## Topological Methods in Nonlinear Analysis

### A semilinear elliptic equation with convex and concave nonlinearities

Elliot Tonkes

#### Abstract

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

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

Dates
First available in Project Euclid: 29 September 2016

https://projecteuclid.org/euclid.tmna/1475178881

Mathematical Reviews number (MathSciNet)
MR1742223

Zentralblatt MATH identifier
0991.35022

#### Citation

Tonkes, Elliot. A semilinear elliptic equation with convex and concave nonlinearities. Topol. Methods Nonlinear Anal. 13 (1999), no. 2, 251--271. https://projecteuclid.org/euclid.tmna/1475178881

#### References

• 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)