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January  2013, 33(1): 123-140. doi: 10.3934/dcds.2013.33.123

## Existence and qualitative properties of solutions for nonlinear Dirichlet problems

 1 Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, United States 2 Escuela de Matemáticas, Universidad Nacional de Colombia, Sede Medellĺn, Apartado Aéreo 3840, Medellín, Colombia 3 Escuela de Matemáticas, Universidad Nacional de Colombia, Sede Medellín, Apartado Aéreo 3840, Medellín, Colombia

Received  May 2011 Revised  October 2011 Published  September 2012

sign-changing solutions to semilinear elliptic problems in connection with their Morse indices. To this end, we first establish a priori bounds for one-sign solutions. Secondly, using abstract saddle point principles we find large augmented Morse index solutions. In this part, extensive use is made of critical groups, Morse index arguments, Lyapunov-Schmidt reduction, and Leray-Schauder degree. Finally, we provide conditions under which these solutions necessarily change sign and we comment about further qualitative properties.
Citation: Alfonso Castro, Jorge Cossio, Carlos Vélez. Existence and qualitative properties of solutions for nonlinear Dirichlet problems. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 123-140. doi: 10.3934/dcds.2013.33.123
##### References:
 [1] R. A. Adams, "Sobolev Spaces," New York, AcademicPress, 1975. [2] A. Aftalion and F. Pacella, Qualitative propertiesof nodal solutions of semilinear elliptic equations in radiallysymmetric domains, C. R. Math. Acad. Sci. Paris, 339 (2004), 339-344. doi: 10.1016/j.crma.2004.07.004. [3] T. Bartsch, Z. Liu and T. Weth, Sign-changing solutions ofsuperlinear Schrödinger equations, Comm. Partial Diff. Eq., 29 (2004), 25-42. doi: 10.1081/PDE-120028842. [4] T. Bartsch, K. C. Chang and Z. Q. Wang, On the Morse indices of sign changing solutions of nonlinear ellipticproblems, Math. Z., 233 (2000), 655-677. doi: 10.1007/s002090050492. [5] T. Bartsch and Z. Q. Wang, On the existence of sign changing solutions for semilinear Dirichlet problems, Topol. Methods Nonlinear Anal., 7 (1996), 115-131. [6] T. Bartsch and T. Weth, A note on additional properties ofsign-changing solutions to superlinear elliptic equations, Topol. Methods Nonlinear Anal., 22 (2003), 1-14. [7] H. Brezis and R. E. L. Turner, On a class of superlinearelliptic problems, Comm. in Partial Diff. Equations, 2 (1977), 601-614. [8] A. Castro and J. Cossio, Multiple solutions for a nonlinear Dirichlet problem, SIAM J. Math. Anal., 25 (1994), 1554-1561. doi: 10.1137/S0036141092230106. [9] A. Castro, J. Cossio, and J. M. Neuberger, A sign changing solution for a superlinear Dirichlet problem, Rocky Mountain J. M., 27 (1997), 1041-1053. doi: 10.1216/rmjm/1181071858. [10] A. Castro, J. Cossio and J. M. Neuberger, A minmax principle,index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems, Electronic Journal Diff. Eqns., 1998 (1998), 