions for a critical quasilinear elliptic system with concave–convex nonlinearities , 2009 .

[10]  Tsing-San Hsu,et al.  Multiple Positive Solutions for a Quasilinear Elliptic System Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions , 2012, Int. J. Math. Math. Sci..

[11]  Vicenţiu D. Rădulescu,et al.  Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems , 2010 .

[12]  Ching-yu Chen,et al.  Multiple positive solutions for indefinite semilinear elliptic problems involving a critical Sobolev exponent , 2014, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[13]  Tsung-fang Wu,et al.  Three positive solutions for Dirichlet problems involving critical Sobolev exponent and sign-changing weight , 2010 .

[14]  Enzo Mitidieri,et al.  Positive Solutions of Semilinear Elliptic Systems , 1992 .

[15]  J. Gossez,et al.  Local superlinearity and sublinearity for indefinite semilinear elliptic problems , 2003 .

[16]  Haim Brezis,et al.  Positive solutions of nonlinear elliptic equations involving critical sobolev exponents , 1983 .

[17]  Yuri Bozhkov,et al.  Existence of multiple solutions for quasilinear systems via fibering method , 2003 .

[18]  Elliott H. Lieb,et al.  A Relation Between Pointwise Convergence of Functions and Convergence of Functionals , 1983 .

[19]  J. Velin.  Existence results for some nonlinear elliptic system with lack of compactness , 2003 .

[20]  Claudianor O. Alves,et al.  On systems of elliptic equations involving subcritical or critical Sobolev exponents , 2000 .

[21]  A. Ambrosetti,et al.  Multiplicity Results for Some Nonlinear Elliptic Equations , 1996 .

[22]  A. El Hamidi,et al.  Existence results to elliptic systems with nonstandard growth conditions , 2004 .

[23]  Tsung‐fang Wu.  Multiplicity Results for a Semi-Linear Elliptic Equation Involving Sign-Changing Weight Function , 2009 .

[24]  Multiple solutions for Schrödinger equations with magnetic field , 2009 .

[25]  K. J. Brown,et al.  A fibering map approach to a potential operator equation and its applications , 2009, Differential and Integral Equations.

[26]  Ke Chen,et al.  Applied Mathematics and Computation , 2022 .

[27]  V. Lubyshev.  Multiple positive solutions of an elliptic equation with a convex-concave nonlinearity containing a sign-changing term , 2010 .

[28]  I. Ekeland.  On the variational principle , 1974 .

[29]  Tsung-fang Wu,et al.  A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function , 2008 .

[30]  Tsung‐fang Wu.  On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function , 2006 .

[31]  Dušan D. Repovš,et al.  Combined effects in nonlinear problems arising in the study ofanisotropic continuous media , 2012 .

[32]  Tsung-fang Wu,et al.  The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions , 2008 .

[33]  Haim Brezis,et al.  Combined Effects of Concave and Convex Nonlinearities in Some Elliptic Problems , 1994 .