## Abstract

The aim of the present paper is to study the existence, localization and multiplicity of positive solutions for a (2m)th-order boundary value problem subject to the Dirichlet conditions. Our approach is based on critical point theory in conical shells and Harnack type inequalities.

## Authors

**Habiba Boulaiki
**Faculty of Mathematics USTHB, PO. BOX 32, El-Alia Bab-Ezzouar Algiers, Algeria

**Toufik Moussaoui
**Laboratory of fixed point theory and applications ecole normale superieure, Kouba, Algiers, Algeria

**Radu Precup**

Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

## Keywords

Higher order boundary value problem; critical point; mountain pass geometry; compression-expansion; cone; positive solution; Harnack inequality.

## Paper coordinates

H. Boulaiki, T. Moussaoui, R. Precup, *Multiple positive solutions to a (2m)th-order boundary value problem, *Carpathian J. Math. 34 (2018), 167-182, https://www.jstor.org/stable/26898725

## About this paper

##### Print ISSN

1584 – 2851

##### Online ISSN

1843 – 4401

google scholar link

[1] Avery, R. I. and Henderson, J., Two positive fixed points of nonlinear operators on ordered Banach spaces, Comm. Appl. Nonlin. Anal., 8 (2001), 27–36

[2] Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2010

[3] Cabada, A., Precup, R., Saavedra, L. and Tersian, S., Multiple positive solutions to a fourth order boundary value problem, Electron. J. Differential Equations, 2016 (2016), No. 254, 1–18

[4] Cabada, A. and Saavedra, L., Existence of solutions for nth-order nonlinear differential boundary value problem by means of new fixed points theorems , Nonlinear Anal. Real World Appl., 42 (2018), 180-206

[5] Chyan, C. J. and Henderson, J. Positive solutions of 2m th-order boundary value problems, Appl. Math. Lett. 15 (2002), 767–774

[6] Graef, J. R. and Henderson, J., Double solutions of boundary value problems for 2m th-order differential equations and difference equations, Comput. Math. Appl., 45 (2003), 873–885

[7] Graef, J. R., Qian, C. and Yang , B., Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations, Proc. Amer. Math. Soc., 131 (2003), 577–585

[8] Liu, J., Existence and uniqueness of positive solution for 2m th-order nonlinear differential equation with boundary conditions , J. Appl. Math. Phys., 3 (2015), 1178–1185

[9] O’Regan, D. and Precup, R., Theorems of Leray-Schauder Type and Applications, Taylor and Francis, 2001

[10] Precup, R., A compression type mountain pass theorem in conical shells, J. Math. Anal. Appl., 338 (2008), 1116–1130

[11] Precup, R., Critical point theorems in cones and multiple positive solutions of elliptic problems, Nonlinear Anal., 75 (2012), 834–851

[12] Shi, G. and Chen, S., Positive solutions of even higher-order singular superlinear boundary value problems, Comput. Math. Appl., 45 (2003), 593–603

[13] Twaty, A. Al and Eloe, P. W., Concavity of solutions of a 2m-th order problem with symmetry, Opuscula Math., 33 (2013), 603–613

[14] Yang, X., Green’s function and positive solutions for higher-order ODE, Appl. Math. Comput., 136 (2003), 379-373.