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July  2016, 36(7): 3775-3789. doi: 10.3934/dcds.2016.36.3775

A Schechter type critical point result in annular conical domains of a Banach space and applications

1. 

Babeş-Bolyai University, Faculty of Mathematics and Computer Science, Str. Kogălniceanu nr. 1, RO - 400084 Cluj-Napoca, Romania, Romania, Romania

Received  April 2015 Revised  November 2015 Published  March 2016

Using Ekeland's variational principle we obtain a critical point theorem of Schechter type for extrema of a functional in an annular conical domain of a Banach space. The result can be seen as a variational analogue of Krasnoselskii's fixed point theorem in cones and can be applied for the existence, localization and multiplicity of the positive solutions of variational problems. The result is then applied to $p$-Laplace equations, where the geometric condition on the boundary of the annular conical domain is established via a weak Harnack type inequality given in terms of the energetic norm. This method can be applied also to other homogeneous operators in order to obtain existence, multiplicity or infinitely many solutions for certain classes of quasilinear equations.
Citation: Hannelore Lisei, Radu Precup, Csaba Varga. A Schechter type critical point result in annular conical domains of a Banach space and applications. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3775-3789. doi: 10.3934/dcds.2016.36.3775
References:
[1]

G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl., (2009), Art. ID 670675, 20pp.

[2]

M. Belloni, V. Ferone and B. Kawohl, Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys., 54 (2003), 771-783. doi: 10.1007/s00033-003-3209-y.

[3]

F. Della Pietra and N. Gavitone, Anisotropic elliptic problems involving Hardy-type potential, J. Math. Anal. Appl., 397 (2013), 800-813. doi: 10.1016/j.jmaa.2012.08.008.

[4]

G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with $p$-Laplacian, Port. Math. (N.S.), 58 (2001), 339-378.

[5]

J. Diestel, Geometry of Banach Spaces - Selected Topics, Lecture Notes in Mathematics, Vol. 485. Springer-Verlag, Berlin-New York, 1975.

[6]

I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.), 1 (1979), 443-474. doi: 10.1090/S0273-0979-1979-14595-6.

[7]

F. Faraci and A. Kristály, One-dimensional scalar field equations involving an oscillatory nonlinear term, Discrete Contin. Dyn. Syst., 18 (2007), 107-120. doi: 10.3934/dcds.2007.18.107.

[8]

V. Ferone and B. Kawohl, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc., 137 (2009), 247-253. doi: 10.1090/S0002-9939-08-09554-3.

[9]

M. Frigon, On a new notion of linking and application to elliptic problems at resonance, J. Differential Equations, 153 (1999), 96-120. doi: 10.1006/jdeq.1998.3540.

[10]

N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 321-330.

[11]

R. Glowinski and A. Marrocco, Sur l'approximation par éléments finis d'ordre un et la resolution par penalisation-dualité d'une classe de problemes de Dirichlet non linéaires, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 9 (1975), 41-76.

[12]

D. Guo, J. Sun and G. Qi, Some extensions of the mountain pass lemma, Differential Integral Equations, 1 (1988), 351-358.

[13]

M. A. Krasnoselskii, Positive Solutions of Operator Equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd., Groningen 1964.

[14]

A. Kristály, Infinitely many solutions for a differential inclusion problem in $\mathbb{R}^N2$, J. Differential Equations, 220 (2006), 511-530. doi: 10.1016/j.jde.2005.02.007.

[15]

L. Ma, Mountain pass on a closed convex set, J. Math. Anal. Appl., 205 (1997), 531-536. doi: 10.1006/jmaa.1997.5227.

[16]

S. A. Marano and D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the $p$-Laplacian, J. Differential Equations, 182 (2002), 108-120. doi: 10.1006/jdeq.2001.4092.

[17]

M. Marcus and V. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal., 33 (1979), 217-229. doi: 10.1016/0022-1236(79)90113-7.

[18]

R. Precup, The Leray-Schauder boundary condition in critical point theory, Nonlinear Anal., 71 (2009), 3218-3228. doi: 10.1016/j.na.2009.01.195.

[19]

R. Precup, On a bounded critical point theorem of Schechter, Stud. Univ. Babeş-Bolyai Math., 58 (2013), 87-95.

[20]

R. Precup, Critical point localization theorems via Ekeland's variational principle, Dynam. Systems Appl., 22 (2013), 355-370.

[21]

P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations, 60 (1985), 142-149. doi: 10.1016/0022-0396(85)90125-1.

[22]

B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401-410. doi: 10.1016/S0377-0427(99)00269-1.

