Positive solutions and eigenvalue intervals for nonlinear systems

AbstractThis paper deals with the existence of positive solutions for the nonlinear system $$ (q(t)\phi (p(t)u'_i (t)))' + f^i (t,u) = 0, 0 < t < 1, i = 1,2, \ldots ,n $$. This system often arises in the study of positive radial solutions of nonlinear elliptic system. Here u = (u1, …, un) and fi, i = 1, 2, …, n are continuous and nonnegative functions, p(t), q(t): [0, 1] → (0, ∞) are continuous functions. Moreover, we characterize the eigenvalue intervals for $$ (q(t)\phi (p(t)u'_i (t)))' + \lambda h_i (t)g^i (u) = 0, 0 < t < 1, i = 1,2, \ldots ,n $$. The proof is based on a well-known fixed point theorem in cones.

[1]  Kunquan Lan,et al.  Positive Solutions of Semilinear Differential Equations with Singularities , 1998 .

[2]  Donal O'Regan,et al.  Some general existence principles and results for f y ′ =qf t,y,y ′ ,0 , 1993 .

[3]  C. Coster.  Pairs of positive solutions for the one-dimensional p -Laplacian , 1994 .

[4]  Haiyan Wang,et al.  Existence and multiplicity of positive solutions for elliptic systems , 1997 .

[5]  Ravi P. Agarwal,et al.  Positive solutions and eigenvalues of conjugate boundary value problems , 1999, Proceedings of the Edinburgh Mathematical Society.

[6]  Abdou Kouider Ben-Naoum,et al.  On the existence and multiplicity of positive solutions of the $p$-Laplacian separated boundary value problem , 1997, Differential and Integral Equations.

[7]  J. Henderson,et al.  Nonlinear eigenvalue problems for quasilinear systems , 2005 .

[8]  Song-Sun Lin.  On the existence of positive radial solutions for nonlinear elliptic equations in annular domains , 1989 .

[9]  Fan Yong.  Existence and Multiplicity of Positive Radial Solutions for Elliptic Systems , 2000 .

[10]  Leo F. Boron,et al.  Positive solutions of operator equations , 1964 .

[11]  Yong-Hoon Lee,et al.  Multiplicity of Positive Radial Solutions for Multiparameter Semilinear Elliptic Systems on an Annulus , 2001 .

[12]  Haiyan Wang,et al.  On the number of positive solutions of nonlinear systems , 2003 .

[13]  Junyu Wang,et al.  Multiple positive solutions for the one-dimensional p -Laplacian , 2000 .

[14]  Ravi P. Agarwal,et al.  Eigenvalues and the One-Dimensional p-Laplacian☆☆☆ , 2002 .

[15]  Kunquan Lan,et al.  Multiple Positive Solutions of Semilinear Differential Equations with Singularities , 2001 .

[16]  Song-Sun Lin.  Positive radial solutions and non-radial bifurcation for semilinear elliptic equations in annular domains , 1990 .

[17]  POSITIVE SOLUTIONS AND EIGENVALUES OF NONLOCAL BOUNDARY-VALUE PROBLEMS , 2005 .

[18]  Daoyi Xu,et al.  Asymptotic behavior of a class of reaction-diffusion equations with delays ✩ , 2003 .

[19]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[20]  Haiyan Wang,et al.  Multiplicity of positive radial solutions for an elliptic system on an annulus , 2000 .

[21]  Junyu Wang.  The existence of positive solutions for the one-dimensional $p$-Laplacian , 1997 .

[22]  D. O’Regan,et al.  On the number of positive solutions of elliptic systems , 2007 .

[23]  Johnny Henderson,et al.  Positive Solutions for Nonlinear Eigenvalue Problems , 1997 .

[24]  Jifeng Chu,et al.  Eigenvalues and discrete boundary value problems for the one-dimensional p-laplacian , 2005 .

[25]  C. Bandle,et al.  Nonlinear elliptic problems in annular domains , 1987 .