## Abstract

An integral inequality is deduced from the negation of the geometrical condition in the bounded mountain pass theorem of Schechter, in a situation where this theorem does not apply. Also two localization results of non-zero solutions to a superlinear boundary value problem are established.

## Authors

**Radu Precup**

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

## Keywords

Integral inequality, Mountain pass theorem, Laplacean, Boundary value problem, Sobolev space.

## Paper coordinates

Radu Precup, *An inequality wich arises in the absence of the mountain pass geometry, *Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 3, Issue 3, Article 32, 2002

## About this paper

##### Journal

Journal of Inequalities in Pure and Applied Mathematics

##### Publisher Name

##### DOI

##### Print ISSN

14435756

##### Online ISSN

MR 1917791, Zbl 1010.26013

google scholar link

[1] R.A. ADAMS, Sobolev Spaces, Academic Press, London, 1978.

[2] H. BREZIS, Analyse fonctionnelle, Masson, Paris, 1983.

[3] L. GAJEK, M. KAŁUSZKA AND A. LENIC, The law of iterated logarithm for Lp-norms of empirical processes, Stat. Prob. Letters, 28 (1996), 107–110.

[4] D. GILBARG AND N.S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New YorkTokyo, 1983.

[5] M.A. KRASNOSELSKII, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, Oxford-London-New York-Paris, 1964.

[6] A.-M. MATEI, First eigenvalue for the p-Laplace operator, Nonlinear Anal., 39 (2000), 1051–1068.

[7] D.S. MITRINOVIC, ´ Analytic Inequalities, Springer-Verlag, BerlinHeidelberg-New York, 1970.

[8] D. O’REGAN AND R. PRECUP, Theorems of Leray-Schauder Type and Applications, Gordon and Breach Science Publishers, Amsterdam, 2001.

[9] G. PÓLYA, Two more inequalities between physical and geometrical quantities, J. Indian Math. Soc., 24 (1960), 413–419. An Inequality which Arises in the Absence of the Mountain Pass Geometry

[10] G. PÓLYA AND G. SZEGÖ, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951.

[11] R. PRECUP, Partial Differential Equations (Romanian), Transilvania Press, Cluj, 1997.

[12] R. PRECUP, An isoperimetric type inequality, In: Proc. of the “Tiberiu Popoviciu” Itinerant Seminar of Functional Equations, Approximation and Convexity (E. Popoviciu, ed.), Srima, Cluj, 2001, 199-204.

[13] R. PRECUP, Inequalities and compactness, to appear in Proc. 6 th International Conference on Nonlinear Functional Analysis and Applications, Korea, 2000, Nova Science Publishers, New York.

[14] M. SCHECHTER, Linking Methods in Critical Point Theory, Birkhäuser, Boston-Basel-Berlin, 1999.

[15] M. STRUWE, Variational Methods, Springer-Verlag, Berlin-HeidelbergNew York-London-Paris-Tokyo-Hong Kong-Barcelona, 1990.