Abstract and Applied Analysis

Single blow-up solutions for a slightly subcritical biharmonic equation

Khalil El Mehdi

Abstract

We consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent ($P_\varepsilon$): $\Delta^2u=u^{9-\varepsilon}$, $u>0$ in $\Omega$ and $u=\Delta u=0$ on $\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^5$, $\varepsilon>0$. We study the asymptotic behavior of solutions of ($P_\varepsilon$) which are minimizing for the Sobolev quotient as $\varepsilon$ goes to zero. We show that such solutions concentrate around a point $x_0 \in \Omega$ as $\varepsilon\rightarrow 0$, moreover $x_0$ is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical point $x_0$ of the Robin's function, there exist solutions of ($P_\varepsilon$) concentrating around $x_0$ as $\varepsilon\rightarrow 0$.

Article information

Source
Abstr. Appl. Anal., Volume 2006 (2006), Article ID 18387, 20 pages.

Dates
First available in Project Euclid: 30 January 2007

https://projecteuclid.org/euclid.aaa/1170202976

Digital Object Identifier
doi:10.1155/AAA/2006/18387

Mathematical Reviews number (MathSciNet)
MR2211670

Zentralblatt MATH identifier
1155.35331

Citation

El Mehdi, Khalil. Single blow-up solutions for a slightly subcritical biharmonic equation. Abstr. Appl. Anal. 2006 (2006), Article ID 18387, 20 pages. doi:10.1155/AAA/2006/18387. https://projecteuclid.org/euclid.aaa/1170202976

References

• F. V. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, Journal of Differential Equations 70 (1987), no. 3, 349–365.
• A. Bahri, Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, vol. 182, Longman Scientific & Technical, Harlow, 1989.
• A. Bahri, Y. Li, and O. Rey, On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Calculus of Variations and Partial Differential Equations 3 (1995), no. 1, 67–93.
• M. Ben Ayed and K. El Mehdi, Existence of conformal metrics on spheres with prescribed Paneitz curvature, Manuscripta Mathematica 114 (2004), no. 2, 211–228.
• ––––, On a biharmonic equation involving nearly critical exponent, to appear in Nonlinear Differential Equations and Applications, 2006.
• ––––, The Paneitz curvature problem on lower dimensional spheres, to appear in Annals of Global Analysis and Geometry.
• M. Ben Ayed, K. El Mehdi, and M. Hammami, Some existence results for a Paneitz type problem via the theory of critical points at infinity, Journal de Mathématiques Pures et Appliquées. Neuvième Série 84 (2005), no. 2, 247–278.
• M. Ben Ayed and M. Hammami, On a fourth order elliptic equation with critical nonlinearity in dimension six, to appear in Nonlinear Analysis TMA.
• H. Brezis and L. A. Peletier, Asymptotics for elliptic equations involving critical growth, Partial Differential Equations and the Calculus of Variations, Vol. 1, Progr. Nonlinear Differential Equations Appl., vol. 1, Birkhäuser Boston, Massachusetts, 1989, pp. 149–192.
• S.-Y. A. Chang, On a fourth-order partial differential equation in conformal geometry, Harmonic Analysis and Partial Differential Equations (Chicago, Ill, 1996) (M. Christ, C. Kenig, and C. Sadorsky, eds.), Chicago Lectures in Math., University of Chicago Press, Illinois, 1999, pp. 127–150, essays in honor of Alberto P. Calderon.
• K.-S. Chou and D. Geng, Asymptotics of positive solutions for a biharmonic equation involving critical exponent, Differential and Integral Equations. An International Journal for Theory &Applications 13 (2000), no. 7-9, 921–940.
• Z. Djadli, E. Hebey, and M. Ledoux, Paneitz-type operators and applications, Duke Mathematical Journal 104 (2000), no. 1, 129–169.
• Z. Djadli, A. Malchiodi, and M. O. Ahmedou, Prescribing a fourth order conformal invariant on the standard sphere. I. A perturbation result, Communications in Contemporary Mathematics 4 (2002), no. 3, 375–408.
• ––––, Prescribing a fourth order conformal invariant on the standard sphere. II. Blow up analysis and applications, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 1 (2002), no. 2, 387–434.
• F. Ebobisse and M. O. Ahmedou, On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 52 (2003), no. 5, 1535–1552.
• D. E. Edmunds, D. Fortunato, and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Archive for Rational Mechanics and Analysis 112 (1990), no. 3, 269–289.
• V. Felli, Existence of conformal metrics on $S^n$ with prescribed fourth-order invariant, Advances in Differential Equations 7 (2002), no. 1, 47–76.
• J. García Azorero and I. Peral Alonso, On limits of solutions of elliptic problems with nearly critical exponent, Communications in Partial Differential Equations 17 (1992), no. 11-12, 2113–2126.
• F. Gazzola, H.-C. Grunau, and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calculus of Variations and Partial Differential Equations 18 (2003), no. 2, 117–143.
• D. Geng, On blow-up of positive solutions for a biharmonic equation involving nearly critical exponent, Communications in Partial Differential Equations 24 (1999), no. 11-12, 2333–2370.
• Z.-C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 8 (1991), no. 2, 159–174.
• R. Lewandowski, Little holes and convergence of solutions of $- \Delta u = u^{(n + 2)/(n - 2)}$, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 14 (1990), no. 10, 873–888.
• C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbf{R}^n$, Commentarii Mathematici Helvetici 73 (1998), no. 2, 206–231.
• O. Rey, Proof of two conjectures of H. Brézis and L. A. Peletier, Manuscripta Mathematica 65 (1989), no. 1, 19–37.
• ––––, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, Journal of Functional Analysis 89 (1990), no. 1, 1–52.
• ––––, Blow-up points of solutions to elliptic equations with limiting nonlinearity, Differential and Integral Equations. An International Journal for Theory and Applications 4 (1991), no. 6, 1155–1167.
• ––––, The topological impact of critical points at infinity in a variational problem with lack of compactness: the dimension $3$, Advances in Differential Equations 4 (1999), no. 4, 581–616.
• R. C. A. M. Van der Vorst, Fourth-order elliptic equations with critical growth, Comptes Rendus de l'Académie des Sciences. Série I. Mathématique 320 (1995), no. 3, 295–299.