×

Non-simple blow-up solutions for the Neumann two-dimensional sinh-Gordon equation. (English) Zbl 1169.35049

The authors prove that there exists small \(\rho_0 > 0\) such that for any \(0 < \rho \leq \rho_0\) the problem
\[ -\Delta u = \rho^2 \bigg(e^u - \frac{1}{\pi}\int_B e^u\bigg)- \rho^2 \bigg(e^{-u} -\frac{1}{\pi}\int_B e^{-u}\bigg) \]
in the unit two-dimensional ball \(B\) and
\[ \frac{\partial u}{\partial \nu} = 0 \]
on \(\partial B\) has a solution \(u_\rho\) such that as \(\rho \to 0\)
\[ \rho^2 e^{u_\rho} \to 8\pi \delta(x), \qquad \rho^2 e^{-u_\rho} \to 24\pi \delta(x) \]
hold weakly in the sense of measure on \(\overline{B}\).
Therewith they present the first nontrivial example of non-simple blow-up of solutions of \(\sinh\)-Gordon equations and partially answered a question posed by J. Jost, G. Wang, D. Ye and C. Zhou [Calc. Var. Partial Differ. Equ. 31, No. 2, 263–276 (2008; Zbl 1137.35061)].

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35J60 Nonlinear elliptic equations
35B33 Critical exponents in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Citations:

Zbl 1137.35061
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baraket S., Pacard F.: Construction of singular limits for a semilinear elliptic equation in dimension 2. Calc. Var. Partial Diff. Equ. 6(1), 1–38 (1998) · Zbl 0890.35047
[2] Bartolucci D., Pistoia A.: Existence and qualitative properties of concentrating solutions for the sinh-Poisson equation. IMA J. Appl. Math. 72(6), 706–729 (2007) · Zbl 1154.35072
[3] Chen X.: Remarks on the existence of branch bubbles on the blowup analysis of equation {\(\Delta\)}u = e 2u in dimension two. Comm. Anal. Geom. 7, 295–302 (1999) · Zbl 0928.35051
[4] del Pino M., Kowalczyk M., Musso M.: Singular limits in Liouville-type equations. Calc. Var. Partial Diff. Equ. 24(1), 47–81 (2005) · Zbl 1088.35067
[5] Esposito P., Grossi M., Pistoia A.: On the existence of blowing-up solutions for a mean field equation. Ann. IHP Anal. Non Lineaire 22(2), 127–157 (2005) · Zbl 1077.76060
[6] Esposito P., Musso M., Pistoia A.: Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent. J. Diff. Equ. 227(1), 29–68 (2006) · Zbl 1254.35083
[7] Esposito P., Pistoia A., Wei J.: Concentrating solutions for the Hénon equation in \({\mathbb {R}^2}\) . J. Anal. Math. 100, 249–280 (2006) · Zbl 1173.35504
[8] Jost J., Wang G., Ye D., Zhou C.: The blow up analysis of solutions to the elliptic sinh-Gordon equation. Calc. Var. Partial Diff. Equ. 31(2), 263–276 (2008) · Zbl 1137.35061
[9] Li Y.Y., Shafrir I.: Blow up analysis for solutions of {\(\Delta\)}u = V(x)e u in dimension two. Indiana Univ. Math. J. 43, 1255–1270 (1994) · Zbl 0842.35011
[10] Ohtsuka H., Suzuki T.: Mean field equation for the equilibrium turbulence and a related functional inequality. Adv. Diff. Equ. 11, 281–304 (2006) · Zbl 1109.26014
[11] Wei J., Ye D., Zhou F.: Bubbling solutions for anisotropic Emden–Fowler equation. Calc. Var. Partial Diff. Equ. 28(2), 217–247 (2007) · Zbl 1159.35402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.