Abstract and Applied Analysis

Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains

Tsung-Fang Wu

Full-text: Open access

Abstract

We consider the elliptic problem Δ u + u = b ( x ) | u | p 2 u + h ( x ) in Ω , u H 0 1 ( Ω ) , where 2 < p < ( 2 N / ( N 2 ) )   ( N 3 ) ,   2 < p <   ( N = 2 ) ,   Ω is a smooth unbounded domain in N ,   b ( x ) C ( Ω ) , and h ( x ) H 1 ( Ω ) . We use the shape of domain Ω to prove that the above elliptic problem has a ground-state solution if the coefficient b ( x ) satisfies b ( x ) b > 0 as | x | and b ( x ) c for some suitable constants c ( 0 , b ) , and h ( x ) 0 . Furthermore, we prove that the above elliptic problem has multiple positive solutions if the coefficient b ( x ) also satisfies the above conditions, h ( x ) 0 and 0 < h H 1 < ( p 2 ) ( 1 / ( p 1 ) ) ( p 1 ) / ( p 2 ) [ b sup S p ( Ω ) ] 1 / ( 2 p ) , where S ( Ω ) is the best Sobolev constant of subcritical operator in H 0 1 ( Ω ) and b sup = sup x Ω b ( x ) .

Article information

Source
Abstr. Appl. Anal., Volume 2007 (2007), Article ID 18187, 25 pages.

Dates
First available in Project Euclid: 27 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1204126588

Digital Object Identifier
doi:10.1155/2007/18187

Mathematical Reviews number (MathSciNet)
MR2302187

Zentralblatt MATH identifier
1157.35377

Citation

Wu, Tsung-Fang. Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains. Abstr. Appl. Anal. 2007 (2007), Article ID 18187, 25 pages. doi:10.1155/2007/18187. https://projecteuclid.org/euclid.aaa/1204126588


Export citation

References

  • P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986.
  • A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” Journal of Functional Analysis, vol. 14, no. 4, pp. 349–381, 1973.
  • M. Willem, Minimax Theorems, vol. 24 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 1996.
  • H.-C. Wang and T.-F. Wu, “Symmetry breaking in a bounded symmetry domain,” NoDEA. Nonlinear Differential Equations and Applications, vol. 11, no. 3, pp. 361–377, 2004.
  • H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations. I. Existence of a ground state,” Archive for Rational Mechanics and Analysis, vol. 82, no. 4, pp. 313–345, 1983.
  • W. C. Lien, S. Y. Tzeng, and H.-C. Wang, “Existence of solutions of semilinear elliptic problems on unbounded domains,” Differential and Integral Equations, vol. 6, no. 6, pp. 1281–1298, 1993.
  • K.-J. Chen and H.-C. Wang, “A necessary and sufficient condition for Palais-Smale conditions,” SIAM Journal on Mathematical Analysis, vol. 31, no. 1, pp. 154–165, 1999.
  • M. A. Del Pino and P. L. Felmer, “Local mountain passes for semilinear elliptic problems in unbounded domains,” Calculus of Variations and Partial Differential Equations, vol. 4, no. 2, pp. 121–137, 1996.
  • M. A. Del Pino and P. L. Felmer, “Least energy solutions for elliptic equations in unbounded domains,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 126, no. 1, pp. 195–208, 1996.
  • M. K. Kwong, “Uniqueness of positive solutions of ${\Delta{}}u-u+{u}^{p}=0$ in ${{\mathbb{R}}}^{N}$,” Archive for Rational Mechanics and Analysis, vol. 105, no. 3, pp. 243–266, 1989.
  • A. Bahri and P.-L. Lions, “On the existence of a positive solution of semilinear elliptic equations in unbounded domains,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 14, no. 3, pp. 365–413, 1997.
  • V. Benci and G. Cerami, “Positive solutions of some nonlinear elliptic problems in exterior domains,” Archive for Rational Mechanics and Analysis, vol. 99, no. 4, pp. 283–300, 1987.
  • M. J. Esteban and P.-L. Lions, “Existence and nonexistence results for semilinear elliptic problems in unbounded domains,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 93, no. 1-2, pp. 1–14, 1982-1983.
  • D.-M. Cao, “Positive solution and bifurcation from the essential spectrum of a semilinear elliptic equation on ${{\mathbb{R}}}^{N}$,” Nonlinear Analysis, vol. 15, no. 11, pp. 1045–1052, 1990.
  • A. Bahri and Y. Y. Li, “On a min-max procedure for the existence of a positive solution for certain scalar field equations in ${{\mathbb{R}}}^{N}$,” Revista Matemática Iberoamericana, vol. 6, no. 1-2, pp. 1–15, 1990.
  • T.-F. Wu, “Multiplicity of single-bump solutions for semilinear elliptic equations in multi-bump domains,” Nonlinear Analysis, vol. 59, no. 6, pp. 973–992, 2004.
  • X. P. Zhu, “A perturbation result on positive entire solutions of a semilinear elliptic equation,” Journal of Differential Equations, vol. 92, no. 2, pp. 163–178, 1991.
  • D.-M. Cao and H.-S. Zhou, “Multiple positive solutions of nonhomogeneous semilinear elliptic equations in ${{\mathbb{R}}}^{N}$,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 126, no. 2, pp. 443–463, 1996.
  • L. Jeanjean, “Two positive solutions for a class of nonhomogeneous elliptic equations,” Differential and Integral Equations, vol. 10, no. 4, pp. 609–624, 1997.
  • S. Adachi and K. Tanaka, “Multiple positive solutions for nonhomogeneous elliptic equations,” Nonlinear Analysis, vol. 47, no. 6, pp. 3783–3793, 2001.
  • S. Adachi and K. Tanaka, “Four positive solutions for the semilinear elliptic equation: $-{\Delta{}}u+u=a(x){u}^{p}+f(x)$ in ${{\mathbb{R}}}^{N}$,” Calculus of Variations and Partial Differential Equations, vol. 11, no. 1, pp. 63–95, 2000.
  • H.-L. Lin, H.-C. Wang, and T.-F. Wu, “A Palais-Smale approach to Sobolev subcritical operators,” Topological Methods in Nonlinear Analysis, vol. 20, no. 2, pp. 393–407, 2002.
  • G. Tarantello, “On nonhomogeneous elliptic equations involving critical Sobolev exponent,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 9, no. 3, pp. 281–304, 1992.
  • H. Brézis and E. Lieb, “A relation between pointwise convergence of functions and convergence of functionals,” Proceedings of the American Mathematical Society, vol. 88, no. 3, pp. 486–490, 1983.
  • I. Ekeland, “On the variational principle,” Journal of Mathematical Analysis and Applications, vol. 47, no. 2, pp. 324–353, 1974.