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2013, 2013(special): 729-736. doi: 10.3934/proc.2013.2013.729

Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data

Futoshi Takahashi 1,

1. 

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

Received  August 2012 Revised  March 2013 Published  November 2013

Let $\Omega \subset \mathbb{R}^2$ be a smooth bounded domain and let $\Gamma = \left \{ p_1, \cdots, p_N \right \} \subset \Omega$ be the set of prescribed points. Consider the Liouville type equation \[ -\delta u = \lambda \Pi_{j = 1}^{N} |x - p_j|^{2\alpha_j} V(x) e^u \quad \mbox{in} \; \Omega, \quad u = 0 \quad \mbox{on} \; \partial \Omega, \] where $\alpha_j \; (j=1,\cdots, N)$ are positive numbers, $V(x) > 0$ is a given smooth function on $\bar{\Omega}$, and $\lambda > 0$ is a parameter. Let $\{ u_n \}$ be a blowing up solution sequence for $\lambda = \lambda_n \downarrow 0$ having the $m$-points blow up set $S = \{ q_1, \cdots, q_m \} \subset \Omega$, i.e., \[ \lambda_n \prod_{j = 1}^N |x - p_j|^{2 \alpha_j} V(x) e^{u_n} dx \rightharpoonup \sum_{i=1}^m b_i \delta_{q_i} \] in the sense of measures, where $b_i = 8\pi$ if $q_i \notin \Gamma$, $b_i = 8\pi(1 + \alpha_j)$ if $q_i = p_j$ for some $p_j \in \Gamma$. We show that the number of blow up points $m$ is less than or equal to the Morse index of $u_n$ for $n$ sufficiently large, provided $\alpha_j \in (0,+\infty) \setminus \mathbb{N}$ for all $j = 1, \cdots, N$. This is a generalization of the result [13] in which nonsingular case ($\alpha_j = 0$ for all $j$) was studied.
Keywords: Morse indices., Liouville equation, concentration compactness result, singular data, blow up points.
Mathematics Subject Classification: Primary: 35B40, 35J20, 35J2.
Citation: Futoshi Takahashi. Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data. Conference Publications, 2013, 2013 (special) : 729-736. doi: 10.3934/proc.2013.2013.729
References:
[1]

D. Bartolucci, C.C. Chen, C.S. Lin and G. Tarantello:, Profile of blow-up solutions to mean field equations with singular data, Comm. Partial Differential Equations 29 no. 7-8 (2004), 1241-1265.

[2]

D. Bartolucci, and G. Tarantello:, The Liouville equation with singular data: a concentration-compactness principle via a local representation formula, J. Differential Equations 185 (2002), 161-180.

[3]

D. Bartolucci, and G. Tarantello:, Liouville type equations with singular data and their applications to periodic multivortices for the Electroweak Theory, Comm. Math. Pfys. 229 (2002), 3-47.

[4]

H. Brezis, and F. Merle:, Uniform estimates and blow-up behavior for solutions of $-\Delta u = V(x)e^u$ in two dimensions, Comm. Partial Differential Equations 16 (1991), 1223-1253.

[5]

P. Esposito:, A Class of Liouville-Type Equations Arising in Chern-Simons Vortex Theory: Asymptotics and Construction of Blowing Up Solutions, Ph. D. thesis, Universitá degli Studi Roma "Tor Vergata", Roma, Italy, 2003.

[6]

P. Esposito:, Blowup solutions for a Liouville equation with singular data, SIAM. J. Math. Anal. 36 (2005), 1310-1345.

[7]

P. Esposito:, Blowup solutions for a Liouville equation with singular data, in Proceedings of the International Conference " Recent Advances in Elliptic and Parabolic Problems" (C.C. Chen, M. Chipot, C.S. Lin (ed.)), World Scientific, (2005), 61-79.

[8]

Y. Y. Li, and I. Shafrir:, Blow-up analysis for solutions of $-\Delta u = V e^u$ in dimension two, Indiana Univ. Math. J. 43 (1994), 1255-1270.

