Sign-changing solutions of nonlinear elliptic equations

Abstract

In this survey article, we recall some known results on existence and multiplicity of sign-changing solutions of elliptic equations. Methods for obtaining sign-changing solutions developed in the last two decades will also be briefly revisited.

Keywords

Elliptic equation / invariant set / sign-changing solution

References

[1] Ackermann N, Bartsch T, Kaplický P, et al. Priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems. Trans Amer Math Soc (in press)

[2] Aftalion A., Pacella F. Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains. C R Math Acad Sci Paris, 2004, 339: 339-344.

[3] Amann H. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev, 1976, 18: 620-709.

[4] Ambrosetti A., Rabinowitz P. H. Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14: 349-381.

[5] Bartsch T. Critical point theory on partially ordered Hilbert spaces. J Funct Anal, 2001, 186: 117-152.

[6] Bartsch T., Chang K.-C., Wang Z.-Q. On the Morse indices of sign changing solutions of nonlinear elliptic problems. Math Z, 2000, 233: 655-677.

[7] Bartsch T., Liu Z. Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the p-Laplacian. Comm Contemp Math, 2004, 6: 245-258.

[8] Bartsch T., Liu Z. On a superlinear elliptic p-Laplacian equation. J Differential Equations, 2004, 198: 149-175.

[9] Bartsch T., Liu Z., Weth T. Sign changing solutions of superlinear Schrödinger equations. Comm Partial Differential Equations, 2004, 29: 25-42.

[10] Bartsch T., Liu Z., Weth T. Nodal solutions of a p-Laplacian equation. Proc London Math Soc, 2005, 91: 129-152.

[11] Bartsch T., Wang Z.-Q. On the existence of sign changing solutions for semilinear Dirichlet problems. Topol Methods Nonlinear Anal, 1996, 7: 115-131.

[12] Bartsch T., Wang Z.-Q. Sign changing solutions of nonlinear Schrödinger equations. Topol Methods Nonlinear Anal, 1999, 13: 191-198.

[13] Bartsch T., Wang Z.-Q., Willem M. Chipot M., QuittnerEds P. The Dirichlet problem for superlinear elliptic equations. Handbook of Differential Equations: Stationary Partial Differential Equations, 2005, Amsterdam: Elsevier, 1-55.

[14] Bartsch T., Weth T. Three nodal solutions of singularly perturbed elliptic equations on domains without topology. Ann Inst H Poincaré Anal Non Linéaire, 2005, 22: 259-281.

[15] Bartsch T., Willem M. Infinitely many nonradial solutions of a Euclidean scalar field equation. J Funct Anal, 1993, 117: 447-460.

[16] Bartsch T., Willem M. Infinitely many radial solutions of a semilinear elliptic problem on ℝ^N. rch Rational Mech Anal, 1993, 124: 261-276.

[17] Brezis H., Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm Pure Appl Math, 1983, 36: 437-477.

[18] Cao D., Noussair E. S. Multiple positive and nodal solutions for semilinear elliptic problems with critical exponents. Indiana Univ Math J, 1995, 44: 1249-1271.

[19] Cao D., Peng S. A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms. J Differential Equations, 2003, 193: 424-434.

[20] Castro A., Cossio J., Neuberger J. M. A sign-changing solution for a superlinear Dirichlet problem. Rocky Mountain J Math, 1997, 27: 1041-1053.

[21] Cerami G., Solimini S., Struwe M. Some existence results for superlinear elliptic boundary value problems involving critical exponents. J Funct Anal, 1986, 69: 289-306.

[22] Chang K.-C. A variant mountain pass lemma. Sci Sinica, Ser A, 1983, 26: 1241-1255.

[23] Chang K.-C. Infinite-dimensional Morse Theory and Multiple Solution Problems. Progress in Nonlinear Differential Equations and Their Applications, No 6, 1993, Boston: Birkhäuser.

[24] Chang K.-C. Morse theory in nonlinear analysis. Nonlinear Functional Analysis and Applications to Differential Equations (Trieste, 1997), 1998, River Edge: World Sci Publ, 60-101.

[25] Chang K.-C. Heat method in nonlinear elliptic equations. Topological Methods, Variational Methods and Their Applications (Taiyuan, 2002), 2003, River Edge: World Sci Publ, 65-76.

[26] Chang K.-C., Jiang M. Dirichlet problem with indefinite nonlinearities. Calc Var Partial Differential Equations, 2004, 20: 257-282.

[27] Conti M., Merizzi L., Terracini S. Remarks on variational methods and lower-upper solutions. Nonlinear Differential Equations Appl, 1999, 6: 371-393.

[28] Conti M., Terracini S., Verzini G. Nehari’s problem and competing species systems. Ann Inst H Poincaré Anal Non Linéaire, 2002, 19: 871-888.

[29] Costa D., Wang Z.-Q. Multiplicity results for a class of superlinear elliptic problems. Proc Amer Math Soc, 2005, 133: 787-794.

[30] Dancer E., Du Y. Competing species equations with diffusion, large interaction, and jumping nonlinearities. J Differential Equations, 1994, 114: 434-475.

[31] Dancer E., Du Y. On sign-changing solutions of certain semilinear elliptic problems. Appl Anal, 1995, 56: 193-206.

[32] Dancer E., Du Y. Multiple solutions of some semilinear elliptic equations via the generalied Conley index. J Math Anal Appl, 1995, 189: 848-871.

[33] Dancer E., Du Y. A note on multiple solutions of some semilinear elliptic problems. J Math Anal Appl, 1997, 211: 626-640.

[34] Dancer E., Wei J. Sign-changing solutions for supercritical elliptic problems in domains with small holes. Manuscripta Math, 2007, 123: 493-511.

