Sign-changing solutions of nonlinear elliptic equations

Zhaoli Liu1, Zhi–Qiang Wang2
1School of Mathematical Sciences, Capital Normal University, Beijing, China
2Department of Mathematics and Statistics, Utah State University, Logan, USA

Tóm tắt

Từ khóa


Tài liệu tham khảo

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)

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

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

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

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

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

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

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

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

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

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

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

Bartsch T, Wang Z-Q, Willem M. The Dirichlet problem for superlinear elliptic equations. In: Chipot M, QuittnerEds P, eds. Handbook of Differential Equations: Stationary Partial Differential Equations, Vol 2. Amsterdam: Elsevier, 2005, 1–55

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

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

Bartsch T, Willem M. Infinitely many radial solutions of a semilinear elliptic problem on ℝN. Arch Rational Mech Anal, 1993, 124: 261–276

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Rabinowitz P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Conf Ser in Math, No 65. Providence: Amer Math Soc, 1986

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

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

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

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

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

Sun J. The Schauder condition in the critical point theory. Kexue Tongbao, 1986, 31: 1157–1162

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

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

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

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

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

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

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

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

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