Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential

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November 2016, 36(11): 6133-6166. doi: 10.3934/dcds.2016068

Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential

Genni Fragnelli ^1, , Dimitri Mugnai ^2, and Nikolaos S. Papageorgiou ^3,

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

Received November 2015 Revised April 2016 Published August 2016

Abstract
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We consider a parametric nonlinear Robin problem driven by the $p -$Laplacian plus an indefinite potential and a Carathéodory reaction which is $(p-1) -$ superlinear without satisfying the Ambrosetti - Rabinowitz condition. We prove a bifurcation-type result describing the dependence of the set of positive solutions on the parameter. We also prove the existence of nodal solutions. Our proofs use tools from critical point theory, Morse theory and suitable truncation techniques.

Keywords: nonlinear regularity and maximum principle, Picone identity, nodal solutions, superlinear reaction, Indefinite potential, critical groups., bifurcation-type theorem for positive solutions.

Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 58E0.

Citation: Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6133-6166. doi: 10.3934/dcds.2016068

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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