[9]

L. Ma, and J. Wei:, Convergence for a Liouville equation, Comment. Math. Helv. 76 (2001), 506-514.

[10]

K. Nagasaki, and T. Suzuki:, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptotic Anal. 3 (1990), 173-188.

[11]

J. Prajapat, and G. Tarantello:, On a class of elliptic problems in $\mathbb{R}^2$: symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh 131 A (2001), 967-985.

[12]

F. Takahashi:, Blow up points and the Morse indices of solutions to the Liouville equation in two-dimension, Advances in Nonlinear Stud. 12 no.1, (2012), 115-122.

[13]

F. Takahashi:, Blow up points and the Morse indices of solutions to the Liouville equation : inhomogeneous case, submitted.

[14]

G. Tarantello:, " Selfdual Gauge Field Vortices: An Analytical Approach," Progress in Nonlinear Differential Equations and Their Applications 72, Birkhäuser, Boston (2008)

[15]

Y. Yang:, "Solitons in Field Theory and Nonlinear Analysis," Springer Monographs in Mathematics, Springer-Verlag, New York (2001)

show all references

References:
[1]

D. Bartolucci, C.C. Chen, C.S. Lin and G. Tarantello:, Profile of blow-up solutions to mean field equations with singular data, Comm. Partial Differential Equations 29 no. 7-8 (2004), 1241-1265.

[2]

D. Bartolucci, and G. Tarantello:, The Liouville equation with singular data: a concentration-compactness principle via a local representation formula, J. Differential Equations 185 (2002), 161-180.

[3]

D. Bartolucci, and G. Tarantello:, Liouville type equations with singular data and their applications to periodic multivortices for the Electroweak Theory, Comm. Math. Pfys. 229 (2002), 3-47.

[4]

H. Brezis, and F. Merle:, Uniform estimates and blow-up behavior for solutions of $-\Delta u = V(x)e^u$ in two dimensions, Comm. Partial Differential Equations 16 (1991), 1223-1253.

[5]

P. Esposito:, A Class of Liouville-Type Equations Arising in Chern-Simons Vortex Theory: Asymptotics and Construction of Blowing Up Solutions, Ph. D. thesis, Universitá degli Studi Roma "Tor Vergata", Roma, Italy, 2003.

[6]

P. Esposito:, Blowup solutions for a Liouville equation with singular data, SIAM. J. Math. Anal. 36 (2005), 1310-1345.

[7]

P. Esposito:, Blowup solutions for a Liouville equation with singular data, in Proceedings of the International Conference " Recent Advances in Elliptic and Parabolic Problems" (C.C. Chen, M. Chipot, C.S. Lin (ed.)), World Scientific, (2005), 61-79.

[8]

Y. Y. Li, and I. Shafrir:, Blow-up analysis for solutions of $-\Delta u = V e^u$ in dimension two, Indiana Univ. Math. J. 43 (1994), 1255-1270.

[9]

L. Ma, and J. Wei:, Convergence for a Liouville equation, Comment. Math. Helv. 76 (2001), 506-514.

[10]

K. Nagasaki, and T. Suzuki:, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptotic Anal. 3 (1990), 173-188.

[11]

J. Prajapat, and G. Tarantello:, On a class of elliptic problems in $\mathbb{R}^2$: symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh 131 A (2001), 967-985.

[12]

F. Takahashi:, Blow up points and the Morse indices of solutions to the Liouville equation in two-dimension, Advances in Nonlinear Stud. 12 no.1, (2012), 115-122.

[13]

F. Takahashi:, Blow up points and the Morse indices of solutions to the Liouville equation : inhomogeneous case, submitted.

[14]

G. Tarantello:, " Selfdual Gauge Field Vortices: An Analytical Approach," Progress in Nonlinear Differential Equations and Their Applications 72, Birkhäuser, Boston (2008)

[15]

Y. Yang:, "Solitons in Field Theory and Nonlinear Analysis," Springer Monographs in Mathematics, Springer-Verlag, New York (2001)

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