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Singular limits in Liouville-type equations. (English) Zbl 1088.35067

The authors consider Liouville-type equations \(\Delta u+\varepsilon^2 k(x) e^u=0\), \(\varepsilon>0\), in a bounded smooth domain \(\Omega\) in the plane, with homogeneous Dirichlet boundary conditions. This type of equation arises in a wide range of applications, for example in astrophysics and combustion theory. They find conditions under which there exists a solution \(u_\varepsilon\) which blows up at exactly \(m\) points as \(\varepsilon \rightarrow0\) and satisfies \(\lim_{\varepsilon\rightarrow0}\varepsilon^2 \int_\Omega k(x) e^{u_\varepsilon}=8m\pi\). One of the aims of the authors here is to manage without a certain hard-to-check nondegeneracy assumption.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
80A25 Combustion
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[1] Bahri, A., Li, Y.-Y., Rey, O.: On a variational problem with lack of compactness: the topological effect of the critical points at infinity. Calc. Var. 3, 67-93 (1995) · Zbl 0814.35032
[2] Baraket, S., Pacard, F.: Construction of singular limits for a semilinear elliptic equation in dimension \(2\) . Calc. Var. 6, 1-38 (1998) · Zbl 0890.35047
[3] Bartolucci, D., Tarantello, G.: The Liouville equation with singular data: a concentration-compactness principle via a local representation formula. J. Differential Equations 185, 161-180 (2002) · Zbl 1247.35032
[4] Brezis, H., Merle, F.: Uniform estimates and blow-up behavior for solutions of \(-\Delta u=V(x)e^ u\) in two dimensions. Comm. Partial Differential Equations 16, 1223-1253 (1991) · Zbl 0746.35006
[5] Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36, 437-47 (1983) · Zbl 0541.35029
[6] Caffarelli, L., Yang, Y.-S.: Vortex condensation in the Chern-Simons Higgs model: an existence theorem. Comm. Math. Phys. 168, 321-336 (1995) · Zbl 0846.58063
[7] Fowler, R.H.: Further studies on Emden’s and similar differential equations. Quart. J. Math. 2, 259-288 (1931) · Zbl 0003.23502
[8] Caglioti, E., Lions, P.L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, I & II. Comm. Math. Phys. 143, 501-525 (1992) & 174, 229-260 (1995) · Zbl 0745.76001
[9] Chae, D., Imanuvilov, O.: The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory. Comm. Math. Phys. 215, 119-142 (2000) · Zbl 1002.58015
[10] Chandrasekhar, S.: An introduction to the study of stellar structure. Dover, New York 1957 · Zbl 0079.23901
[11] Crandall, M., Rabinowitz, P.H.: Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Rational Mech. Anal. 58, 207-218 (1975) · Zbl 0309.35057
[12] Chang, S.-Y.A., Yang, P.; Conformal deformation of metrics on S2. J. Diff. Geom. 27, 259-296 (1988)
[13] Chang, S.-Y., Gursky, M., Yang, P.: The scalar curvature equation on \(2\) - and \(3\) -spheres. Calc. Var. 1, 205-229 (1993) · Zbl 0822.35043
[14] Chanillo, S., Kiessling, M.: Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry. Comm. Math. Phys. 160, 217-238 (1994) · Zbl 0821.35044
[15] Chen, C.-C., Lin, C.-S.: Topological degree for a mean field equation on Riemann surfaces. Comm. Pure Appl. Math. 56, 1667-1727 (2003) · Zbl 1032.58010
[16] del Pino, M., Felmer, P.: Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal. 149, 245-265 (1997) · Zbl 0887.35058
[17] del Pino, M., Felmer, P., Musso, M.: Two-bubble solutions in the super-critical Bahri-Coron’s problem. Calc. Var. 16, 113-145 (2003) · Zbl 1142.35421
[18] del Pino, M., Dolbeault, J., Musso, M.: ”Bubble-tower” radial solutions in the slightly supercritical Brezis-Nirenberg problem. J. Differential Equations 193, 280-306 (2003) · Zbl 1140.35413
[19] Esposito, P.: A class of Liouville-type equations arising in Chern-Simons vortex theory: asymptotics and construction of blowing-up solutions. Ph.D. Thesis, Universitá di Roma ”Tor Vergata” (2004)
[20] Esposito, P., Grossi, M., Pistoia, A.: On the existence of blowing-up solutions for a mean field equation. Preprint 2004 · Zbl 1129.35376
[21] Gelfand, I.M.: Some problems in the theory of quasilinear equations. Amer. Math. Soc. Transl. 29, 295-381 (1963) · Zbl 0127.04901
[22] Joseph, D.D., Lundgren, T.S.: Quasilinear problems driven by positive sources. Arch. Rat. Mech. Anal. 49, 241-269 (1973) · Zbl 0266.34021
[23] Kazdan, J., Warner, F.: Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures. Ann. Math. 101, 317-331 (1975) · Zbl 0297.53020
[24] Li, Y.-Y., Shafrir, I.: Blow-up analysis for solutions of \(-\Delta u=Ve^ u\) in dimension two. Indiana Univ. Math. J. 43, 1255-1270 (1994) · Zbl 0842.35011
[25] Lin, C.-S.: Topological degree for mean field equations on S2. Duke Math. J. 104, 501-536 (2000) · Zbl 0964.35038
[26] Liouville, J.: Sur L’ Equation aux Difference Partielles \(\frac{d^2 \log \lambda}{dudv} \pm \frac{\lambda}{2a^2} = 0\) . C.R. Acad. Sci. Paris 36, 71-72 (1853)
[27] Ma, L., Wei, J.: Convergence for a Liouville equation. Comment. Math. Helv. 76, 506-514 (2001) · Zbl 0987.35056
[28] Mignot, F., Murat, F., Puel, J.: Variation d’un point de retournement par rapport au domaine. Comm. Partial Differential Equations 4, 1263-1297 (1979) · Zbl 0422.35039
[29] Nagasaki, K., Suzuki, T.: Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities. Asymptotic Anal. 3, 173-188 (1990) · Zbl 0726.35011
[30] Struwe, M., Tarantello, G.: On multivortex solutions in Chern-Simons gauge theory. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8, 109-121 (1998) · Zbl 0912.58046
[31] Suzuki, T.: Two-dimensional Emden-Fowler equation with exponential nonlinearity. Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), pp. 493-512. Progr. Nonlinear Differential Equations Appl. 7. Birkhäuser Boston, Boston, MA 1992
[32] Rey, O.: The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent J. Funct. Anal. 89, 1-52 (1990) · Zbl 0786.35059
[33] Tarantello, G.: A quantization property for blow-up solutions of singular Liouville-type equations. Preprint 2003 · Zbl 1174.35379
[34] Weston, V.H.: On the asymptotic solution of a partial differential equation with an exponential nonlinearity. SIAM J. Math. Anal. 9, 1030-1053 (1978) · Zbl 0402.35038
[35] Ye, D., Zhou, F.: A generalized two dimensional Emden-Fowler equation with exponential nonlinearity. Calc. Var. 13, 141-158 (2001) · Zbl 1077.35048
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