# American Institute of Mathematical Sciences

2015, 2015(special): 1025-1033. doi: 10.3934/proc.2015.1025

## Singular extremal solutions to a Liouville-Gelfand type problem with exponential nonlinearity

 1 Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585

Received  September 2014 Revised  July 2015 Published  November 2015

We consider a Liouville-Gelfand type problem $-\Delta u = e^u - 1 + \lambda f(x) \quad \text{in} \; \Omega, \quad u > 0 \quad \text{in} \; \Omega, \quad u = 0 \quad \text{on} \; \partial\Omega,$ where $\Omega \subset \mathbb{R}^N \; (N \ge 1)$ is a smooth bounded domain, $f \ge 0$, $f \not\equiv 0$ is a given smooth function, and $\lambda \ge 0$ is a parameter. We are concerned with the regularity property of extremal solutions to the problem, and prove that there exists a domain $\Omega$ and a smooth nonnegative function $f$ such that the extremal solution of the problem is singular when the dimension $N \ge 10$. This result is sharp in the sense that the extremal solution is always regular (bounded) for any $f$ and $\Omega$ when $1 \le N \le 9$.
Citation: Futoshi Takahashi. Singular extremal solutions to a Liouville-Gelfand type problem with exponential nonlinearity. Conference Publications, 2015, 2015 (special) : 1025-1033. doi: 10.3934/proc.2015.1025
##### References:
 [1] H. Beresticki, L. Nirenberg, and S. R. S. Varadhan, The principal eigenvalue and the maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92. [2] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_t - \Delta u = g(u)$ revisited, Adv. Differential Equations. 1 (1996), 73-90. [3] H. Brezis, and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Compl. Madrid, 10 (1997), 443-469. [4] M.G. Crandall, and R. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58 (1975), 207-218. [5] J. Dávila, Some extremal singular solutions of a nonlinear elliptic equation, Differential Integral Equations, 14 no.3 (2001), 289-304. [6] J. Dávila, and L. Dupaigne, Perturbing singular solutions of the Gelfand problem, Commun. Contemp. Math., 9 (2007), 639-680. [7] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Monographs and Surveys in Pure and Applied Mathematics 143, Chapman & Hall/CRC Press, xiv+321 pp. (2011) [8] Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math., 23 (1997), 161-168. [9] F. Mignot and J.P. Puel, Sur une classe de problèmes non linéaires avec nonlinéairité positive, croissante, convexe, Comm. Partial Differential Equations, 5 (1980), 791-836. [10] Y. Miyamoto, Classification of bifurcation diagrams for elliptic equations with exponential growth in a ball, Ann. Mat. Pura Appl. (4), 194 no.4 (2015), 931-952.

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##### References:
 [1] H. Beresticki, L. Nirenberg, and S. R. S. Varadhan, The principal eigenvalue and the maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92. [2] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_t - \Delta u = g(u)$ revisited, Adv. Differential Equations. 1 (1996), 73-90. [3] H. Brezis, and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Compl. Madrid, 10 (1997), 443-469. [4] M.G. Crandall, and R. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58 (1975), 207-218. [5] J. Dávila, Some extremal singular solutions of a nonlinear elliptic equation, Differential Integral Equations, 14 no.3 (2001), 289-304. [6] J. Dávila, and L. Dupaigne, Perturbing singular solutions of the Gelfand problem, Commun. Contemp. Math., 9 (2007), 639-680. [7] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Monographs and Surveys in Pure and Applied Mathematics 143, Chapman & Hall/CRC Press, xiv+321 pp. (2011) [8] Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math., 23 (1997), 161-168. [9] F. Mignot and J.P. Puel, Sur une classe de problèmes non linéaires avec nonlinéairité positive, croissante, convexe, Comm. Partial Differential Equations, 5 (1980), 791-836. [10] Y. Miyamoto, Classification of bifurcation diagrams for elliptic equations with exponential growth in a ball, Ann. Mat. Pura Appl. (4), 194 no.4 (2015), 931-952.
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