Pairs of positive solutions for the one-dimensional \(p\)-Laplacian. (English) Zbl 0813.34021

The paper studies the boundary value problem (1) \((\varphi_ p (u'))' + f(t,u) = 0\), \(u(0) = u(T) = 0\), where \(f\) is a Carathéodory function and \(\varphi_ p(s) = | s |^{p - 1} s\), \(p > 1\), (or more generally with an increasing homeomorphism \(\varphi : \mathbb{R} \to \mathbb{R}\) satisfying \(\varphi (0) = 0)\). The author proves the existence of at least two positive \(W^{2,1}\)-solutions of the problem. The argument is based on the Leray-Schauder degree theory and the upper and lower solutions method. The results are proved for example under the assumption that the behaviour of \(f\) near 0 and near \(\infty\) is controlled from a growth condition \(f(t,u) \geq b(t) \varphi_ p (u)\), where \(b\) is a positive \(L^ \infty\)-function, that \(\lambda_ 1 < 1\), where \(\lambda_ 1\) is the first eigenvalue of the spectral problem \((\varphi_ p (u'))' + \lambda b(t) \varphi_ p (u) = 0\), \(u(0) = u(T) = 0\), and that there exists a certain class of upper solutions of (1).


34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


[1] Figueiredo, D. G.de; Lions, P. L., On pairs of positive solutions for a class of semilinear elliptic problems, Indiana Univ. math. J., 34, 591-606 (1985) · Zbl 0587.35033
[2] Correa, F. J.S. A., On pairs of positive solutions for a class of sub-superlinear elliptic problems, Diff. Integral Eqns, 5, 387-392 (1992) · Zbl 0755.35031
[3] Brown, K. J.; Budin, H., Multiple positive solutions for a class of nonlinear boundary value problem, J. math. Analysis Applic., 60, 329-338 (1977) · Zbl 0361.35023
[4] Adje, A., Sur et sous-solutions généralisées et problémes aux limites du second ordre, Bull. Soc. math. Belg., 42, 3, 347-368 (1990), ser. B · Zbl 0724.34018
[5] Costa, D. G.; Goncalves, J. V.A., On the existence of positive solutions for a class of non-selfadjoint elliptic boundary value problems, Applicable Analysis, 31, 309-320 (1989) · Zbl 0645.35034
[6] Habets, P.; Sanchez, L., Periodic solutions of some Liénard equations with singularities, Proc. Am. math. Soc., 109, 1035-1044 (1990) · Zbl 0695.34036
[7] Nieto, J. J., Nonlinear second order periodic boundary value problems with Carathéodory functions, Applicable Analysis, 34, 111-128 (1989) · Zbl 0662.34022
[8] Deuel, J.; Hess, P., A criterion for the existence of solutions to nonlinear elliptic boundary value problems, Proc. R. Soc. Edinb., 74, 49-54 (1974) · Zbl 0331.35028
[10] Anane, A., Simplicité et isolation de la premiére valeur propre du \(p\)-Laplacien avec poids, C.r. Acad. Sci. Paris, Série 1, 305, 725-728 (1987) · Zbl 0633.35061
[11] Manasevich, R.; Zanolin, F., Time mappings and multiplicity of solutions for the one-dimensional \(p\)-Laplacian, Nonlinear Analysis, 21, 4, 269-291 (1993) · Zbl 0792.34021
[14] Kaper, H. G.; Knapp, M.; Kwong, M. K., Existence theorems for second-order boundary value problems, Diff. Integral Eqns, 4, 543-554 (1991) · Zbl 0732.34019
[15] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. funct. Analysis, 14, 349-381 (1973) · Zbl 0273.49063
[16] Brezis, H.; Turner, R. E.L., On a class of superlinear elliptic problems, Communs partial diff. Eqns, 2, 601-614 (1977) · Zbl 0358.35032
[17] Figueiredo, D. G.de, Positive solutions of semilinear elliptic problems, (Differential Equations, Proc. Sao Paolo. Differential Equations, Proc. Sao Paolo, Lecture Notes in Mathematics, Vol. 957 (1982), Springer: Springer Berlin), 34-87
[18] Njoku, F. I.; Zanolin, F., Positive solutions for two-point BVPs: existence and multiplicity results, Nonlinear Analysis, 13, 1329-1338 (1989) · Zbl 0704.34020
[19] Mawhin, J., Boundary value problems with nonlinearities having infinite jumps, Commentat. math. Univ. Carol., 25, 401-414 (1984) · Zbl 0562.34010
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.