Topological Methods in Nonlinear Analysis

Nodal solutions of perturbed elliptic problem

Yi Li, Zhaoli Liu, and Cunshan Zhao

Full-text: Open access


Multiple nodal solutions are obtained for the elliptic problem $$ \begin{alignat}{2} -\Delta u&=f(x, u)+\varepsilon g(x, u)&\quad& \text{in } \Omega,\\ u&=0&\quad& \text{on } \partial \Omega , \end{alignat} $$ where $\varepsilon $ is a parameter, $\Omega $ is a smooth bounded domain in ${{\mathbb R}}^{N}$, $f\in C(\overline{\Omega }\times {{\mathbb R}})$, and $g\in C(\overline{\Omega }\times {{\mathbb R}})$. For a superlinear $C^{1}$ function $f$ which is odd in $u$ and for any $C^{1}$ function $g$, we prove that for any $j\in {\mathbb N}$ there exists $\varepsilon _{j}> 0$ such that if $|\varepsilon |\leq \varepsilon _{j}$ then the above problem possesses at least $j$ distinct nodal solutions. Except $C^{1}$ continuity no further condition is needed for $g$. We also prove a similar result for a continuous sublinear function $f$ and for any continuous function $g$. Results obtained here refine earlier results of S. J. Li and Z. L. Liu in which the nodal property of the solutions was not considered.

Article information

Topol. Methods Nonlinear Anal., Volume 32, Number 1 (2008), 49-68.

First available in Project Euclid: 13 May 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Li, Yi; Liu, Zhaoli; Zhao, Cunshan. Nodal solutions of perturbed elliptic problem. Topol. Methods Nonlinear Anal. 32 (2008), no. 1, 49--68.

Export citation


  • A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications , J. Funct. Anal., 14 (1973), 349–381 \ref\key 2
  • F. V. Atkinson, H. Brézis and L. A. Peletier, Nodal solutions of elliptic equations with critical Sobolev exponents , J. Differential Equations, 85 (1990), 151–170 \ref\key 3
  • A. Bahri, Topological results on a certain class of functionals and application , J. Funct. Anal., 41 (1981), 397–427 \ref\key 4
  • A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications , Trans. Amer. Math. Soc., 267 (1981), 1–32 \ref\key 5
  • A. Bahri and P.-L. Lions, Morse index of some min-max critical points, \romI: Application to multiplicity results, Comm. Pure Appl. Math., 41 (1988), 1027–1037 \ref\key 6
  • T. Bartsch, Critical point theory on partially ordered Hilbert spaces , J. Funct. Anal., 186 (2001), 117–152 \ref\key 7
  • T. Bartsch, Z. L. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations , Comm. Partial Differential Equations, 29 (2004), 25–42 \ref\key 8
  • H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials , J. Math. Pures Appl. (9), 58 (1979), 137–151 \ref\key 9
  • A. Castro and M. Clap, The effect of the domain topology on the number of minimal nodal solutions of an elliptic equation at critical growth in a symmetric domain , Nonlinearity 16 (2003), 579–590 \ref\key 10
  • G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents , J. Funct. Anal., 69 (1986), 289–306 \ref\key 11
  • C. Chambers and N. Ghoussoub, Deformation from symmetry and multiplicity of solutions in non-homogeneous problems , Discrete Contin. Dynam. Systems, 8 (2002), 267–281 \ref\key 12
  • K. C. Chang, A variant mountain pass lemma , Sci. Sinica Ser. A, 26 (1983), 1241–1255 \ref\key 13 ––––, Variational methods and sub- and supersolutions , Sci. Sinica Ser. A, 26 (1983), 1256–1265 \ref\key 14
  • M. Degiovanni and S. Lancelotti, Perturbations of even nonsmooth functionals , Differential Integral Equations, 8 (1995), 981–992 \ref\key 15
  • M. Degiovanni and V. Rǎdulescu, Perturbations of nonsmooth symmetric nonlinear eigenvalue problems , C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 281–286 \ref\key 16
  • D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order , Reprint Edition, 224, Grundlehren der Mathematischen Wissenschaften , Springer–Verlag, Berlin (1998) \ref\key 17
  • E. Hebey and M. Vaugon, Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev growth , J. Funct. Anal., 119 (1994), 298–318 \ref\key 18
  • H. Hofer, Variational and topological methods in partially ordered Hilbert spaces , Math. Ann., 261 (1982), 493–514 \ref\key 19
  • S. J. Li and Z. L. Liu, Perturbations from symmetric elliptic boundary value problems , J. Differential Equations, 185 (2002), 271–280 \ref\key 20
  • S. J. Li and Z.-Q. Wang, Lusternik–Schnirelman theory in partially ordered Hilbert spaces , Trans. Amer. Math. Soc., 354 (2002), 3207–3227 \ref\key 21
  • Z. L. Liu, Positive solutions of superlinear elliptic equations , J. Funct. Anal. 167 (1999), 370–398 \ref\key 22
  • P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals , Trans. Amer. Math. Soc., 272 (1982), 753–769 \ref\key 23 ––––, Minimax Methods in Critical Point Theory with Applications to Differential Equations , 65, CBMS Regional Conference Series in Mathematics , American Mathematical Society, Providence (1986) \ref\key 24
  • M. Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems , Manuscripta Math., 32 (1980), 335–364 \ref\key 25
  • Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin , NoDEA Nonlinear Differential Equations Appl., 8 (2001), 15–33