[35] Hofer H. Variational and topological methods in partially ordered Hilbert spaces. Math Ann, 1982, 261: 493-514.

[36] Jiang M. Critical groups and multiple solutions of the p-Laplacian equations. Nonlinear Anal, 2004, 59: 1221-1241.

[37] Li C., Li S. Multiple solutions and sign-changing solutions of a class of nonlinear elliptic equations with Neumann boundary condition. J Math Anal Appl, 2004, 298: 14-32.

[38] Li S., Wang Z.-Q. Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems. J Anal Math, 2000, 81: 373-396.

[39] Li S., Wang Z.-Q. Ljusternik-Schnirelman theory in partially ordered Hilbert spaces. Trans Amer Math Soc, 2002, 354: 3207-3227.

[40] Li S., Zhang Z. Sign-changing solution and multiple solutions theorems for semilinear elliptic boundary value problems with jumping nonlinearities. Acta Math Sinica, 2001, 44: 507-516.

[41] Li Y., Liu Z. Multiple and sign changing solutions of an elliptic eigenvalue problem with constraint. Sci China, Ser A, 2001, 44: 48-57.

[42] Liu J., Wang Y., Wang Z.-Q. Solutions for quasilinear Schrödinger equations via the Nehari method. Comm Partial Differential Equations, 2004, 29: 879-901.

[43] Liu Z. Multiple Solutions of Differential Equations. Ph D Thesis, 1992, Jinan: Shandong Univ.

[44] Liu Z., Li Y. Solutions of an elliptic eigenvalue problem involving subcritical or critical exponents. Comm Partial Differential Equations, 2001, 26: 2227-2248.

[45] Liu Z., Sun J. Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations. J Differential Equations, 2001, 172: 257-299.

[46] Liu Z., Sun J. Brezis H., Li S. J., Liu J. Q. Number of invariant sets of descending flow with applications in critical point theory. Morse Theory, Minimax Theory and Their Applications to Nonlinear Differential Equations, 2003, Boston: Int Press, 139-156.

[47] Liu Z., Sun J. Four versus two solutions of semilinear elliptic boundary value problems. Calc Var Partial Differential Equations, 2002, 14: 319-327.

[48] Liu Z., van Heerden F. A., Wang Z.-Q. Nodal type bound states of Schrödinger equations via invariant set and minimax methods. J Differential Equations, 2005, 214: 358-390.

[49] Liu Z., Wang Z.-Q. On the Ambrosetti-Rabinowitz superlinear condition. Adv Nonlinear Stud, 2004, 4: 563-574.

[50] Liu Z., Wang Z.-Q. Schrödinger equations with concave and convex nonlinearities. Z Angew Math Phys, 2005, 56: 609-629.

[51] Liu Z., Wang Z.-Q. Multi-bump type nodal solutions having a prescribed number of nodal domains. I. Ann Inst H Poincaré Anal Non Linéaire, 2005, 22: 597-608.

[52] Liu Z., Wang Z.-Q. Multi-bump type nodal solutions having a prescribed number of nodal domains. II. Ann Inst H Poincaré Anal Non Linéaire, 2005, 22: 609-631.

[53] Liu Z., Wang Z.-Q., Weth T. Multiple solutions of nonlinear Schrödinger equations via flow invariance and Morse theory. Proc Roy Soc Edinburgh Sect A, 2006, 136: 945-969.

[54] Rabinowitz P. H. Minimax Methods in Critical Point Theory with Applications to Differential Equations, 1986, Providence: Amer Math Soc.

[55] Rabinowitz P. H., Su J., Wang Z.-Q. Multiple solutions of a superlinear elliptic equation. Rend Lincei di Matematica, 2007, 18: 97-108.

[56] Schecheter M., Wang Z.-Q., Zou W. New linking theorem and sign-changing solutions. Comm Partial Differential Equations, 2004, 29: 471-488.

[57] Schechter M., Zou W. Infinitely many solutions to perturbed elliptic equations. J Funct Anal, 2005, 228: 1-38.

[58] Schechter M., Zou W. Sign-changing critical points from linking type theorems. Trans Amer Math Soc, 2006, 358: 5293-5318.

[59] Sun J. Topics on Nonlinear Operators. Ph D Thesis, 1984, Jinan: Shandong Univ.

[60] Sun J. The Schauder condition in the critical point theory. Kexue Tongbao, 1986, 31: 1157-1162.

[61] Sun J., Liu Z. Calculus of variations and super-and sub-solutions in reverse order. Acta Math Sinica, 1994, 37: 512-514.

[62] Wang Z.-Q. On a superlinear elliptic equation. Ann Inst H Poincaré Anal Non Linéaire, 1991, 8: 43-57.

[63] Wang Z.-Q. Nonlinear boundary value problems with concave nonlinearities near the origin. Nonlinear Differential Equations Appl, 2001, 8: 15-33.

[64] Wang Z.-Q. Minimax methods, invariant sets, and applications to nodal solutions of nonlinear elliptic problems. Proceedings of EquaDiff 03, Hasselt 2003, 2005, Singapore: World Scientific, 561-566.

[65] Wang Z.-Q., Zhou J. A local minimax-Newton method for finding multiple saddle points with symmetries. SIAM J Numer Anal, 2004, 42: 1745-1759.

[66] Wang Z.-Q., Zhou J. An efficient and stable method for computing multiple saddle points with symmetries. SIAM J Numer Anal, 2005, 43: 891-907.

[67] Zhang Z., Li S. On sign-changing and multiple solutions of the p-Laplacian. J Funct Anal, 2003, 197: 447-468.

[68] Zou W. Sign-changing saddle point. J Funct Anal, 2005, 219: 433-468.

[69] Zou W. On finding sign-changing solutions. J Funct Anal, 2006, 234: 364-419.