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Existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function. (English) Zbl 1188.34019

The paper deals with the existence, multiplicity and stability of positive solutions for the following boundary value problem
\[ u''(t)+\lambda a(t)f(u)=0, \;\;t\in (0,1), \]
\[ u(0)=u(1)=0, \]
where \(a\in C[0,1]\) may change sign and \(f\in C(\mathbb R, \mathbb R)\). The proof of the main result is based on global bifurcation techniques.

MSC:

34B09 Boundary eigenvalue problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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[1] Afrouzi, G.A.; Brown, K.J., Positive solutions for a semilinear elliptic problem with sign-changing nonlinearity, Nonlinear anal., 36, 4, 507-510, (1999) · Zbl 0930.35068
[2] Binding, P., Variational principles for indefinite eigenvalue problems, Linear algebra appl., 218, 251-262, (1995) · Zbl 0821.15005
[3] Bôcher, M., The smallest characteristic numbers in a certain exceptional case, Bull. amer. math. soc., 21, 6-9, (1914) · JFM 45.0491.02
[4] Brown, K.J.; Hess, P., Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem, Diff. integral equat., 3, 2, 201-207, (1990) · Zbl 0729.35046
[5] Brown, K.J.; Lin, S.S., On the existence of positive eigenfunction for an eigenvalue problem with indefinite weight function, J. math. anal. appl., 75, 112-120, (1980) · Zbl 0437.35058
[6] Brown, K.J., Local and global bifurcation results for a semilinear boundary value problem, J. diff. equat., 239, 296-310, (2007) · Zbl 1331.35129
[7] Các, N.P.; Fink, A.M.; Gatica, J.A., Nonnegative solutions of the radial Laplacian with nonlinearity that changes sign, Proc. amer. math. soc., 123, 5, 1393-1398, (1995) · Zbl 0826.34021
[8] Các, N.P.; Gatica, J.A.; Li, Y., Positive solutions to semilinear problems with coefficient that changes sign, Nonlinear anal., 37, 4, 501-510, (1999) · Zbl 0930.35069
[9] Erbe, L.H.; Wang, H.Y., On the existence of positive solutions of ordinary differential equations, Proc. amer. math. soc., 120, 3, 743-748, (1994) · Zbl 0802.34018
[10] Fink, A.M.; Gatica, J.A., Positive solutions of second order systems of boundary value problems, J. math. anal. appl., 180, 1, 93-108, (1993) · Zbl 0807.34024
[11] Fink, A.M.; Gatica, J.A.; Hernández, G.E., Eigenvalues of generalized Gelfand models, Nonlinear anal., 20, 12, 1453-1468, (1993) · Zbl 0790.34021
[12] Fleming, W.H., A selection – migration model in population genetics, J. math. biol., 2, 3, 219-233, (1975) · Zbl 0325.92009
[13] Lions, P.-L., On the existence of positive solutions of semilinear elliptic equations, SIAM rev., 24, 4, 441-467, (1982) · Zbl 0511.35033
[14] Hai, D.D., Positive solutions to a class of elliptic boundary value problems, J. math. anal. appl., 227, 195-199, (1998) · Zbl 0915.35043
[15] Henderson, J.; Wang, H.Y., Positive solutions for nonlinear eigenvalue problems, J. math. anal. appl., 208, 252-259, (1997) · Zbl 0876.34023
[16] Ince, E.L., Ordinary differential equations, (1926), Dover New York · Zbl 0063.02971
[17] Kielhöfer, H., Bifurcation theory: an introduction with applications to pdes, (2004), Springer-Verlag New York · Zbl 1032.35001
[18] Lan, K.Q.; Webb, J.R.L., Positive solutions of semilinear differential equations with singularities, J. diff. equat., 148, 2, 407-421, (1998) · Zbl 0909.34013
[19] Ma, R.Y., Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems, Appl. math. lett., 21, 7, 754-760, (2008) · Zbl 1152.34319
[20] Ma, R.Y.; Han, X.L., Existence of nodal solutions of a nonlinear eigenvalue problem with indefinite weight function, Nonlinear anal., 71, 2119-2125, (2009) · Zbl 1173.34310
[21] Protter, M.H.; Weinberger, H.F., Maximum principles in differential equations, (1984), Springer-Verlag New York · Zbl 0153.13602
[22] Rabinowitz, P.H., Some global results for nonlinear eigenvalue problems, J. funct. anal., 7, 487-513, (1971) · Zbl 0212.16504
[23] Rabinowitz, P.H., On bifurcation from infinity, J. diff. equat., 14, 462-475, (1973) · Zbl 0272.35017
[24] Wang, H., On the existence of positive solutions for semilinear elliptic equations in the annulus, J. diff. equat., 109, 1, 1-7, (1994) · Zbl 0798.34030
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