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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 & Continuous Dynamical Systems - A, 2013, 33 (1) : 123-140. doi: 10.3934/dcds.2013.33.123
References:
[1]

R. A. Adams, "Sobolev Spaces,", New York, (1975).   Google Scholar

[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.  doi: 10.1016/j.crma.2004.07.004.  Google Scholar

[3]

T. Bartsch, Z. Liu and T. Weth, Sign-changing solutions ofsuperlinear Schrödinger equations,, Comm. Partial Diff. Eq., 29 (2004), 25.  doi: 10.1081/PDE-120028842.  Google Scholar

[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.  doi: 10.1007/s002090050492.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[7]

H. Brezis and R. E. L. Turner, On a class of superlinearelliptic problems,, Comm. in Partial Diff. Equations, 2 (1977), 601.   Google Scholar

[8]

A. Castro and J. Cossio, Multiple solutions for a nonlinear Dirichlet problem,, SIAM J. Math. Anal., 25 (1994), 1554.  doi: 10.1137/S0036141092230106.  Google Scholar

[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.  doi: 10.1216/rmjm/1181071858.  Google Scholar

[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.   Google Scholar

[11]

A. Castro, J. Cossio and C. Vélez, Existence of seven solutions for an asymptotically linear Dirichlet problem,, Annali di Mat. Pura et Appl., ().   Google Scholar

[12]

A. Castro, P. Drabek and J. M. Neuberger, Asign-changing solution for a superlinear Dirichlet problem,, 101-107 (electronic), (2003), 101.   Google Scholar

[13]

K. C. Chang, "Infinite Dimensional Morse Theory and MultipleSolution Problems,", Birkhäuser, (1993).   Google Scholar

[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.  doi: 10.1007/s10884-004-6695-5.  Google Scholar

[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.  doi: 10.1016/j.na.2008.10.031.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[18]

J. Cossio and C. Vélez, Soluciones no triviales para un problema de Dirichlet asintóticamente lineal,, Rev. Colombiana Mat., 37 (2003), 25.   Google Scholar

[19]

E. N. Dancer, Counterexamples to some conjectures on the number of solutions of nonlinear equations,, Mathematische Annalen, 272 (1985), 421.  doi: 10.1007/BF01455568.  Google Scholar

[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.   Google Scholar

[21]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209.  doi: 0.1007/BF01221125.  Google Scholar

[22]

V. Guillemin and A. Pollack, "Differential Topology,", New York, (1974).   Google Scholar

[23]

D. Gilbart and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer Verlag, (1977).   Google Scholar

[24]

H. Hofer, The topological degree at a critical point of mountain pass type,, Proc. Sympos. Pure Math., 45 (1986), 501.   Google Scholar

[25]

S. Kesavan, "Nonlinear Functional Analysis (A First Course),", Text and readings in mathematics 28. Hindustan Book Agency, (2004).   Google Scholar

[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.  doi: 10.1016/0362-546X(88)90037-5.  Google Scholar

[27]

S. Liu, Remarks on multiple solutions for elliptic resonant problems,, J. Math. Anal. Appl., 336 (2007), 498.  doi: 10.1016/j.jmaa.2007.01.051.  Google Scholar

[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.  doi: 10.1016/j.jde.2004.08.023.  Google Scholar

[29]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,", Regional Conference Series in Mathematics, (1986).   Google Scholar

[30]

X. F. Yang, Nodal sets and Morse indices of solutions of super-linear elliptic PDEs,, Journal of Functional Analysis, 160 (1998), 223.  doi: 10.1006/jfan.1998.3301.  Google Scholar

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces,", New York, (1975).   Google Scholar

[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.  doi: 10.1016/j.crma.2004.07.004.  Google Scholar

[3]

T. Bartsch, Z. Liu and T. Weth, Sign-changing solutions ofsuperlinear Schrödinger equations,, Comm. Partial Diff. Eq., 29 (2004), 25.  doi: 10.1081/PDE-120028842.  Google Scholar

[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.  doi: 10.1007/s002090050492.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[7]

H. Brezis and R. E. L. Turner, On a class of superlinearelliptic problems,, Comm. in Partial Diff. Equations, 2 (1977), 601.   Google Scholar

[8]

A. Castro and J. Cossio, Multiple solutions for a nonlinear Dirichlet problem,, SIAM J. Math. Anal., 25 (1994), 1554.  doi: 10.1137/S0036141092230106.  Google Scholar

[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.  doi: 10.1216/rmjm/1181071858.  Google Scholar

[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.   Google Scholar

[11]

A. Castro, J. Cossio and C. Vélez, Existence of seven solutions for an asymptotically linear Dirichlet problem,, Annali di Mat. Pura et Appl., ().   Google Scholar

[12]

A. Castro, P. Drabek and J. M. Neuberger, Asign-changing solution for a superlinear Dirichlet problem,, 101-107 (electronic), (2003), 101.   Google Scholar

[13]

K. C. Chang, "Infinite Dimensional Morse Theory and MultipleSolution Problems,", Birkhäuser, (1993).   Google Scholar

[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.  doi: 10.1007/s10884-004-6695-5.  Google Scholar

[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.  doi: 10.1016/j.na.2008.10.031.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[18]

J. Cossio and C. Vélez, Soluciones no triviales para un problema de Dirichlet asintóticamente lineal,, Rev. Colombiana Mat., 37 (2003), 25.   Google Scholar

[19]

E. N. Dancer, Counterexamples to some conjectures on the number of solutions of nonlinear equations,, Mathematische Annalen, 272 (1985), 421.  doi: 10.1007/BF01455568.  Google Scholar

[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.   Google Scholar

[21]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209.  doi: 0.1007/BF01221125.  Google Scholar

[22]

V. Guillemin and A. Pollack, "Differential Topology,", New York, (1974).   Google Scholar

[23]

D. Gilbart and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer Verlag, (1977).   Google Scholar

[24]

H. Hofer, The topological degree at a critical point of mountain pass type,, Proc. Sympos. Pure Math., 45 (1986), 501.   Google Scholar

[25]

S. Kesavan, "Nonlinear Functional Analysis (A First Course),", Text and readings in mathematics 28. Hindustan Book Agency, (2004).   Google Scholar

[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.  doi: 10.1016/0362-546X(88)90037-5.  Google Scholar

[27]

S. Liu, Remarks on multiple solutions for elliptic resonant problems,, J. Math. Anal. Appl., 336 (2007), 498.  doi: 10.1016/j.jmaa.2007.01.051.  Google Scholar

[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.  doi: 10.1016/j.jde.2004.08.023.  Google Scholar

[29]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,", Regional Conference Series in Mathematics, (1986).   Google Scholar

[30]

X. F. Yang, Nodal sets and Morse indices of solutions of super-linear elliptic PDEs,, Journal of Functional Analysis, 160 (1998), 223.  doi: 10.1006/jfan.1998.3301.  Google Scholar

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