# American Institute of Mathematical Sciences

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

## 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

Received  November 2015 Revised  April 2016 Published  August 2016

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.
Citation: Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential. Discrete & Continuous Dynamical Systems - A, 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. Google Scholar [2] W. Allegretto and Y. X. Huang, A Picone's identity for the $p -$Laplacian and applications,, Nonlinear Anal., 32 (1998), 819. doi: 10.1016/S0362-546X(97)00530-0. Google Scholar [3] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Anal., 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [4] D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p-$Laplacian operator,, Comm. Partial Differential Equations, 31 (2006), 849. doi: 10.1080/03605300500394447. Google Scholar [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. doi: 10.1007/s00023-008-0386-4. Google Scholar [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. doi: 10.1007/s10231-012-0263-0. Google Scholar [7] J. N. Corvellec and A. Hantoute, Homotopical stability of isolated critical points of continuous functionals,, Set-Valued Analysis, 10 (2002), 143. doi: 10.1023/A:1016544301594. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jde.2008.07.004. Google Scholar [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), (). doi: 10.1007/s10231-016-0582-7. Google Scholar [11] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis,, Series in Mathematical Analysis and Applications 9, (2006). Google Scholar [12] L. Gasinski and N. S. Papageorgiou, Bifurcation - type results for nonlinear parametric elliptic equations,, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 595. doi: 10.1017/S0308210511000126. Google Scholar [13] M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents,, Nonlinear Anal., 13 (1989), 879. doi: 10.1016/0362-546X(89)90020-5. Google Scholar [14] S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis,, Kluwer, (1997). Google Scholar [15] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, Nonlinear Anal., 12 (1988), 1203. doi: 10.1016/0362-546X(88)90053-3. Google Scholar [16] V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory,, Topol. Methods Nonlinear Anal., 10 (1997), 387. Google Scholar [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. Google Scholar [18] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems,, Springer, (2014). doi: 10.1007/978-1-4614-9323-5. Google Scholar [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. doi: 10.1007/s00030-004-2016-2. Google Scholar [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. Google Scholar [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. doi: 10.1090/S0002-9947-2013-06124-7. Google Scholar [22] D. Mugnai and N. S. Papageorgiou, Bifurcation for positive solutions of nonlinear diffusive logistic equations in $\mathbbR^N$ with indefinite weight,, Indiana Univ. Math. J., 63 (2014), 1397. doi: 10.1512/iumj.2014.63.5369. Google Scholar [23] N. S. Papageorgiou and V. Radulescu, Multiple solutions with precise sign information for nonlinear Robin problems,, J. Differential Equations, 256 (2014), 2449. doi: 10.1016/j.jde.2014.01.010. Google Scholar [24] N.S. Papageorgiou and V. Radulescu, Positive solutions for nonlinear Robin eigenvalue problems,, Proc. Amer. Math. Soc., (). doi: 10.1090/proc/13107. Google Scholar [25] N. S. Papageorgiou and V. Radulescu, Nonlinear, nonhomogeneous Robin problems with superlinear reaction term,, submitted., (). Google Scholar [26] N. S. Papageorgiou and P. Winkert, Resonant $(p,2) -$ equations with concave terms,, Appl. Anal., 94 (2015), 342. doi: 10.1080/00036811.2014.895332. Google Scholar [27] P. Pucci and J. Serrin, The Maximum Principle,, Progress in Nonlinear Differential Equations and their Applications 73, (2007). Google Scholar [28] S. Takeuchi, Positive solutions of a degenerate elliptic equation with a logistic reaction,, Proc. Amer. Math. Soc., 129 (2001), 433. doi: 10.1090/S0002-9939-00-05723-3. Google Scholar [29] S. Takeuchi, Multiplicity results for a degenerate elliptic equation with a logistic reaction,, J. Differential Equations, 173 (2001), 138. doi: 10.1006/jdeq.2000.3914. Google Scholar [30] P. Winkert, $L^\infty$-estimates for nonlinear elliptic Neumann boundary value problems,, NoDEA. Nonlinear Differential Equations Appl., 17 (2010), 289. doi: 10.1007/s00030-009-0054-5. Google Scholar

