Topological Methods in Nonlinear Analysis

Multiplicity of solutions for some elliptic equations involving critical and supercritical Sobolev exponents

Shujie Li and Zhaoli Liu

Full-text: Open access


We study multiplicity of solutions of the following elliptic problems in which critical and supercritical Sobolev exponents are involved: $$ \begin{alignat}{2} -\Delta u& =g(x, u)+\lambda h(x, u) &\quad& \text{in } \Omega \text{ and } u=0 \text{ on } \partial\Omega, \\ -{\rm div}(|\nabla u|^{p-2}\nabla u)&=g(x, u)+\lambda h(x, u) &\quad& \text{in } \Omega \text{ and } u=0 \text{ on } \partial\Omega, \end{alignat} $$ where $\Omega$ is a smooth bounded domain in ${\mathbb R}^N$, $p> 1$, $\lambda$ is a parameter, and $\lambda h(x, u)$ is regarded as a perturbation term of the problems. Except oddness with respect to $u$ in some cases, we do not assume any condition on $h$. For the first problem, we get a result on existence of three nontrivial solutions for $|\lambda|$ small in the case where $g$ is superlinear and $\limsup_{|t| \to\infty}g(x, t)/|t|^{2^*-1}$ is suitably small. We also prove that the first problem has $2k$ distinct solutions for $|\lambda|$ small when $g$ and $h$ are odd and there are $k$ eigenvalues between $\lim_{t\to0}g(x, t)/t$ and $\lim_{|t|\to\infty}g(x, t)/t$. For the second problem, we prove that it has more and more distinct solutions as $\lambda$ tends to $0$ assuming that $g$ and $h$ are odd and $g$ is superlinear and $\lim_{|t| \to\infty}g(x, t)/|t|^{p^*-1}=0$.

Article information

Topol. Methods Nonlinear Anal., Volume 28, Number 2 (2006), 235-261.

First available in Project Euclid: 13 May 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Li, Shujie; Liu, Zhaoli. Multiplicity of solutions for some elliptic equations involving critical and supercritical Sobolev exponents. Topol. Methods Nonlinear Anal. 28 (2006), no. 2, 235--261.

