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]

Guillaume Warnault. Regularity of the extremal solution for a biharmonic problem with general nonlinearity. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1709-1723. doi: 10.3934/cpaa.2009.8.1709

[2]

Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227

[3]

Jagmohan Tyagi, Ram Baran Verma. Positive solution to extremal Pucci's equations with singular and gradient nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2637-2659. doi: 10.3934/dcds.2019110

[4]

Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011

[5]

Juan Dávila, Louis Dupaigne, Marcelo Montenegro. The extremal solution of a boundary reaction problem. Communications on Pure & Applied Analysis, 2008, 7 (4) : 795-817. doi: 10.3934/cpaa.2008.7.795

[6]

Eugenia N. Petropoulou. On some difference equations with exponential nonlinearity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2587-2594. doi: 10.3934/dcdsb.2017098

[7]

Manuel del Pino, Jean Dolbeault, Monica Musso. Multiple bubbling for the exponential nonlinearity in the slightly supercritical case. Communications on Pure & Applied Analysis, 2006, 5 (3) : 463-482. doi: 10.3934/cpaa.2006.5.463

[8]

A. Adam Azzam. Scattering for the two dimensional NLS with (full) exponential nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1071-1101. doi: 10.3934/cpaa.2018052

[9]

Canghua Jiang, Kok Lay Teo, Ryan Loxton, Guang-Ren Duan. A neighboring extremal solution for an optimal switched impulsive control problem. Journal of Industrial & Management Optimization, 2012, 8 (3) : 591-609. doi: 10.3934/jimo.2012.8.591

[10]

Sandra Carillo. Materials with memory: Free energies & solution exponential decay. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1235-1248. doi: 10.3934/cpaa.2010.9.1235

[11]

Soohyun Bae, Yūki Naito. Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4537-4554. doi: 10.3934/dcds.2018198

[12]

Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613

[13]

Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335

[14]

Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101

[15]

Sami Aouaoui. On some semilinear equation in $R^4$ containing a Laplacian term and involving nonlinearity with exponential growth. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2185-2201. doi: 10.3934/cpaa.2015.14.2185

[16]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273

[17]

Abdelwahab Bensouilah, Van Duong Dinh, Mohamed Majdoub. Scattering in the weighted $ L^2 $-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2735-2755. doi: 10.3934/cpaa.2019122

[18]

John A. D. Appleby, John A. Daniels. Exponential growth in the solution of an affine stochastic differential equation with an average functional and financial market bubbles. Conference Publications, 2011, 2011 (Special) : 91-101. doi: 10.3934/proc.2011.2011.91

[19]

Olivier Goubet. Regularity of extremal solutions of a Liouville system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 339-345. doi: 10.3934/dcdss.2019023

[20]

Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082

 Impact Factor: 

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]