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

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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

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

Abstract
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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), 29 (2004), 7.
[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), 185* (2002), 161.
[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), 229* (2002), 3.
[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), 16 (1991), 1223.
[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, (2003).
[6]	P. Esposito:, Blowup solutions for a Liouville equation with singular data,, SIAM. J. Math. Anal. 36 (2005), 36 (2005), 1310.
[7]	P. Esposito:, Blowup solutions for a Liouville equation with singular data,, in Proceedings of the International Conference, (2005), 61.
[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), 43 (1994), 1255.
[9]	L. Ma, and J. Wei:, Convergence for a Liouville equation,, Comment. Math. Helv. 76 (2001), 76 (2001), 506.
[10]	K. Nagasaki, and T. Suzuki:, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities,, Asymptotic Anal. 3 (1990), 3 (1990), 173.
[11]	J. Prajapat, and G. Tarantello:, On a class of elliptic problems in $\mathbbR^2$: symmetry and uniqueness results,, Proc. Roy. Soc. Edinburgh 131 A* (2001), 131 A* (2001), 967.
[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, 12 (2012), 115.
[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, (2008).
[15]	Y. Yang:, "Solitons in Field Theory and Nonlinear Analysis,", Springer Monographs in Mathematics, (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), 29 (2004), 7.
[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), 185* (2002), 161.
[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), 229* (2002), 3.
[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), 16 (1991), 1223.
[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, (2003).
[6]	P. Esposito:, Blowup solutions for a Liouville equation with singular data,, SIAM. J. Math. Anal. 36 (2005), 36 (2005), 1310.
[7]	P. Esposito:, Blowup solutions for a Liouville equation with singular data,, in Proceedings of the International Conference, (2005), 61.
[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), 43 (1994), 1255.
[9]	L. Ma, and J. Wei:, Convergence for a Liouville equation,, Comment. Math. Helv. 76 (2001), 76 (2001), 506.
[10]	K. Nagasaki, and T. Suzuki:, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities,, Asymptotic Anal. 3 (1990), 3 (1990), 173.
[11]	J. Prajapat, and G. Tarantello:, On a class of elliptic problems in $\mathbbR^2$: symmetry and uniqueness results,, Proc. Roy. Soc. Edinburgh 131 A* (2001), 131 A* (2001), 967.
[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, 12 (2012), 115.
[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, (2008).
[15]	Y. Yang:, "Solitons in Field Theory and Nonlinear Analysis,", Springer Monographs in Mathematics, (2001).

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