Export citation


  • Adimurthi and S. L. Yadava, Nonexistence of positive radial solutions of a quasilinear Neumann problem with a critical Sobolev exponent , Arch. Rational Mech. Anal., 139 (1997), 239–253 \ref\key 2
  • A. Ambrosetti, H. Brézis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems , J. Funct. Anal., 122 (1994), 519–543 \ref\key 3
  • A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications , J. Funct. Anal., 14 (1973), 349–381 \ref\key 4
  • F. V. Atkinson and L. A. Peletier, Oscillation of solutions of perturbed autonomous equations with an application to nonlinear elliptic eigenvalue problems involving critical Sobolev exponents , Differential Integral Equations, 3 (1990), 401–433 \ref\key 5
  • A. Bahri, Topological results on a certain class of functionals and applications , J. Funct. Anal., 41 (1981), 397–427 \ref\key 6
  • A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications , Trans. Amer. Math. Soc., 267 (1981), 1–32 \ref\key 7
  • A. Bahri and P. L. Lions, Morse-index of some min-max critical points , Comm. Pure Appl. Math., 41 (1988), 1027–1037 \ref\key 8
  • T. Bartsch, K. C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems , Math. Z., 233 (2000), 655–677 \ref\key 9
  • T. Bartsch and Z.-Q. Wang, On the existence of sign changing solutions for semilinear Dirichlet problem , Topol. Methods Nonlinear Anal., 7 (1996), 115–131 \ref\key 10
  • T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities , Proc. Amer. Math. Soc., 123 (1995), 3555–3561 \ref\key 11
  • E. di Benedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations , Nonlinear Anal., 7 (1983), 827–850 \ref\key 12
  • H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I: Existence of a ground state, II: Existence of infinitely many solutions , Arch. Rational Mech. Anal., 82 (1983), 313–345, 347–375 \ref\key 13
  • H. Brézis, Some variational problems with lack of compactness , Nonlinear Functional Analysis and its Applications (F. E. Browder, ed.), Proceeding of Symposia in Pure Mathematics, 45 (1983), 165–201 \ref\key 14
  • H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials , J. Math. Pure Appl., 58 (1979), 137–151 \ref\key 15
  • H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents , Comm. Pure Appl. Math., 36 (1983), 437–477 \ref\key 16
  • A. Castro, J. Cossio and M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem , Rocky Mountain J. Math., 27 (1997), 1041–1053 \ref\key 17
  • J. Chabrowski and J. F. Yang, Existence theorems for elliptic equations involving supercritical Sobolev exponent , Adv. Differential Equations, 2 (1997), 231–256 \ref\key 18
  • K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston (1993) \ref\key 19
  • F.-C. St. Cirstea and V. D. Radulescu, On a double bifurcation quasilinear problem arising in the study of anisotropic continuous media , Proc. Edinburgh Math. Soc., 44 (2001), 527–548 \ref\key 20
  • D. C. Clark, A variant of Lusternik–Schnirelman theory , Indiana Univ. Math. J., 22 (1972), 65–74 \ref\key 21
  • E. N. Dancer and Y. Du, Existence of changing sign solutions for some semilinear problems with jumping nonlinearities at zero , Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 1165–1176 \ref\key 22 ––––, Multiple solutions of some semilinear elliptic equations via the generalized Conley index , J. Math. Anal. Appl., 189 (1995), 848–871 \ref\key 23
  • G. C. Dong and S. J. Li, On the existence of infinitely many solutions of the Dirichlet problem for some nonlinear elliptic equations , Sci. Sinica Ser. A, 25 (1982), 468–475 \ref\key 24
  • Y. Du, Exact multiplicity and $S$-shaped bifurcation curve for some semilinear elliptic problems from combustion theory , SIAM J. Math. Anal., 32 (2000), 707–733 \ref\key 25
  • P. Drábek, Nonlinear eigenvalue problems for $p$-Laplacian in ${\Bbb R}^N$ , Math. Nachr., 173 (1995), 131–139 \ref\key 26
  • P. Drábek and C. Simader, Nonlinear eigenvalue problems for quasilinear equations in unbounded domains , Math. Nachr., 203 (1999), 5–30 \ref\key 27
  • D. G. de Figueiredo and P. L. Lions, On pairs of positive solutions for a class of semilinear elliptic problems , Indiana Univ. Math. J., 34 (1985), 591–606 \ref\key 28
  • D. G. de Figueiredo, P. L. Lions, and R. D. Nussbaum, A priori estimates and existence results for positive solutions of semilinear elliptic equations , J. Math. Pure Appl., 61 (1982), 41–63 \ref\key 29
  • J. Garcia Azorero and I. Peral Alonso, Existence and nonuniqueness for the $p$-Laplacian: nonlinear eigenvalues , Comm. Partial Differential Equations, 12 (1987), 1389–1430 \ref\key 30 ––––, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term , Trans. Amer. Math. Soc., 323 (1991), 877–895 \ref\key 31
  • D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, second edition, Spinger–Verlag, Berlin (1998) \ref\key 32
  • Z. Guo and J. R. L. Webb, Large and small solutions of a class of quasilinear elliptic eigenvalue problems , J. Differential Equations, 180 (2002), 1–50 \ref\key 33
  • S. J. Li and Z.-Q. Wang, Mountain pass theorem in order interval and multiple solutions for semilinear elliptic Dirichlet problems , J. Anal. Math., 81 (2000), 373–396 \ref\key 34
  • S. J. Li and Z. T. Zhang, Sign-changing and multiple solutions theorems for semilinear elliptic boundary value problems with jumping nonlinearities , Acta Math. Sinica Engl. Ser., 16 (2000), 113–122 \ref\key 35
  • G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations , Nonlinear Anal., 12 (1988), 1203–1219 \ref\key 36
  • Z. L. Liu, Positive solutions of superlinear elliptic equations , J. Funct. Anal., 167 (1999), 370–398 \ref\key 37
  • Z. L. Liu and J. X. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations , J. Differential Equations, 172 (2001), 257–299 \ref\key 38
  • P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems , J. Funct. Anal., 7 (1971), 183–200 \ref\key 39 ––––, Multiple critical points of perturbed symmetric functionals , Trans. Amer. Math. Soc., 272 (1982), 753–769 \ref\key 40 ––––, Minimax Methods in Critical Point Theory with Applications to Differential Equations , CBMS, No. 65, Amer. Math. Soc., Providence (1985) \ref\key 41
  • 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 42
  • A. Szulkin and M. Willem, Eigenvalue problems with indefinite weight , Studia Math., 135 (1999), 191–201 \ref\key 43
  • P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations , J. Differential Equations, 51 (1984), 126–150 \ref\key 44
  • Z.-Q. Wang, On a semilinear elliptic equation , Ann. Inst. H. Poincaré, Anal. Non Linéaire, 8 (1991), 43–57 \ref\key 45 ––––, Nonlinear boundary value problems with concave nonlinearities near the origion , NoDEA Nonlinear Differential Equations and Appl., 8 (2001), 15–33 \ref\key 46
  • M. Willem, Minimax Theorems, Birkhäuser, Boston (1996)