Topological Methods in Nonlinear Analysis

A semilinear elliptic equation with convex and concave nonlinearities

Elliot Tonkes

Full-text: Open access


In this paper we establish the existence of multiple solutions for a semilinear elliptic equation with competing convex and concave nonlinearities. With either a subcritical or critical exponent in the nonlinearity, the existence of solutions is determined with critical point theorems based on the symmetric mountain pass theorem.

Article information

Topol. Methods Nonlinear Anal., Volume 13, Number 2 (1999), 251-271.

First available in Project Euclid: 29 September 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Tonkes, Elliot. A semilinear elliptic equation with convex and concave nonlinearities. Topol. Methods Nonlinear Anal. 13 (1999), no. 2, 251--271.

Export citation


  • C. O. Alves, J. V. Goncalves and O. H. Miyagaki, On elliptic equations in $\R{N}$ with critical exponents , Electron. J. Differential Equations (1996, 9 ), 1–11 \ref \key 2 ––––, Multiple solutions for semilinear elliptic equations in $\R{N}$ involving critical exponents , Nonlinear Anal., 34 (1998), 593–616 \ref \key 3
  • 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 4
  • A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications , J. Funct. Anal., 14 (1973), 349–381 \ref \key 5
  • A. Ambrosetti, J. Garcia Azorero and I. Peral, Quasilinear equations with a multiple bifurcation , Differential Integral Equations, 10 (1997), 37–50 \ref \key 6
  • A. Bahri and H. Berestycki, Existence of forced oscillations for some nonlinear differential equations , Comm. Pure Appl. Math., 37 (1984), 403–442 \ref \key 7
  • T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem , Nonlinear Anal., 20 (1993), 1205–1216 \ref \key 8
  • T. Bartsch and M. Clapp, Critical point theory for indefinite functionals with symmetries , J. Funct. Anal., 138 (1996), 107–136 \ref \key 9
  • T. Bartsch and Zhi-Qiang Wang, Existence and multiplicity results for some superlinear elliptic problems on $\R{N}$ , Comm. Partial Differential Equations, 20 (1995), 1725–1741 \ref \key 10
  • T. Bartsch and M. Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation , J. Funct. Anal., 117 (1993), 447–460 \ref \key 11 ––––, Periodic solutions of non-autonomous Hamiltonian systems with symmetries, J. Reine Angew. Math., 451 (1994), 149-159 \ref \key 12 ––––, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123 (1995), 3555–3561 \ref \key 13
  • H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486–490 \ref \key 14
  • 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 15
  • K. J. Brown, C. Cosner and J. Fleckinger, Principal eigenvalues for problems with indefinite weight function on $\R{N}$ , Proc. Amer. Math. Soc., 109 (1990), 147–155 \ref \key 16
  • K. J. Brown and N. M. Stavrakakis, Global bifurcation results for a semilinear elliptic equation on all of $\RR^N$, Duke Math. J., 85 (1996), 77–94 \ref \key 17
  • S. Cingolani and J. L. Gámez, Positive solutions of a semilinear elliptic equation on ${\Bbb R}^N$ with indefinite nonlinearity , Adv. Differential Equations, 1 (1996), 773–791 \ref \key 18
  • P. Drábek and Y. X. Huang, Bifurcation problems for the $p$-Laplacian in $\R{N}$, Trans. Amer. Math. Soc., 349 (1997), 171–188 \ref \key 19
  • J. Garcia Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term , Trans. Amer. Math. Soc., 323 (1991), 877–895 \ref \key 20
  • Z. Jin, Multiple solutions for a class of semilinear elliptic equations, Proc. Amer. Math. Soc., 125 (1997), 3659–3667 \ref \key 21
  • S. Li and M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl., 189 (1995), 6–32 \ref \key 22
  • S. J. Li and J.Q. Liu, Some existence theorems on multiple critical points and theire applications, Kexue Tongbao, 17 (1984), (Chinese) \ref \key 23
  • S. Li and W. Zou, Remarks on a class of elliptic problems with critical exponent, Nonlinear Anal., 32 (1998), 769–774 \ref \key 24
  • O. H. Miyagaki, On a class of semilinear elliptic problems in ${\Bbb R}^N$ with critical growth, Nonlinear Anal., 29 (1997), 773–781 \ref \key 25
  • V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory, Top. Methods Nonlinear Anal., 10 (1997), 387–397 \ref \key 26
  • M. Reed and B. Simon, Functional Analysis, Academic Press, New York (1972) \ref \key 27
  • H. J. Ruppen, Multiplicity results for a semilinear elliptic differential equation with conflicting nonlinearities, J. Differential Equations, 147 (1998), 79–122 \ref \key 28
  • A. Taylor, Introduction to Functional Analysis, Wiley, New York, (1958) \ref \key 29
  • A. Tertikas, Critical phenomena in linear elliptic problems , J. Funct. Anal., 154 (1998), 42–66 \ref \key 30
  • S.B. Tshinanga, On multiple solutions of semilinear elliptic equation on unbounded domains with concave and convex nonlinearities, Nonlinear Anal., 28 (1997), 809–814 \ref \key 31
  • M. Willem, Minimax Theorems, Progress in nonlinear differential equations and their applications, 24, Birkhäuser, Boston (1996)