# American Institute of Mathematical Sciences

November  2011, 10(6): 1645-1662. doi: 10.3934/cpaa.2011.10.1645

## Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains

 1 Instituto de Matemáticas, Universidad Nacional Autónoma de Mexico, Circuito Exterior, C.U., 04510 México D.F., Mexico

Received  May 2010 Revised  April 2011 Published  May 2011

We consider the nonlinear and nonlocal problem

$A_{1/2}u=|u|^{2^{\sharp}-2}u$ in $\Omega, \quad u=0$ on $\partial\Omega$

where $A_{1/2}$ represents the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions, $\Omega$ is a bounded smooth domain in $R^n$, $n\ge 2$ and $2^{\sharp}=2n/(n-1)$ is the critical trace-Sobolev exponent. We assume that $\Omega$ is annular-shaped, i.e., there exist $R_2>R_1>0$ constants such that $\{ x \in R^n$ s.t. $R_1 < |x| < R_2 \}\subset\Omega$ and $0\notin\Omega$, and invariant under a group $\Gamma$ of orthogonal transformations of $R^n$ without fixed points. We establish the existence of positive and multiple sign changing solutions in the two following cases: if $R_1/R_2$ is arbitrary and the minimal $\Gamma$-orbit of $\Omega$ is large enough, or if $R_1/R_2$ is small enough and $\Gamma$ is arbitrary.

Citation: Antonio Capella. Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1645-1662. doi: 10.3934/cpaa.2011.10.1645
##### References:
 [1] A. Bahri and J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294. doi: 10.1002/cpa.3160410302.  Google Scholar [2] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar [3] X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.  Google Scholar [4] M. Clapp and F. Pacella, Multiple solutions to the pure exponent problem in domains with a hole of arbitrary size, Math. Z., 259 (2008), 575-589. doi: 10.1007/s00209-007-0238-9.  Google Scholar [5] M. Clapp and T. Weth, Minimal nodal solutions of the pure critical exponent problem on a symmetric domain, Calc. Var., 21 (2004), 1-14. doi: 10.1007/s00526-003-0241-x.  Google Scholar [6] M. Clapp and T. Weth, Two solutions of the Bahri-Coron problem in punctured domains via the fixed point transfer, Commun. Contemp. Math., 10 (2008), 81-101. doi: 10.1142/S0219199708002715.  Google Scholar [7] J. M. Coron, Topologie et cas limite des injections de Sobolev, C.R. Acad. Sci. Paris Ser. I, 299 (1984), 209-212.  Google Scholar [8] K. Deimling, "Ordinary Differential Equations in Banach Spaces," Lect. Notes Math. 596, Springer-Verlag, Berlin-Heidelberg-New York, 1977.  Google Scholar [9] J. Escobar, Sharp constant in a Sobolev trace inequality, Indiana Univ. Math. J., 37 (1988), 687-698. doi: 10.1512/iumj.1988.37.37033.  Google Scholar [10] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana, 1.2 (1985), 45-121.  Google Scholar [11] M. V. Marchi and F. Pacella, On the existence of nodal solutions of the equation $-\Delta u = |u|^{2^{\ast}-2} u$ with Dirichlet boundary conditions, Diff. Int. Eq., 6 (1993), 849-862.  Google Scholar [12] R. E. Megginson, "An Introduction to Banach Space Theory," Graduate Texts in Mathematics, Vol. 183, Springer-Verlag, New York, 1998.  Google Scholar [13] D. Pohozaev, Eigenfunctions of the equation $\Delta u+ f(u)=0$, (Russian) Soviet Math. Dokl. 6 (1965), 1408-1411.  Google Scholar [14] M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," 1st edition, Springer-Verlag, Berlin, 1990.  Google Scholar [15] J. Tan, "Nonlinear Analysis for an elastic Lattice and a Fractional Laplacian," Ph.D thesis, Universitat Politècnica de Catalunya, Spain, 2008 Google Scholar [16] M. Willem, "Minimax Theorems," PNLDE 24, Birkhäuser, Boston, 1996.  Google Scholar

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##### References:
 [1] A. Bahri and J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294. doi: 10.1002/cpa.3160410302.  Google Scholar [2] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar [3] X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.  Google Scholar [4] M. Clapp and F. Pacella, Multiple solutions to the pure exponent problem in domains with a hole of arbitrary size, Math. Z., 259 (2008), 575-589. doi: 10.1007/s00209-007-0238-9.  Google Scholar [5] M. Clapp and T. Weth, Minimal nodal solutions of the pure critical exponent problem on a symmetric domain, Calc. Var., 21 (2004), 1-14. doi: 10.1007/s00526-003-0241-x.  Google Scholar [6] M. Clapp and T. Weth, Two solutions of the Bahri-Coron problem in punctured domains via the fixed point transfer, Commun. Contemp. Math., 10 (2008), 81-101. doi: 10.1142/S0219199708002715.  Google Scholar [7] J. M. Coron, Topologie et cas limite des injections de Sobolev, C.R. Acad. Sci. Paris Ser. I, 299 (1984), 209-212.  Google Scholar [8] K. Deimling, "Ordinary Differential Equations in Banach Spaces," Lect. Notes Math. 596, Springer-Verlag, Berlin-Heidelberg-New York, 1977.  Google Scholar [9] J. Escobar, Sharp constant in a Sobolev trace inequality, Indiana Univ. Math. J., 37 (1988), 687-698. doi: 10.1512/iumj.1988.37.37033.  Google Scholar [10] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana, 1.2 (1985), 45-121.  Google Scholar [11] M. V. Marchi and F. Pacella, On the existence of nodal solutions of the equation $-\Delta u = |u|^{2^{\ast}-2} u$ with Dirichlet boundary conditions, Diff. Int. Eq., 6 (1993), 849-862.  Google Scholar [12] R. E. Megginson, "An Introduction to Banach Space Theory," Graduate Texts in Mathematics, Vol. 183, Springer-Verlag, New York, 1998.  Google Scholar [13] D. Pohozaev, Eigenfunctions of the equation $\Delta u+ f(u)=0$, (Russian) Soviet Math. Dokl. 6 (1965), 1408-1411.  Google Scholar [14] M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," 1st edition, Springer-Verlag, Berlin, 1990.  Google Scholar [15] J. Tan, "Nonlinear Analysis for an elastic Lattice and a Fractional Laplacian," Ph.D thesis, Universitat Politècnica de Catalunya, Spain, 2008 Google Scholar [16] M. Willem, "Minimax Theorems," PNLDE 24, Birkhäuser, Boston, 1996.  Google Scholar
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