[23]

B. Ricceri, Infinitely many solutions of the Neumann problem for elliptic equations involving the $p$-Laplacian, Bull. London Math. Soc., 33 (2001), 331-340. doi: 10.1017/S0024609301008001.

[24]

J. Saint Raymond, On the multiplicity of solutions of the equation $-\Delta u=\lambda f(u),$ J. Differential Equations, 180 (2002), 65-88. doi: 10.1006/jdeq.2001.4057.

[25]

M. Schechter, A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc., 331 (1992), 681-703. doi: 10.1090/S0002-9947-1992-1064270-1.

[26]

M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-1596-7.

show all references

References:
[1]

G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl., (2009), Art. ID 670675, 20pp.

[2]

M. Belloni, V. Ferone and B. Kawohl, Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys., 54 (2003), 771-783. doi: 10.1007/s00033-003-3209-y.

[3]

F. Della Pietra and N. Gavitone, Anisotropic elliptic problems involving Hardy-type potential, J. Math. Anal. Appl., 397 (2013), 800-813. doi: 10.1016/j.jmaa.2012.08.008.

[4]

G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with $p$-Laplacian, Port. Math. (N.S.), 58 (2001), 339-378.

[5]

J. Diestel, Geometry of Banach Spaces - Selected Topics, Lecture Notes in Mathematics, Vol. 485. Springer-Verlag, Berlin-New York, 1975.

[6]

I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.), 1 (1979), 443-474. doi: 10.1090/S0273-0979-1979-14595-6.

[7]

F. Faraci and A. Kristály, One-dimensional scalar field equations involving an oscillatory nonlinear term, Discrete Contin. Dyn. Syst., 18 (2007), 107-120. doi: 10.3934/dcds.2007.18.107.

[8]

V. Ferone and B. Kawohl, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc., 137 (2009), 247-253. doi: 10.1090/S0002-9939-08-09554-3.

[9]

M. Frigon, On a new notion of linking and application to elliptic problems at resonance, J. Differential Equations, 153 (1999), 96-120. doi: 10.1006/jdeq.1998.3540.

[10]

N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 321-330.

[11]

R. Glowinski and A. Marrocco, Sur l'approximation par éléments finis d'ordre un et la resolution par penalisation-dualité d'une classe de problemes de Dirichlet non linéaires, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 9 (1975), 41-76.

[12]

D. Guo, J. Sun and G. Qi, Some extensions of the mountain pass lemma, Differential Integral Equations, 1 (1988), 351-358.

[13]

M. A. Krasnoselskii, Positive Solutions of Operator Equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd., Groningen 1964.

[14]

A. Kristály, Infinitely many solutions for a differential inclusion problem in $\mathbb{R}^N2$, J. Differential Equations, 220 (2006), 511-530. doi: 10.1016/j.jde.2005.02.007.

[15]

L. Ma, Mountain pass on a closed convex set, J. Math. Anal. Appl., 205 (1997), 531-536. doi: 10.1006/jmaa.1997.5227.

[16]

S. A. Marano and D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the $p$-Laplacian, J. Differential Equations, 182 (2002), 108-120. doi: 10.1006/jdeq.2001.4092.

[17]

M. Marcus and V. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal., 33 (1979), 217-229. doi: 10.1016/0022-1236(79)90113-7.

[18]

R. Precup, The Leray-Schauder boundary condition in critical point theory, Nonlinear Anal., 71 (2009), 3218-3228. doi: 10.1016/j.na.2009.01.195.

[19]

R. Precup, On a bounded critical point theorem of Schechter, Stud. Univ. Babeş-Bolyai Math., 58 (2013), 87-95.

[20]

R. Precup, Critical point localization theorems via Ekeland's variational principle, Dynam. Systems Appl., 22 (2013), 355-370.

[21]

P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations, 60 (1985), 142-149. doi: 10.1016/0022-0396(85)90125-1.

[22]

B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401-410. doi: 10.1016/S0377-0427(99)00269-1.

[23]

B. Ricceri, Infinitely many solutions of the Neumann problem for elliptic equations involving the $p$-Laplacian, Bull. London Math. Soc., 33 (2001), 331-340. doi: 10.1017/S0024609301008001.

[24]

J. Saint Raymond, On the multiplicity of solutions of the equation $-\Delta u=\lambda f(u),$ J. Differential Equations, 180 (2002), 65-88. doi: 10.1006/jdeq.2001.4057.

[25]

M. Schechter, A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc., 331 (1992), 681-703. doi: 10.1090/S0002-9947-1992-1064270-1.

[26]

M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-1596-7.

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