    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

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  in which nonsingular case ($\alpha_j = 0$ for all $j$) was studied.
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:
  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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).   Google Scholar  P. Esposito:, Blowup solutions for a Liouville equation with singular data,, SIAM. J. Math. Anal. 36 (2005), 36 (2005), 1310. Google Scholar  P. Esposito:, Blowup solutions for a Liouville equation with singular data,, in Proceedings of the International Conference, (2005), 61. Google Scholar  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. Google Scholar  L. Ma, and J. Wei:, Convergence for a Liouville equation,, Comment. Math. Helv. 76 (2001), 76 (2001), 506. Google Scholar  K. Nagasaki, and T. Suzuki:, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities,, Asymptotic Anal. 3 (1990), 3 (1990), 173. Google Scholar  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. Google Scholar  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. Google Scholar  F. Takahashi:, Blow up points and the Morse indices of solutions to the Liouville equation : inhomogeneous case,, submitted., ().   Google Scholar  G. Tarantello:, " Selfdual Gauge Field Vortices: An Analytical Approach,", Progress in Nonlinear Differential Equations and Their Applications 72, (2008). Google Scholar  Y. Yang:, "Solitons in Field Theory and Nonlinear Analysis,", Springer Monographs in Mathematics, (2001). Google Scholar

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##### References:
  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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).   Google Scholar  P. Esposito:, Blowup solutions for a Liouville equation with singular data,, SIAM. J. Math. Anal. 36 (2005), 36 (2005), 1310. Google Scholar  P. Esposito:, Blowup solutions for a Liouville equation with singular data,, in Proceedings of the International Conference, (2005), 61. Google Scholar  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. Google Scholar  L. Ma, and J. Wei:, Convergence for a Liouville equation,, Comment. Math. Helv. 76 (2001), 76 (2001), 506. Google Scholar  K. Nagasaki, and T. Suzuki:, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities,, Asymptotic Anal. 3 (1990), 3 (1990), 173. Google Scholar  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. Google Scholar  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. Google Scholar  F. Takahashi:, Blow up points and the Morse indices of solutions to the Liouville equation : inhomogeneous case,, submitted., ().   Google Scholar  G. Tarantello:, " Selfdual Gauge Field Vortices: An Analytical Approach,", Progress in Nonlinear Differential Equations and Their Applications 72, (2008). Google Scholar  Y. Yang:, "Solitons in Field Theory and Nonlinear Analysis,", Springer Monographs in Mathematics, (2001). Google Scholar
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