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. Google Scholar [2] W. Allegretto and Y. X. Huang, A Picone's identity for the $p -$Laplacian and applications,, Nonlinear Anal., 32 (1998), 819. doi: 10.1016/S0362-546X(97)00530-0. Google Scholar [3] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Anal., 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [4] D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p-$Laplacian operator,, Comm. Partial Differential Equations, 31 (2006), 849. doi: 10.1080/03605300500394447. Google Scholar [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. doi: 10.1007/s00023-008-0386-4. Google Scholar [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. doi: 10.1007/s10231-012-0263-0. Google Scholar [7] J. N. Corvellec and A. Hantoute, Homotopical stability of isolated critical points of continuous functionals,, Set-Valued Analysis, 10 (2002), 143. doi: 10.1023/A:1016544301594. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jde.2008.07.004. Google Scholar [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), (). doi: 10.1007/s10231-016-0582-7. Google Scholar [11] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis,, Series in Mathematical Analysis and Applications 9, (2006). Google Scholar [12] L. Gasinski and N. S. Papageorgiou, Bifurcation - type results for nonlinear parametric elliptic equations,, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 595. doi: 10.1017/S0308210511000126. Google Scholar [13] M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents,, Nonlinear Anal., 13 (1989), 879. doi: 10.1016/0362-546X(89)90020-5. Google Scholar [14] S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis,, Kluwer, (1997). Google Scholar [15] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, Nonlinear Anal., 12 (1988), 1203. doi: 10.1016/0362-546X(88)90053-3. Google Scholar [16] V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory,, Topol. Methods Nonlinear Anal., 10 (1997), 387. Google Scholar [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. Google Scholar [18] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems,, Springer, (2014). doi: 10.1007/978-1-4614-9323-5. Google Scholar [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. doi: 10.1007/s00030-004-2016-2. Google Scholar [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. Google Scholar [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. doi: 10.1090/S0002-9947-2013-06124-7. Google Scholar [22] D. Mugnai and N. S. Papageorgiou, Bifurcation for positive solutions of nonlinear diffusive logistic equations in $\mathbbR^N$ with indefinite weight,, Indiana Univ. Math. J., 63 (2014), 1397. doi: 10.1512/iumj.2014.63.5369. Google Scholar [23] N. S. Papageorgiou and V. Radulescu, Multiple solutions with precise sign information for nonlinear Robin problems,, J. Differential Equations, 256 (2014), 2449. doi: 10.1016/j.jde.2014.01.010. Google Scholar [24] N.S. Papageorgiou and V. Radulescu, Positive solutions for nonlinear Robin eigenvalue problems,, Proc. Amer. Math. Soc., (). doi: 10.1090/proc/13107. Google Scholar [25] N. S. Papageorgiou and V. Radulescu, Nonlinear, nonhomogeneous Robin problems with superlinear reaction term,, submitted., (). Google Scholar [26] N. S. Papageorgiou and P. Winkert, Resonant $(p,2) -$ equations with concave terms,, Appl. Anal., 94 (2015), 342. doi: 10.1080/00036811.2014.895332. Google Scholar [27] P. Pucci and J. Serrin, The Maximum Principle,, Progress in Nonlinear Differential Equations and their Applications 73, (2007). Google Scholar [28] S. Takeuchi, Positive solutions of a degenerate elliptic equation with a logistic reaction,, Proc. Amer. Math. Soc., 129 (2001), 433. doi: 10.1090/S0002-9939-00-05723-3. Google Scholar [29] S. Takeuchi, Multiplicity results for a degenerate elliptic equation with a logistic reaction,, J. Differential Equations, 173 (2001), 138. doi: 10.1006/jdeq.2000.3914. Google Scholar [30] P. Winkert, $L^\infty$-estimates for nonlinear elliptic Neumann boundary value problems,, NoDEA. Nonlinear Differential Equations Appl., 17 (2010), 289. doi: 10.1007/s00030-009-0054-5. Google Scholar
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