## Taiwanese Journal of Mathematics

### MULTIPLE SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS IN UNBOUNDED CYLINDER DOMAINS

#### Abstract

In this paper, we show that if $Q(x)$ satisfies some suitable conditions, then the quasilinear elliptic Dirichlet problem $-\Delta_p u + |u|^{p-2} u = Q(x)|u|^{q-2}u$ in an unbounded cylinder domain $\Omega$ has at least two solutions in which one is a positive ground state solution and the other is a nodal solution.

#### Article information

Source
Taiwanese J. Math., Volume 16, Number 2 (2012), 409-428.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406593

Digital Object Identifier
doi:10.11650/twjm/1500406593

Mathematical Reviews number (MathSciNet)
MR2892890

Zentralblatt MATH identifier
1247.35022

#### Citation

Hsu, Tsing-San; Lin, Huei-Li. MULTIPLE SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS IN UNBOUNDED CYLINDER DOMAINS. Taiwanese J. Math. 16 (2012), no. 2, 409--428. doi:10.11650/twjm/1500406593. https://projecteuclid.org/euclid.twjm/1500406593

#### References

• C. O. Alves, P. C. Carri$\rm\tilde{a}$o and E. S. Medeiros, Multiplicity of solutions for a class of quasilinear problem in exterior domains with Neumann conditions, Abstr. Appl. Anal., 3 (2004), 251-268.
• C. O. Alves, Existence of positive solutions for a problem with lack of compactness involving the $p$-Laplacian, Nonlinear Analysis T.M.A., 51 (2002), 1187-1206.
• \label1 A. Bahri and P. L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. I. H. P. Analyse Non Lineaire, 14 (1997), 365-413.
• \label2 A. Bahri and Y. Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $\Bbb R^N$, Revi. Mate. Iber., 6 (1990), 1-15.
• V. Benci and G. Cerami, Positive solution of semilinear elliptic equation in exterior domains, Arch. Rat. Mech. Anal., 99 (1987), 283-300.
• \label3 H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I, II, Arch. Rati. Mech. Anal., 82 (1983), 313-376.
• \label4 W. Ding and W. M. Ni, On the existence of positive entire solutions of semilinear elliptic equations, Arch. Rati. Mech. Anal., 91 (1986), 288-308.
• I. Ekeland, Non-convex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474.
• \label9 D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983.
• T. S. Hsu, Multiple solutions for semilinear elliptic equations in unbounded cylinder domains, Proc. Roy. Soc. Edin., 134A (2004), 719-731.
• T. S. Hsu and H. L. Lin, Multiple solutions for some Neumann problems in exterior domains, Bull. Austral. Math. Soc., 73 (2006), 353-364.
• D. Huang and Y. Li, A concentration-compactness principle at infinity and positive solutions of some quasilinear elliptic equations in unbounded domains, J. Math. Anal. Appl., 304 (2005), 58-73.
• Y. Jianfu, Positive solutions of quasilinear elliptic obstacle problems with critical exponents, Nonlinear Analysis T.M.A., 25 (1995), 1283-1306.
• G. Li and S. Yan, Eigenvalue problems for quasilinear elliptic equations on $\mathbb{R}^N$, Comm. Partial Differential Equations, 14 (1989), 1291-1314.
• \label5 P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case I, Ann. Inst. H. Poincaré, Analyse Nonlinéaire, 1 (1984), 109-145.
• \label6 P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case II, Ann. Inst. H. Poincaré, Analyse nonlinéaire, 1 (1984), 223-283.
• \label7 P. L. Lions, On positive solutions of semilinear elliptic equation in unbounded domains, in: Nonlinear Diffusion Equations and their Equilibrium States, (Ni, Peletier and Serrin, eds.), Springer-Verlarg, Berlin, 1988.
• \label10 M. $\rm \hat{O}$tani, Existence and nonexistence of nontrivial solutions for some nonlinear degenerate elliptic equations, J. Functional Anal., 76 (1988), 140-159.
• P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, in: Eigenvalues of nonlinear problems, Rome: Ediz. Cremonese, 1972.
• \label11 J. Serrin, Local behavior of solutions of quasilinear elliptic equations, Acta. Math., 111 (1964), 247-302.
• J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana University Mathematics Journal, 49 (2000), 897-923.
• \label12 P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Diff. Equ., 51 (1984), 126-150.
• N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math., XX (1967), 721-747.
• \label13 J. L. V\vaa zquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
• \label8 X. P. Zhu, Multiple entire solutions of a semilinear elliptic equation, Nonlinear Analysis T.M.A., 12 (1988), 1297-1316.