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 Mexico, 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.  doi: 10.1002/cpa.3160410302.  Google Scholar

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H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437.  doi: 10.1002/cpa.3160360405.  Google Scholar

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X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052.  doi: 10.1016/j.aim.2010.01.025.  Google Scholar

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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.  doi: 10.1007/s00209-007-0238-9.  Google Scholar

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M. Clapp and T. Weth, Minimal nodal solutions of the pure critical exponent problem on a symmetric domain,, Calc. Var., 21 (2004), 1.  doi: 10.1007/s00526-003-0241-x.  Google Scholar

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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.  doi: 10.1142/S0219199708002715.  Google Scholar

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J. M. Coron, Topologie et cas limite des injections de Sobolev,, C.R. Acad. Sci. Paris Ser. I, 299 (1984), 209.   Google Scholar

[8]

K. Deimling, "Ordinary Differential Equations in Banach Spaces,", Lect. Notes Math. 596, (1977).   Google Scholar

[9]

J. Escobar, Sharp constant in a Sobolev trace inequality,, Indiana Univ. Math. J., 37 (1988), 687.  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.   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.   Google Scholar

[12]

R. E. Megginson, "An Introduction to Banach Space Theory,", Graduate Texts in Mathematics, (1998).   Google Scholar

[13]

D. Pohozaev, Eigenfunctions of the equation $\Delta u+ f(u)=0$, (Russian), Soviet Math. Dokl. 6 (1965), 6 (1965), 1408.   Google Scholar

[14]

M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", 1st edition, (1990).   Google Scholar

[15]

J. Tan, "Nonlinear Analysis for an elastic Lattice and a Fractional Laplacian,", Ph.D thesis, (2008).   Google Scholar

[16]

M. Willem, "Minimax Theorems,", PNLDE 24, (1996).   Google Scholar

show all references

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.  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.  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.  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.  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.  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.  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.   Google Scholar

[8]

K. Deimling, "Ordinary Differential Equations in Banach Spaces,", Lect. Notes Math. 596, (1977).   Google Scholar

[9]

J. Escobar, Sharp constant in a Sobolev trace inequality,, Indiana Univ. Math. J., 37 (1988), 687.  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.   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.   Google Scholar

[12]

R. E. Megginson, "An Introduction to Banach Space Theory,", Graduate Texts in Mathematics, (1998).   Google Scholar

[13]

D. Pohozaev, Eigenfunctions of the equation $\Delta u+ f(u)=0$, (Russian), Soviet Math. Dokl. 6 (1965), 6 (1965), 1408.   Google Scholar

[14]

M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", 1st edition, (1990).   Google Scholar

[15]

J. Tan, "Nonlinear Analysis for an elastic Lattice and a Fractional Laplacian,", Ph.D thesis, (2008).   Google Scholar

[16]

M. Willem, "Minimax Theorems,", PNLDE 24, (1996).   Google Scholar

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