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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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