The singular extremal solutions of the bi-Laplacian with exponential nonlinearity
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Abstract:
Consider the problem \begin{eqnarray*} \left \{ \begin {array}{ll} \Delta ^2 u= \lambda e^{u} &\text {in } B,\\ u=\frac {\partial u}{\partial n}=0 &\text {on }\partial B, \end{array} \right . \end{eqnarray*} where $B$ is the unit ball in ${\mathbb {R}}^N$ and $\lambda$ is a parameter. Unlike the Gelfand problem the natural candidate $u=-4\ln (|x|)$, for the extremal solution, does not satisfy the boundary conditions, and hence showing the singular nature of the extremal solution in large dimensions close to the critical dimension is challenging. Recently a computer-assisted proof was used to show that the extremal solution is singular in dimensions $13\leq N\leq 31$. Here by an improved Hardy-Rellich inequality we overcome this difficulty and give a simple mathematical proof to show that the extremal solution is singular in dimensions $N\geq 13$.References
- Gianni Arioli, Filippo Gazzola, Hans-Christoph Grunau, and Enzo Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, SIAM J. Math. Anal. 36 (2005), no. 4, 1226–1258. MR 2139208, DOI 10.1137/S0036141002418534
- Haim Brezis and Juan Luis Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 443–469. MR 1605678
- C. Cowan, P. Esposito, N. Ghoussoub, A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal., to appear.
- Michael G. Crandall and Paul H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal. 58 (1975), no. 3, 207–218. MR 382848, DOI 10.1007/BF00280741
- Juan Dávila, Louis Dupaigne, Ignacio Guerra, and Marcelo Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal. 39 (2007), no. 2, 565–592. MR 2338421, DOI 10.1137/060665579
- N. Ghoussoub, A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, submitted.
- Nassif Ghoussoub and Amir Moradifam, On the best possible remaining term in the Hardy inequality, Proc. Natl. Acad. Sci. USA 105 (2008), no. 37, 13746–13751. MR 2443723, DOI 10.1073/pnas.0803703105
- Zongming Guo and Juncheng Wei, On a fourth order nonlinear elliptic equation with negative exponent, SIAM J. Math. Anal. 40 (2008/09), no. 5, 2034–2054. MR 2471911, DOI 10.1137/070703375
- Fulbert Mignot and Jean-Pierre Puel, Solution radiale singulière de $-\Delta u=\lambda e^u$, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 8, 379–382 (French, with English summary). MR 965802
- A. Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent, Journal of Differential Equations 248 (2010), 594-616.
- Juncheng Wei, Asymptotic behavior of a nonlinear fourth order eigenvalue problem, Comm. Partial Differential Equations 21 (1996), no. 9-10, 1451–1467. MR 1410837, DOI 10.1080/03605309608821234
Additional Information
- Amir Moradifam
- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2
- MR Author ID: 781850
- Email: a.moradi@math.ubc.ca
- Received by editor(s): April 23, 2009
- Published electronically: December 3, 2009
- Additional Notes: This work is supported by a Killam Predoctoral Fellowship and is part of the author’s Ph.D. dissertation in preparation under the supervision of N. Ghoussoub.
- Communicated by: Matthew J. Gursky
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 1287-1293
- MSC (2010): Primary 35J65; Secondary 35J40
- DOI: https://doi.org/10.1090/S0002-9939-09-10257-5
- MathSciNet review: 2578522