1-18. [11] A. Castro, J. Cossio and C. Vélez, Existence of seven solutions for an asymptotically linear Dirichlet problem, Annali di Mat. Pura et Appl., to appear. [12] A. Castro, P. Drabek and J. M. Neuberger, Asign-changing solution for a superlinear Dirichlet problem, 101-107 (electronic), Electron. J. Diff. Eqns., Conf. 10, Southwest Texas State Univ., San Marcos, TX, 2003. [13] K. C. Chang, "Infinite Dimensional Morse Theory and MultipleSolution Problems," Birkhäuser, Boston, 1993. [14] J. Cossio and S. Herrón, Nontrivial Solutions for asemilinear Dirichlet problem with nonlinearity crossing multipleeigenvalues, Journal of Dynamics and Differential Equations, 16 (2004), 795-803. doi: 10.1007/s10884-004-6695-5. [15] J. Cossio, S. Herrón and C. Vélez, Existence of solutions for an asymptotically linearDirichlet problem via Lazer-Solimini results, Nonlinear Anal., 71 (2009), 66-71. doi: 10.1016/j.na.2008.10.031. [16] A. Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. Pura Appl., 70 (1979), 113-137. [17] K. Chang, S. Li and J. Liu, Remarks on multiple solutions for asymptotically linear elliptic boundary value problems, Topol. Methods in Nonlinear Anal., 3 (1994), 179-187. [18] J. Cossio and C. Vélez, Soluciones no triviales para un problema de Dirichlet asintóticamente lineal, Rev. Colombiana Mat., 37 (2003), 25-36. [19] E. N. Dancer, Counterexamples to some conjectures on the number of solutions of nonlinear equations, Mathematische Annalen, 272 (1985), 421-440. doi: 10.1007/BF01455568. [20] D. De Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures et Appl., 61 (1982), 41-63. [21] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 0.1007/BF01221125. [22] V. Guillemin and A. Pollack, "Differential Topology," New York, NY, Prentice-Hall, 1974. [23] D. Gilbart and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer Verlag, Berlin 1977. [24] H. Hofer, The topological degree at a critical point of mountain pass type, Proc. Sympos. Pure Math., 45 (1986), 501-509. [25] S. Kesavan, "Nonlinear Functional Analysis (A First Course)," Text and readings in mathematics 28. Hindustan Book Agency, India, 2004. [26] A. C. Lazer and S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, Nonlinear Anal., 12 (1988), 761-775. doi: 10.1016/0362-546X(88)90037-5. [27] S. Liu, Remarks on multiple solutions for elliptic resonant problems, J. Math. Anal. Appl., 336 (2007), 498-505. doi: 10.1016/j.jmaa.2007.01.051. [28] Z. Liu, F. A. van Heerden, and Z. Q. Wang, Nodal type bound states of Schrödinger equations via invariant sets and minimax methods, Journal of Differential Equations, 214 (2005), 358-390. doi: 10.1016/j.jde.2004.08.023. [29] P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," Regional Conference Series in Mathematics, number 65, AMS, Providence, R.I., 1986. [30] X. F. Yang, Nodal sets and Morse indices of solutions of super-linear elliptic PDEs, Journal of Functional Analysis, 160 (1998), 223-253. doi: 10.1006/jfan.1998.3301.

