Abstract
We obtain critical point variants of the compression fixed point theorem in cones of Krasnoselskii. Critical points are localized in a set defined by means of two norms. In applications to semilinear elliptic boundary value problems this makes possible the use of local Moser–Harnack inequalities for the estimations from below. Multiple solutions are found for problems with oscillating nonlinearity.
Authors
Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Keywords
Critical point; Mountain pass lemma; Compression; Cone; Positive solution; Elliptic problem; Moser–Harnack inequality
Paper coordinates
R. Precup, Critical point theorems in cones and multiple positive solutions of elliptic problems, Nonlinear Anal. 75 (2012), 834-851,
http://doi.org/10.1016/j.na.2011.09.016
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About this paper
Journal
Nonlinear Analysis: Theory, Methods & Applications
Publisher Name
Elsevier
Print ISSN
Online ISSN
0362546X
google scholar link
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[5] R.P. Agarwal, M. Meehan, D. O’Regan, R. Precup, Location of nonnegative solutions for differential equations on finite and semi-infinite intervals, Dynam. Systems Appl., 12 (2003), pp. 323-332, CrossRefGoogle Scholar
[6] R.P. Agarwal, D. O’Regan, P.J.Y. Wong, Positive solutions of differential, Difference and Integral Equations, Kluwer, Dordrecht (1999), Google Scholar
[7] L.H. Erbe, S. Hu, H. Wang, Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl., 184 (1994), pp. 640-648 Google Scholar
[8] L.H. Erbe, H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120 (1994), pp. 743-748, Google Scholar
[9] J. Henderson, H. Wang, Positive solutions for nonlinear eigenvalue problems, J. Math. Anal. Appl., 208 (1997), pp. 252-259, Google Scholar
[10] K. Lan, J.R.L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148 (1998), pp. 407-421, Google Scholar
[11] M. Meehan, D. O’Regan, Positive Lp solutions of Hammerstein integral equations, Arch. Math., 76 (2001), pp. 366-376, Google Scholar
[12] D. O’Regan, H. Wang, Positive periodic solutions of systems of second order ordinary differential equations, Positivity, 10 (2006), pp. 285-298, Google Scholar
[13] H. Wang, On the existence of positive solutions for semilinear elliptic equations in the annulus, J. Differential Equations, 109 (1994), pp. 1-7, Google Scholar
[14] R. Precup, Existence and localization results for the nonlinear wave equation, Fixed Point Theory, 5 (2004), pp. 309-321, Google Scholar
[15] R. Precup, Positive solutions of evolution operator equations, Austral. J. Math. Anal. Appl., 2 (1) (2005), pp. 1-10, (electronic), Google Scholar
[16] R. Precup, Positive solutions of semi-linear elliptic problems via Krasnoselskii type theorems in cones and Harnack’s inequality, Mathematical Analysis and Applications, C.P. Niculescu, V.D. Radulescu (Eds.), AIP Conf. Proc., vol. 835 (2006), pp. 125-132, Google Scholar
[17] M. Schechter, A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc., 331 (1992), pp. 681-703, Google Scholar
[18] M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Basel (1999) Google Scholar
[19] M. Schechter, K. Tintarev, Nonlinear eigenvalues and mountain pass methods, Topol. Methods Nonlinear Anal., 1 (1993), pp. 183-201, CrossRefGoogle Scholar
[20] R. Precup, The Leray–Schauder condition in critical point theory, Nonlinear Anal., 71 (2009), pp. 3218-3228, Google Scholar
[21] R. Precup, Two positive solutions of some singular boundary value problems, Anal. Appl. Singap., 8 (3) (2010), pp. 305-314, Google Scholar
[22] R. Precup, Two positive nontrivial solutions for a class of semilinear elliptic variational systems, J. Math. Anal. Appl., 373 (2011), pp. 138-146, Google Scholar
[23] R. Precup, A compression type mountain pass theorem in conical shells, J. Math. Anal. Appl., 338 (2008), pp. 1116-1130, Google Scholar
[24] K. Deimling, Ordinary Differential Equations in Banach Spaces, Springer, Berlin (1977) Google Scholar
[25] M. Frigon, On a new notion of linking and application to elliptic problems at resonance, J. Differential Equations, 153 (1999), pp. 96-120, Google Scholar
[26] M. Struwe, Variational Methods, Springer, Berlin (1990), Google Scholar
[27] J. Jost, Partial Differential Equations, Springer, New York (2007), Google Scholar