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Quasi-stability property and attractors for a semilinear Timoshenko system
Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential
1. | Department of Mathematics, University of Bari Aldo Moro, Via E.Orabona 4, 70125 Bari |
2. | Department of Mathematics and Computer Sciences, University of Perugia, Via Vanvitelli 1, 06123 Perugia |
3. | Department of Mathematics, National Technical University, Zografou Campus, Athens 15780 |
References:
[1] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints, Mem. Amer. Math. Soc., 196 (2008).
doi: 10.1090/memo/0915. |
[2] |
W. Allegretto and Y. X. Huang, A Picone's identity for the $p -$Laplacian and applications, Nonlinear Anal., 32 (1998), 819-830.
doi: 10.1016/S0362-546X(97)00530-0. |
[3] |
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[4] |
D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p-$Laplacian operator, Comm. Partial Differential Equations, 31 (2006), 849-865.
doi: 10.1080/03605300500394447. |
[5] |
F. Brock, L. Iturriaga and P. Ubilla, A multiplicity result for the $p -$Laplacian involving a parameter, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (2008), 349-371.
doi: 10.1007/s00023-008-0386-4. |
[6] |
T. Cardinali, N. S. Papageorgiou and P. Rubbioni, Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type, Ann. Mat. Pura Appl. (4), 193 (2013), 1-21.
doi: 10.1007/s10231-012-0263-0. |
[7] |
J. N. Corvellec and A. Hantoute, Homotopical stability of isolated critical points of continuous functionals, Set-Valued Analysis, 10 (2002), 143-164.
doi: 10.1023/A:1016544301594. |
[8] |
J. I. Diaz and J. E. Saa, Existence et unicité de solutions positives pur certaines équations elliptiques quasilinéaires, C. R. Math. Acad. Sci. Paris, 305 (1987), 521-524. |
[9] |
M. Filippakis and N. S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the $p -$Laplacian, J. Differential Equations, 245 (2008), 1883-1922.
doi: 10.1016/j.jde.2008.07.004. |
[10] |
G. Fragnelli, D. Mugnai and N. S. Papageorgiou, Superlinear Neumann problems with the $p -$Laplacian plus an indefinite potential, Ann. Mat. Pura Appl. (4), to appear.
doi: 10.1007/s10231-016-0582-7. |
[11] |
L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications 9, Chapman & Hall/CRC, Boca Raton, FL, 2006. |
[12] |
L. Gasinski and N. S. Papageorgiou, Bifurcation - type results for nonlinear parametric elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 595-623.
doi: 10.1017/S0308210511000126. |
[13] |
M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13 (1989), 879-902.
doi: 10.1016/0362-546X(89)90020-5. |
[14] |
S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Kluwer, Dordrecht, The Netherlands, 1997. |
[15] |
G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.
doi: 10.1016/0362-546X(88)90053-3. |
[16] |
V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory, Topol. Methods Nonlinear Anal., 10 (1997), 387-397. |
[17] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), (2011), 729-755. |
[18] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.
doi: 10.1007/978-1-4614-9323-5. |
[19] |
D. Mugnai, Addendum to Multiplicity of critical points in presence of a linking: Application to a superlinear boundary value problem, NoDEA. Nonlinear Differential Equations Appl., 11 (2004), 379-391, and a comment on the generalized Ambrosetti-Rabinowitz condition, NoDEA. Nonlinear Differential Equations Appl., 19 (2011), 299-301.
doi: 10.1007/s00030-004-2016-2. |
[20] |
D. Mugnai and N. S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), XI (2012), 729-788. |
[21] |
D. Mugnai and N. S. Papageorgiou, Wang's multiplicity result for superlinear $(p,q)$ - equations without the Ambrosetti - Rabinowitz condition, Trans. Amer. Math. Soc., 366 (2014), 4919-4937.
doi: 10.1090/S0002-9947-2013-06124-7. |
[22] |
D. Mugnai and N. S. Papageorgiou, Bifurcation for positive solutions of nonlinear diffusive logistic equations in $\mathbb{R}^N2$ with indefinite weight, Indiana Univ. Math. J., 63 (2014), 1397-1418.
doi: 10.1512/iumj.2014.63.5369. |
[23] |
N. S. Papageorgiou and V. Radulescu, Multiple solutions with precise sign information for nonlinear Robin problems, J. Differential Equations, 256 (2014), 2449-2479.
doi: 10.1016/j.jde.2014.01.010. |
[24] |
N.S. Papageorgiou and V. Radulescu, Positive solutions for nonlinear Robin eigenvalue problems, Proc. Amer. Math. Soc., to appear.
doi: 10.1090/proc/13107. |
[25] |
N. S. Papageorgiou and V. Radulescu, Nonlinear, nonhomogeneous Robin problems with superlinear reaction term, submitted. |
[26] |
N. S. Papageorgiou and P. Winkert, Resonant $(p,2) -$ equations with concave terms, Appl. Anal., 94 (2015), 342-360.
doi: 10.1080/00036811.2014.895332. |
[27] |
P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications 73, Birkhäuser Verlag, Basel, 2007. |
[28] |
S. Takeuchi, Positive solutions of a degenerate elliptic equation with a logistic reaction, Proc. Amer. Math. Soc., 129 (2001), 433-441.
doi: 10.1090/S0002-9939-00-05723-3. |
[29] |
S. Takeuchi, Multiplicity results for a degenerate elliptic equation with a logistic reaction, J. Differential Equations, 173 (2001), 138-144.