show all references

##### References:
 [1] R. A. Adams, "Sobolev Spaces," New York, AcademicPress, 1975. [2] A. Aftalion and F. Pacella, Qualitative propertiesof nodal solutions of semilinear elliptic equations in radiallysymmetric domains, C. R. Math. Acad. Sci. Paris, 339 (2004), 339-344. doi: 10.1016/j.crma.2004.07.004. [3] T. Bartsch, Z. Liu and T. Weth, Sign-changing solutions ofsuperlinear Schrödinger equations, Comm. Partial Diff. Eq., 29 (2004), 25-42. doi: 10.1081/PDE-120028842. [4] T. Bartsch, K. C. Chang and Z. Q. Wang, On the Morse indices of sign changing solutions of nonlinear ellipticproblems, Math. Z., 233 (2000), 655-677. doi: 10.1007/s002090050492. [5] T. Bartsch and Z. Q. Wang, On the existence of sign changing solutions for semilinear Dirichlet problems, Topol. Methods Nonlinear Anal., 7 (1996), 115-131. [6] T. Bartsch and T. Weth, A note on additional properties ofsign-changing solutions to superlinear elliptic equations, Topol. Methods Nonlinear Anal., 22 (2003), 1-14. [7] H. Brezis and R. E. L. Turner, On a class of superlinearelliptic problems, Comm. in Partial Diff. Equations, 2 (1977), 601-614. [8] A. Castro and J. Cossio, Multiple solutions for a nonlinear Dirichlet problem, SIAM J. Math. Anal., 25 (1994), 1554-1561. doi: 10.1137/S0036141092230106. [9] A. Castro, J. Cossio, and J. M. Neuberger, A sign changing solution for a superlinear Dirichlet problem, Rocky Mountain J. M., 27 (1997), 1041-1053. doi: 10.1216/rmjm/1181071858. [10] A. Castro, J. Cossio and J. M. Neuberger, A minmax principle,index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems, Electronic Journal Diff. Eqns., 1998 (1998), 1-18. [11] A. Castro, J. Cossio and C. Vélez, Existence of seven solutions for an asymptotically linear Dirichlet problem, Annali di Mat. Pura et Appl., to appear. [12] A. Castro, P. Drabek and J. M. Neuberger, Asign-changing solution for a superlinear Dirichlet problem, 101-107 (electronic), Electron. J. Diff. Eqns., Conf. 10, Southwest Texas State Univ., San Marcos, TX, 2003. [13] K. C. Chang, "Infinite Dimensional Morse Theory and MultipleSolution Problems," Birkhäuser, Boston, 1993. [14] J. Cossio and S. Herrón, Nontrivial Solutions for asemilinear Dirichlet problem with nonlinearity crossing multipleeigenvalues, Journal of Dynamics and Differential Equations, 16 (2004), 795-803. doi: 10.1007/s10884-004-6695-5. [15] J. Cossio, S. Herrón and C. Vélez, Existence of solutions for an asymptotically linearDirichlet problem via Lazer-Solimini results, Nonlinear Anal., 71 (2009), 66-71. doi: 10.1016/j.na.2008.10.031. [16] A. Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. Pura Appl., 70 (1979), 113-137. [17] K. Chang, S. Li and J. Liu, Remarks on multiple solutions for asymptotically linear elliptic boundary value problems, Topol. Methods in Nonlinear Anal., 3 (1994), 179-187. [18] J. Cossio and C. Vélez, Soluciones no triviales para un problema de Dirichlet asintóticamente lineal, Rev. Colombiana Mat., 37 (2003), 25-36. [19] E. N. Dancer, Counterexamples to some conjectures on the number of solutions of nonlinear equations, Mathematische Annalen, 272 (1985), 421-440. doi: 10.1007/BF01455568. [20] D. De Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures et Appl., 61 (1982), 41-63. [21] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 0.1007/BF01221125. [22] V. Guillemin and A. Pollack, "Differential Topology," New York, NY, Prentice-Hall, 1974. [23] D. Gilbart and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer Verlag, Berlin 1977. [24] H. Hofer, The topological degree at a critical point of mountain pass type, Proc. Sympos. Pure Math., 45 (1986), 501-509. [25] S. Kesavan, "Nonlinear Functional Analysis (A First Course)," Text and readings in mathematics 28. Hindustan Book Agency, India, 2004. [26] A. C. Lazer and S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, Nonlinear Anal., 12 (1988), 761-775. doi: 10.1016/0362-546X(88)90037-5. [27] S. Liu, Remarks on multiple solutions for elliptic resonant problems, J. Math. Anal. Appl., 336 (2007), 498-505. doi: 10.1016/j.jmaa.2007.01.051. [28] Z. Liu, F. A. van Heerden, and Z. Q. Wang, Nodal type bound states of Schrödinger equations via invariant sets and minimax methods, Journal of Differential Equations, 214 (2005), 358-390. doi: 10.1016/j.jde.2004.08.023. [29] P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," Regional Conference Series in Mathematics, number 65, AMS, Providence, R.I., 1986. [30] X. F. Yang, Nodal sets and Morse indices of solutions of super-linear elliptic PDEs, Journal of Functional Analysis, 160 (1998), 223-253. doi: 10.1006/jfan.1998.3301.
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