Abstract
We consider the following anisotropic Emden–Fowler equation \(\nabla (a(x) \nabla u)+ \epsilon^{2} a(x) e^{u} = 0 \quad in \quad \Omega, \quad u=0 \quad on \quad \partial \Omega,\) where \(\Omega \subset \mathbb{R}^2\) is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient a(x) on the existence of bubbling solutions. We show that at given local maximum points of a(x), there exists arbitrarily many bubbles. As a consequence, the quantity \(\mathcal{T}_\epsilon = \epsilon^{2} \int_{\Omega} a(x)e^{u} {\rm d}x \) can approach to \( + \infty\) as \(\epsilon \to 0\). These results show a striking difference with the isotropic case [\(a(x) \equiv \) Constant].
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References
Aubin T. (1982) Nonlinear Analysis on Manifolds. Springer, Berlin Heidelberg New York
Baraket S., Pacard F. (1998) Construction of singular limits for a semilinear elliptic equation in dimension 2. Calc. Var. Partial Differ. Equ. 6(1): 1–38
Bartolucci D., Orsina L. (2005) Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Commun. Pure Appl. Anal. 4(3): 499–522
Bates P., Dancer E.N., Shi J. (1999) Multi-spike stationary solutions of the Cahn–Hilliard equation in higher-dimension and instability. Adv. Differ. Equ. 4, 1–69
Brezis H., Merle F. (1991) Uniform estimates and blow-up behavior for solutions of −Δu = V(x)e u in two dimensions. Commun. Partial Differ. Equ. 16(8–9): 1223–1253
Chanilo S., Li Y.Y. (1992) Continuity of solutions of uniformly elliptic equations in \(\mathbb{R}^2\). Manuscr. Math. 77, 415–433
Chen C.C., Lin C.S. (2002) Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces. Commun. Pure Appl. Math. 55(6): 728–771
Chen X. (1999) Remarks on the existence of branch bubbles on the blowup-analysis of equation −Δu = e 2u in dimesnion two. Commun. Anal. Geom. 7, 295–302
Dancer E.N., Yan S. (1999) Multipeak solutions for a singular perturbed Neumann problem. Pac. J. Math. 189, 241–262
Del Pino M., Felmer P. (1997) Semiclassical states for nonlinear Schrodinger equations. J. Funct. Anal. 149, 245–265
Del Pino M., Felmer P., Musso M. (2003) Two-bubble solutions in the super-critical Bahri–Coron’s problem. Calc. Var. Partial Differ. Equ. 16, 113–145
Del Pino M., Kowalczyk M., Musso M. (2005) Singular limits in Liouville-type equations. Calc. Var. Partial Differ. Equ. 24, 47–81
Esposito P., Grossi M., Pistoia A. (2005) On the existence of blowing-up solutions for a mean field equation. Ann. Inst. H. Poincaré Anal. Nonlinéaire 22, 227–257
Gelfand I.M. (1963) Some problems in the theory of quasilinear equations. Am. Math. Soc. Transl. 29, 295–381
Gui C., Wei J. (1999) Multiple interior spike solutions for some singular perturbed Neumann problems. J. Differ. Equ. 158, 1–27
Gui C., Wei J. (2000) On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems. Can. J. Math. 52, 522–538
Gui C., Wei J., Winter M. (2000) Multiple boundary peak solutions for some singularly perturbed Neumann problems. Ann. Inst. H. Poincaré Anal. Nonlinéaire 17, 249–289
Joseph D.D., Lundgren T.S. (1973) Quasilinear problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241–269
Li Y.Y. (1999) Harnack type inequality: the method of moving planes. Commun. Math. Phys. 200, 421–444
Li Y.Y., Shafrir I. (1994) Blow-up analysis for solutions of −Δu = Ve u in dimension two. Indiana Univ. Math. J. 43(4): 1255–1270
Ma L., Wei J. (2001) Convergence for a Liouville equation. Commun. Math. Helv. 76, 506–514
Mignot F., Murat F., Puel J.P. (1979) Variation d’un point retourment par rapport au domaine. Commun. Partial Differ. Equ. 4, 1263–1297
Mizoguchi N., Suzuki T. (1997) Equations of gas combustion: S-shaped bifurcation and mushrooms. J. Differ. Equ. 134, 183–215
Nagasaki K., Suzuki T. (1990) Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities. Asymptot. Anal. 3, 173–188
Rey O., Wei J. (2004) Blow-up solutions for an elliptic Neumann problem with sub-or-supcritical nonlinearity, I: N = 3. J. Funct. Anal. 212(2): 472–499
Wei, J., Ye, D., Zhou, F. Boundary blow-up solution for an anisotropic Emden–Fowler equation, preprint (2006)
Ye D. (1997) Une remarque sur le comportement asymptotique des solutions de \(-\Delta u = \lambda f(u)\). C. R. Acad. Sci. Paris 325, 1279–1282
Ye D., Zhou F. (2001) A generalized two dimensional Emden–Fowler equation with exponential nonlinearity. Calc. Var. Partial Differ. Eq. 13, 141–158
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Wei, J., Ye, D. & Zhou, F. Bubbling solutions for an anisotropic Emden–Fowler equation. Calc. Var. 28, 217–247 (2007). https://doi.org/10.1007/s00526-006-0044-y
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DOI: https://doi.org/10.1007/s00526-006-0044-y