doi: 10.1006/jdeq.2000.3914. |
[30] |
P. Winkert, $L^\infty$-estimates for nonlinear elliptic Neumann boundary value problems, NoDEA. Nonlinear Differential Equations Appl., 17 (2010), 289-302.
doi: 10.1007/s00030-009-0054-5. |
show all references
References:
[1] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints, Mem. Amer. Math. Soc., 196 (2008).
doi: 10.1090/memo/0915. |
[2] |
W. Allegretto and Y. X. Huang, A Picone's identity for the $p -$Laplacian and applications, Nonlinear Anal., 32 (1998), 819-830.
doi: 10.1016/S0362-546X(97)00530-0. |
[3] |
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[4] |
D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p-$Laplacian operator, Comm. Partial Differential Equations, 31 (2006), 849-865.
doi: 10.1080/03605300500394447. |
[5] |
F. Brock, L. Iturriaga and P. Ubilla, A multiplicity result for the $p -$Laplacian involving a parameter, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (2008), 349-371.
doi: 10.1007/s00023-008-0386-4. |
[6] |
T. Cardinali, N. S. Papageorgiou and P. Rubbioni, Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type, Ann. Mat. Pura Appl. (4), 193 (2013), 1-21.
doi: 10.1007/s10231-012-0263-0. |
[7] |
J. N. Corvellec and A. Hantoute, Homotopical stability of isolated critical points of continuous functionals, Set-Valued Analysis, 10 (2002), 143-164.
doi: 10.1023/A:1016544301594. |
[8] |
J. I. Diaz and J. E. Saa, Existence et unicité de solutions positives pur certaines équations elliptiques quasilinéaires, C. R. Math. Acad. Sci. Paris, 305 (1987), 521-524. |
[9] |
M. Filippakis and N. S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the $p -$Laplacian, J. Differential Equations, 245 (2008), 1883-1922.
doi: 10.1016/j.jde.2008.07.004. |
[10] |
G. Fragnelli, D. Mugnai and N. S. Papageorgiou, Superlinear Neumann problems with the $p -$Laplacian plus an indefinite potential, Ann. Mat. Pura Appl. (4), to appear.
doi: 10.1007/s10231-016-0582-7. |
[11] |
L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications 9, Chapman & Hall/CRC, Boca Raton, FL, 2006. |
[12] |
L. Gasinski and N. S. Papageorgiou, Bifurcation - type results for nonlinear parametric elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 595-623.
doi: 10.1017/S0308210511000126. |
[13] |
M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13 (1989), 879-902.
doi: 10.1016/0362-546X(89)90020-5. |
[14] |
S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Kluwer, Dordrecht, The Netherlands, 1997. |
[15] |
G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.
doi: 10.1016/0362-546X(88)90053-3. |
[16] |
V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory, Topol. Methods Nonlinear Anal., 10 (1997), 387-397. |
[17] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), (2011), 729-755. |
[18] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.
doi: 10.1007/978-1-4614-9323-5. |
[19] |
D. Mugnai, Addendum to Multiplicity of critical points in presence of a linking: Application to a superlinear boundary value problem, NoDEA. Nonlinear Differential Equations Appl., 11 (2004), 379-391, and a comment on the generalized Ambrosetti-Rabinowitz condition, NoDEA. Nonlinear Differential Equations Appl., 19 (2011), 299-301.
doi: 10.1007/s00030-004-2016-2. |
[20] |
D. Mugnai and N. S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), XI (2012), 729-788. |
[21] |
D. Mugnai and N. S. Papageorgiou, Wang's multiplicity result for superlinear $(p,q)$ - equations without the Ambrosetti - Rabinowitz condition, Trans. Amer. Math. Soc., 366 (2014), 4919-4937.
doi: 10.1090/S0002-9947-2013-06124-7. |
[22] |
D. Mugnai and N. S. Papageorgiou, Bifurcation for positive solutions of nonlinear diffusive logistic equations in $\mathbb{R}^N2$ with indefinite weight, Indiana Univ. Math. J., 63 (2014), 1397-1418.
doi: 10.1512/iumj.2014.63.5369. |
[23] |
N. S. Papageorgiou and V. Radulescu, Multiple solutions with precise sign information for nonlinear Robin problems, J. Differential Equations, 256 (2014), 2449-2479.
doi: 10.1016/j.jde.2014.01.010. |
[24] |
N.S. Papageorgiou and V. Radulescu, Positive solutions for nonlinear Robin eigenvalue problems, Proc. Amer. Math. Soc., to appear.
doi: 10.1090/proc/13107. |
[25] |
N. S. Papageorgiou and V. Radulescu, Nonlinear, nonhomogeneous Robin problems with superlinear reaction term, submitted. |
[26] |
N. S. Papageorgiou and P. Winkert, Resonant $(p,2) -$ equations with concave terms, Appl. Anal., 94 (2015), 342-360.
doi: 10.1080/00036811.2014.895332. |
[27] |
P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications 73, Birkhäuser Verlag, Basel, 2007. |
[28] |
S. Takeuchi, Positive solutions of a degenerate elliptic equation with a logistic reaction, Proc. Amer. Math. Soc., 129 (2001), 433-441.
doi: 10.1090/S0002-9939-00-05723-3. |
[29] |
S. Takeuchi, Multiplicity results for a degenerate elliptic equation with a logistic reaction, J. Differential Equations, 173 (2001), 138-144.
doi: 10.1006/jdeq.2000.3914. |
[30] |
P. Winkert, $L^\infty$-estimates for nonlinear elliptic Neumann boundary value problems, NoDEA. Nonlinear Differential Equations Appl., 17 (2010), 289-302.
doi: 10.1007/s00030-009-0054-5. |
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