Topological Methods in Nonlinear Analysis

Existence and multiplicity of nontrivial solutions for semilinear elliptic Dirichlet problems across resonance

Xiaojun Chang and Yong Li

Full-text: Open access


With the linear growth of the nonlinearity and a new compactness condition involving the asymptotic behavior of its potential at infinity, we establish the existence and multiplicity results of nontrivial solutions for semilinear elliptic Dirichlet problems. The nonlinearity may cross multiple eigenvalues.

Article information

Topol. Methods Nonlinear Anal., Volume 36, Number 2 (2010), 285-310.

First available in Project Euclid: 21 April 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Chang, Xiaojun; Li, Yong. Existence and multiplicity of nontrivial solutions for semilinear elliptic Dirichlet problems across resonance. Topol. Methods Nonlinear Anal. 36 (2010), no. 2, 285--310.

Export citation


  • \ref S. Ahmad, A. C. Lazer and J. L. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance. , Indiana Univ. Math. J., 25 (1976), 933–944
  • \ref\no2 A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications , J. Func. Anal., 14 (1973), 349–381
  • \ref\no3 H. Amann, A note on degree theory for gradient mappings , Proc. Amer. Math. Soc., 85 (1982), 591–595
  • \ref\no4 H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations , Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (1980), 539–603
  • \ref D. Arcoya and D. G. Costa, Nontrivial solutions for strongly resonant problem , Differential Integral Equations, 8 (1995), 151–159
  • \ref\no6 T. Bartsch, K. C. Chang and Z.-Q. Wang, On the Morse index of sign changing solutions of nonlinear elliptic problems , Math. Z., 233 (2000), 655–677
  • \ref\no7V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals , Invent. Math., 52 (1979), 241–273
  • \ref\no8H. Brezis and L. Nirenberg, Remarks on finding critical points , Comm. Pure. Appl. Math., 44 (1991), 939–963
  • \ref\no9J. Cossio and S. Herrón, Nontrivial solutions for a semilinear Dirichlet problem with nonlinearity crossing multiple eigenvalues , J. Dynam. Differential Equations, 16 (2004), 795–803
  • \ref\no10A. Castro and J. Cossio, Multiple solutions for a nonlinear Dirichlet problem , SIAM J. Math. Anal., 25 (1994), 1554–1561
  • \ref\no11A. Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem , Ann. Mat. Pura Appl., 120 (1979), 113–137
  • \ref\no12D. G. Costa and C. A. Magalhães, Variational elliptic problems which are nonquadratic at infinity , Nonlinear Anal., 23 (1994), 1401–1412 \ref\no13 ––––, A unified approach to a class of strongly indefinite functionals , J. Differential Equations, 125 (1996), 521–547
  • \ref\no14 K. C. Chang, Morse Theory in Nonlinear Analysis, Proceedings of the Symposium on Nonlinear Functional Analysis and Applications (Trieste, 1997), World Scientific Pub., River Edge, NJ,(1998), 60–101
  • \ref\no15 D. G. Costa and A. S. Oliveira, Existence of solution for a class of semilinear elliptic problems at double resonance , Bol. Soc. Bras. Mat., 19 (1988), 21–37
  • \ref\no16 J. Cossio and C. Vélez, Soluciones notriviales para un problema de Dirichlet asintóticamente lineal , Rev. Colombiana Mat., 37 (2003), 25–36 \ref\no17 E. N. Dancer and Y. Du, A note on multiple solutions of some semilinear elliptic problems , J. Math. Anal. Appl., 211 (1997), 626–640
  • \ref\no18 F. O. V. De Paiva, Multiple solutions for asymptotically linear resonant elliptic problems , Topol. Methods Nonlinear Anal., 21 (2003), 227–247
  • \ref\no19 D. Del Santo and P. Omari, Nonresonance conditions on the potential for a semilinear elliptic problem , J. Differential Equations, 108 (1994), 120–138
  • \ref\no20 H. Hofer, A note on the topological degree at a critical point of mountain pass type , Proc. Amer. Math. Soc., 90 (1984), 309–315
  • \ref\no21 E. Landesman and A. C. Lazer, Nonlinear perturbations of linear eigenvalues problem at resonance , J. Math. Mech., 19 (1970), 609–623
  • \ref\no22 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
  • \ref\no23 S. J. Li and M. Willem, Applications of local linking to critical point theory , J. Math. Anal. Appl., 189 (1995), 6–32 \ref\no24 ––––, Multiple solutions for asymptotically linear boundary value problems in which the nonlinearity crosses at least one eivenvalue , NoDEA Nonlinear Differential Equations Appl., 5 (1998), 479–490
  • \ref\no25 S. J. Li and Z. T. Zhang, Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity , Discrete Contin. Dynam. Systems, 5 (1999), 489–493
  • \ref\no26 G. B. Li and H. S. Zhou, Multiple solutions to $p$-Laplacian problems with asymptotic nonlinearity as $u^{p-1}$ at infinity , J. London Math. Soc., 65 (2002), 123–138
  • \ref\no27 S. B. Liu and S. J. Li, Critical groups at infinity, saddle point reduction and elliptic resonant problems , Commun. Contemp. Math., 5 (2003), 761–773
  • \ref\no28 J. Mawhin and J. R. Ward Jr., Nonresonance and existence for nonlinear elliptic boundary value problems , Nonlinear Anal., 5 (1981), 677–684
  • \ref\no29 J. Mawhin, J. R. Ward Jr. and M. Willem, Variational methods and semilinear elliptic equations , Arch. Rational Mech. Anal., 95 (1986), 269–277
  • \ref\no30 K. Perera, Multiplicity results for some elliptic problems with concave nonlinearities , J. Differential Equations, 140 (1997), 134–141
  • \ref \no31P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, AMS Conf. Ser. Math., 65 (1986)
  • \ref\no32 C. A. Stuart, Self-trapping of an electromagnetic field and bifurcation from the essential spectrum , Arch. Rational Mech. Anal., 113 (1991), 65–96
  • \ref\no33 C. A. Stuart, Magnetic field wave equations for TM-modes in nonlinear optical wave guides (Caristi and Mitidieri, Reaction DiTusion Systems, eds.), Marcel Dekker, New York (1997)
  • \ref\no34 Z. T. Zhang, S. J. Li, S. B. Liu and W. J. Feng, On an asymptotically linear elliptic Dirichlet problem , Abstr. Appl. Anal., 7 (2002), 509–516
  • \ref\no35W. M. Zou and J. Q. Liu, Multiple solutions for resonant elliptic equations via local linking theory and Morse theory , J. Differential Equations, 170 (2001), 68–95
  • \ref\no36 H. S. Zhou, Existence of asymptotically linear Dirichlet problem , Nonlinear Anal., 44 (2001), 909–918 \ref\no37 ––––, An application of a mountain pass theorem , Acta Math. Sinica (Engl. Ser.), 18 (2002), 27–36