Critical points of the Trudinger–Moser trace functional with high energy levels

Deng, Shengbing; Musso, Monica

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 59-95

Let Ω be a bounded domain in ${ℝ}^2$ with smooth boundary. In this paper we are concerned with the existence of critical points for the supercritical Trudinger–Moser trace functional

$$\begin{array}{cc}\underset{\partial \Omega }{\int }{e}^{k\pi (1+\mu )u^2}& \text{(0.1)}\end{array}$$

in the set $\{u\in H^1\left(\Omega \right):{\int }_{\Omega }({\left|\nabla u\right|}^2+u^2)dx=1\}$, where $k⩾1$ is an integer and $\mu >0$ is a small parameter. For any integer $k⩾1$ and for any $\mu >0$ sufficiently small, we prove the existence of a pair of k-peaks constrained critical points of the above problem.

Keywords: Trudinger–Moser trace functional, Reduction methods

[1] D. Adams, Traces of potentials. II, Indiana Univ. Math. J. 22 (1973), 907 -918 | MR | Zbl

[2] Adimurthi, S.L. Yadava, Critical exponent problem in ${ℝ}^2$ with Neumann boundary condition, Commun. Partial Differ. Equ. 15 no. 4 (1990), 461 -501 | MR | Zbl

[3] Adimurthi, K. Tintarev, On compactness in the Trudinger–Moser inequality, Ann. Sc. Norm. Super. Pisa (2013), http://dx.doi.org/10.2422/2036-2145.201111_003, http://arxiv.org/abs/1110.3647 | MR | Zbl

[4] Adimurthi, K. Tintarev, On a version of Trudinger–Moser inequality with Möbius shift invariance, Calc. Var. Partial Differ. Equ. 39 (2010), 203 -212 | MR | Zbl

[5] L. Carleson, S.-Y.A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. (2) 110 (1986), 113 -127 | MR | Zbl

[6] P. Cherrier, Problèmes de Neumann non linéaires sur les variétés riemanniennes, C. R. Acad. Sci. Paris Sér. I Math. 292 no. 13 (1981), 637 -640 | MR | Zbl

[7] P. Cherrier, Pascal Meilleures constantes dans des inégalités relatives aux espaces de Sobolev, Bull. Sci. Math. (2) 108 no. 3 (1984), 225 -262 | MR | Zbl

[8] A. Cianchi, Moser–Trudinger trace inequalities, Adv. Math. 217 (2008), 2005 -2044 | MR | Zbl

[9] J. Dávila, M. Del Pino, M. Musso, Concentrating solutions in a two-dimensional elliptic problem with exponential Neumann data, J. Funct. Anal. 227 no. 2 (2005), 430 -490 | MR | Zbl

[10] D.G. De Figueiredo, J.M. Do Ó, B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations, Commun. Pure Appl. Math. 55 (2002), 135 -152 | MR | Zbl

[11] M. Del Pino, M. Musso, B. Ruf, New solutions for Trudinger–Moser critical equations in ${ℝ}^2$ , J. Funct. Anal. 258 (2010), 421 -457 | MR | Zbl

[12] M. Del Pino, M. Musso, B. Ruf, Beyond the Trudinger–Moser supremum, Calc. Var. Partial Differ. Equ. 44 (2012), 543 -576 | MR | Zbl

[13] S.-B. Deng, M. Musso, New solutions for critical Neumann problems ${ℝ}^2$, preprint.

[14] M. Flucher, Extremal functions for the Trudinger–Moser inequality in 2 dimensions, Comment. Math. Helv. 67 (1992), 471 -497 | MR | EuDML | Zbl

[15] T. Lamm, F. Robert, M. Struwe, The heat flow with a critical exponential nonlinearity, J. Funct. Anal. 257 no. 9 (2009), 2951 -2998 | MR | Zbl

[16] Y.-X. Li, Moser–Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differ. Equ. 14 no. 2 (2001), 163 -192 | MR | Zbl

[17] Y.-X. Li, Extremal functions for the Moser–Trudinger inequalities on compact Riemannian manifolds, Sci. China Ser. A 48 no. 5 (2005), 618 -648 | MR | Zbl

[18] Y.-X. Li, P. Liu, A Moser–Trudinger inequality on the boundary of a compact Riemann surface, Math. Z. 250 no. 2 (2005), 363 -386 | MR | Zbl

[19] Y. Li, M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J. 80 (1995), 383 -417 | MR | Zbl

[20] K.C. Lin, Extremal functions for Moser's inequality, Trans. Am. Math. Soc. 348 no. 7 (1996), 2663 -2671 | MR | Zbl

[21] P.L. Lions, The concentration–compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoam. 1 (1985), 145 -201 | MR | EuDML | Zbl

[22] V. Maz'Ya, Sobolev Spaces, Springer Ser. Sov. Math. , Springer-Verlag, Berlin (1985) | MR | Zbl

[23] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970–1971), 1077 -1092 | MR

[24] B. Ou, A uniqueness theorem for harmonic functions on the upper-half plane, Conform. Geom. Dyn. 4 (2000), 120 -125 | MR | Zbl

[25] S.I. Pohozaev, On the Sobolev imbedding theorem for $pl=n$ , Doklady Conference, Section Math. Moscow Power Inst. (1965), 158 -170

[26] M. Struwe, Critical points of embeddings of $H_0^{1,n}$ into Orlicz spaces, Ann. Inst. Henri Poincaré 5 no. 5 (1988), 425 -464 | MR | EuDML | Zbl | Numdam

[27] N.S. Trudinger, On embedding into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473 -483 | MR | Zbl

[28] Y.-Y. Yang, Moser–Trudinger trace inequalities on a compact Riemannian surface with boundary, Pac. J. Math. 227 no. 1 (2006), 177 -200 | MR | Zbl

[29] V.I. Yudovic, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR 138 (1961), 805 -808 | MR

[30] L. Zhang, Classification of conformal metrics on ${ℝ}_{+}^2$ with constant Gauss curvature and geodesic curvature on the boundary under various integral finiteness assumptions, Calc. Var. Partial Differ. Equ. 16 (2003), 405 -430 | MR | Zbl