×

Exact multiplicity of sign-changing solutions for a class of second-order Dirichlet boundary value problem with weight function. (English) Zbl 1470.34064

Summary: Using bifurcation techniques and Sturm comparison theorem, we establish exact multiplicity results of sign-changing or constant sign solutions for the boundary value problems \(u'' + a(t) f(u) = 0\), \(t \in(0, 1)\), \(u(0) = 0\), and \(u(1) = 0\), where \(f \in C(\mathbb{R}, \mathbb{R})\) satisfies \(f(0) = 0\) and the limits \(f_\infty = \lim_{| s | \rightarrow \infty}(f(s) / s)\), \(f_0 = \lim_{| s | \rightarrow 0}(f(s) / s) \in \{0, \infty \}\). Weight function \(a(t) \in C^1 [0, 1]\) satisfies \(a(t) > 0\) on \([0, 1]\).

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ambrosetti, A.; Hess, P., Positive solutions of asymptotically linear elliptic eigenvalue problems, Journal of Mathematical Analysis and Applications, 73, 2, 411-422, (1980) · Zbl 0433.35026
[2] Erbe, L. H.; Wang, H., On the existence of positive solutions of ordinary differential equations, Proceedings of the American Mathematical Society, 120, 3, 743-748, (1994) · Zbl 0802.34018
[3] Asakawa, H., Nonresonant singular two-point boundary value problems, Nonlinear Analysis. Theory, Methods & Applications, 44, 6, 791-809, (2001) · Zbl 0992.34011
[4] Ma, R.; Thompson, B., Multiplicity results for second-order two-point boundary value problems with superlinear or sublinear nonlinearities, Journal of Mathematical Analysis and Applications, 303, 2, 726-735, (2005) · Zbl 1075.34017
[5] Ma, R.; Thompson, B., Nodal solutions for nonlinear eigenvalue problems, Nonlinear Analysis. Theory, Methods & Applications, 59, 5, 707-718, (2004) · Zbl 1059.34013
[6] Chamberlain, J.; Kong, L.; Kong, Q., Nodal solutions of boundary value problems with boundary conditions involving Riemann-Stieltjes integrals, Nonlinear Analysis. Theory, Methods & Applications, 74, 6, 2380-2387, (2011) · Zbl 1220.34027
[7] Jankowski, T., Existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions, Nonlinear Analysis. Theory, Methods & Applications, 75, 2, 913-923, (2012) · Zbl 1235.34179
[8] Ouyang, T.; Shi, J., Exact multiplicity of positive solutions for a class of semilinear problem—II, Journal of Differential Equations, 158, 1, 94-151, (1999) · Zbl 0947.35067
[9] Xu, B., Exact multiplicity and global structure of solutions for a class of semilinear elliptic equations, Journal of Mathematical Analysis and Applications, 341, 2, 783-790, (2008) · Zbl 1137.35368
[10] Wang, Y.; Wang, Y.; Shi, J., Exact multiplicity of solutions to a diffusive logistic equation with harvesting, Applied Mathematics and Computation, 216, 5, 1531-1537, (2010) · Zbl 1189.35105
[11] Adamowicz, T.; Korman, P., Remarks on time map for quasilinear equations, Journal of Mathematical Analysis and Applications, 376, 2, 686-695, (2011) · Zbl 1221.34054
[12] Hung, K.-C.; Wang, S.-H., Classification and evolution of bifurcation curves for a multiparameter p-Laplacian Dirichlet problem, Nonlinear Analysis. Theory, Methods & Applications, 74, 11, 3589-3598, (2011) · Zbl 1220.34025
[13] García-Melián, J., Multiplicity of positive solutions to boundary blow-up elliptic problems with sign-changing weights, Journal of Functional Analysis, 261, 7, 1775-1798, (2011) · Zbl 1387.35308
[14] Shi, J., Exact multiplicity of solutions to superlinear and sublinear problems, Nonlinear Analysis. Theory, Methods & Applications, 50, 5, 665-687, (2002) · Zbl 1004.35043
[15] Bari, R.; Rynne, B. P., Solution curves and exact multiplicity results for 2mth order boundary value problems, Journal of Mathematical Analysis and Applications, 292, 1, 17-22, (2004) · Zbl 1057.34004
[16] Korman, P., Curves of sign-changing solutions for semilinear equations, Nonlinear Analysis. Theory, Methods & Applications, 51, 5, 801-820, (2002) · Zbl 1173.35475
[17] Pan, H.; Xing, R., Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations—II, Nonlinear Analysis. Theory, Methods & Applications, 74, 11, 3751-3768, (2011) · Zbl 1248.34016
[18] An, Y.; Ma, R., Exact multiplicity of solutions for a class of two-point boundary value problems, Electronic Journal of Differential Equations, 27, article 7, (2010)
[19] Pan, H.; Xing, R., Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations, Nonlinear Analysis. Theory, Methods & Applications, 74, 4, 1234-1260, (2011) · Zbl 1218.34020
[20] Naito, Y.; Tanaka, S., On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations, Nonlinear Analysis. Theory, Methods & Applications, 56, 6, 919-935, (2004) · Zbl 1046.34038
[21] Kong, Q., Existence and nonexistence of solutions of second-order nonlinear boundary value problems, Nonlinear Analysis. Theory, Methods & Applications, 66, 11, 2635-2651, (2007) · Zbl 1119.34024
[22] Kong, L.; Kong, Q., Nodal solutions of second order nonlinear boundary value problems, Mathematical Proceedings of the Cambridge Philosophical Society, 146, 3, 747-763, (2009) · Zbl 1189.34043
[23] Anuradha, V.; Shivaji, R., Sign changing solutions for a class of superlinear multi-parameter semi-positone problems, Nonlinear Analysis. Theory, Methods & Applications, 24, 11, 1581-1596, (1995) · Zbl 0824.34024
[24] Korman, P.; Ouyang, T., Solution curves for two classes of boundary-value problems, Nonlinear Analysis. Theory, Methods & Applications, 27, 9, 1031-1047, (1996) · Zbl 0860.34009
[25] Crandall, M. G.; Rabinowitz, P. H., Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8, 321-340, (1971) · Zbl 0219.46015
[26] Rabinowitz, P. H.; Paul, H., On bifurcation from infinity, Journal of Differential Equations, 14, 462-475, (1973) · Zbl 0272.35017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.