## Abstract and Applied Analysis

### Existence of Multiple Solutions for a Singular Elliptic Problem with Critical Sobolev Exponent

Zonghu Xiu

#### Abstract

We consider the existence of multiple solutions of the singular elliptic problem $-\mathrm{div}({|x|}^{-ap}{|\nabla u|}^{p-2}\nabla u)+{|u|}^{p-2}u/{|x|}^{(a+1)p}=f{|u|}^{r-2}u+h{|u|}^{s-2}u+{|x|}^{-b{p}^{\ast}}{|u|}^{{p}^{\ast}-2}u$, $u(x)\to 0$ as $|x|\to +\infty$, where $x\in {\Bbb R}^{N}$, $1, $a<(N-p)/p$, $a\le b\le a+1,r$, $s>1$, ${p}^{\ast}=Np/(N-pd)$, $d=a+1-b$. By the variational method and the theory of genus, we prove that the above-mentioned problem has infinitely many solutions when some conditions are satisfied.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 806397, 15 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365168318

Digital Object Identifier
doi:10.1155/2012/806397

Mathematical Reviews number (MathSciNet)
MR2999909

Zentralblatt MATH identifier
1261.35068

#### Citation

Xiu, Zonghu. Existence of Multiple Solutions for a Singular Elliptic Problem with Critical Sobolev Exponent. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 806397, 15 pages. doi:10.1155/2012/806397. https://projecteuclid.org/euclid.aaa/1365168318

#### References

• R. B. Assunção, P. C. Carrião, and O. H. Miyagaki, “Critical singular problems via concentration-compactness lemma,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 137–154, 2007.
• R. D. S. Rodrigues, “On elliptic problems involving critical Hardy-Sobolev exponents and sign-changing function,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 4, pp. 857–880, 2010.
• Z. Nehari, “On a class of nonlinear second-order differential equations,” Transactions of the American Mathematical Society, vol. 95, pp. 101–123, 1960.
• K. J. Brown, “The Nehari manifold for a semilinear elliptic equation involving a sublinear term,” Calculus of Variations and Partial Differential Equations, vol. 22, no. 4, pp. 483–494, 2005.
• M. L. Miotto and O. H. Miyagaki, “Multiple positive solutions for semilinear Dirichlet problems with sign-changing weight function in infinite strip domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3434–3447, 2009.
• R. B. Assunção, P. C. Carrião, and O. H. Miyagaki, “Subcritical perturbations of a singular quasilinear elliptic equation involving the critical Hardy-Sobolev exponent,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 6, pp. 1351–1364, 2007.
• J. V. Gonçalves and C. O. Alves, “Existence of positive solutions for $m$-Laplacian equations in ${\mathbb{R}}^{N}$ involving critical Sobolev exponents,” Nonlinear Analysis: Theory, Methods & Applications, vol. 32, no. 1, pp. 53–70, 1998.
• T.-F. Wu, “On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 253–270, 2006.
• T.-F. Wu, “On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function,” Communications on Pure and Applied Analysis, vol. 7, no. 2, pp. 383–405, 2008.
• L. Iturriaga, “Existence and multiplicity results for some quasilinear elliptic equation with weights,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1084–1102, 2008.
• H. Q. Toan and Q.-A. Ngô, “Multiplicity of weak solutions for a class of nonuniformly elliptic equations of $p$-Laplacian type,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 4, pp. 1536–1546, 2009.
• L. Caffarelli, R. Kohn, and L. Nirenberg, “First order interpolation inequalities with weights,” Compositio Mathematica, vol. 53, no. 3, pp. 259–275, 1984.
• 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.
• B. Xuan, “The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights,” Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 4, pp. 703–725, 2005.
• P.-L. Lions, “The concentration-compactness čommentComment on ref. [3a?]: We split this reference to [3a, 3b?]. Please check. principle in the calculus of variations. The locally compact case. I,” Annales de l'Institut Henri Poincaré Analyse Non Linéaire, vol. 1, no. 2, pp. 109–145, 1984.
• P.-L. Lions, “The concentration-compactness principle in the calculus of variations. The locally compact case. II,” Annales de l'Institut Henri Poincaré Analyse Non Linéaire, vol. 1, no. 4, pp. 223–283, 1984.
• P.-L. Lions, “The concentration-compactness čommentComment on ref. [4a?]: We split this reference to [4a, 4b?]. Please check. principle in the calculus of variations. The limit case. I,” Revista Matemática Iberoamericana, vol. 1, no. 1, pp. 145–201, 1985.
• P.-L. Lions, “The concentration-compactness principle in the calculus of variations. The limit case. II,” Revista Matemática Iberoamericana, vol. 1, no. 2, pp. 45–121, 1985.
• G. Bianchi, J. Chabrowski, and A. Szulkin, “On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent,” Nonlinear Analysis: Theory, Methods & Applications, vol. 25, no. 1, pp. 41–59, 1995.
• A. K. Ben-Naoum, C. Troestler, and M. Willem, “Extrema problems with critical Sobolev exponents on unbounded domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 26, no. 4, pp. 823–833, 1996.
• J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, vol. 106, Pitman, Boston, Mass, USA, 1985, Elliptic Equations.
• M. Struwe, Variational Methods, vol. 34, Springer, New York, NY, USA, 3rd edition, 2000.
• I. Kuzin and S. Pohozaev, Entire Solutions of Semilinear Elliptic Equations, Birkhäuser, 1997.