# 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.   Google Scholar [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), 1 (1996), 73.   Google Scholar [3] H. Brezis, and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems,, Rev. Mat. Univ. Compl. Madrid, 10 (1997), 443.   Google Scholar [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.   Google Scholar [5] J. Dávila, Some extremal singular solutions of a nonlinear elliptic equation,, Differential Integral Equations, 14 (2001), 289.   Google Scholar [6] J. Dávila, and L. Dupaigne, Perturbing singular solutions of the Gelfand problem,, Commun. Contemp. Math., 9 (2007), 639.   Google Scholar [7] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations,, Monographs and Surveys in Pure and Applied Mathematics 143, (2011).   Google Scholar [8] Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems,, Houston J. Math., 23 (1997), 161.   Google Scholar [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.   Google Scholar [10] Y. Miyamoto, Classification of bifurcation diagrams for elliptic equations with exponential growth in a ball,, Ann. Mat. Pura Appl. (4), 194 (2015), 931.   Google Scholar

show all references

##### 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.   Google Scholar [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), 1 (1996), 73.   Google Scholar [3] H. Brezis, and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems,, Rev. Mat. Univ. Compl. Madrid, 10 (1997), 443.   Google Scholar [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.   Google Scholar [5] J. Dávila, Some extremal singular solutions of a nonlinear elliptic equation,, Differential Integral Equations, 14 (2001), 289.   Google Scholar [6] J. Dávila, and L. Dupaigne, Perturbing singular solutions of the Gelfand problem,, Commun. Contemp. Math., 9 (2007), 639.   Google Scholar [7] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations,, Monographs and Surveys in Pure and Applied Mathematics 143, (2011).   Google Scholar [8] Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems,, Houston J. Math., 23 (1997), 161.   Google Scholar [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.   Google Scholar [10] Y. Miyamoto, Classification of bifurcation diagrams for elliptic equations with exponential growth in a ball,, Ann. Mat. Pura Appl. (4), 194 (2015), 931.   Google Scholar
 [1] Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020031 [2] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [3] João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 [4] Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116 [5] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [6] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [7] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [8] Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118 [9] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264

Impact Factor: