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March  2012, 11(2): 453-464. doi: 10.3934/cpaa.2012.11.453

Solutions for polyharmonic elliptic problems with critical nonlinearities in symmetric domains

Jingbo Dou 1, and  Qianqiao Guo 2,

1. 

Center for Nonlinear Studies, Northwest University, Xi'an, Shaanxi, 710069, China
2. 

Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi, 710129, China

Received  January 2010 Revised  May 2011 Published  October 2011

Consider the polyharmonic elliptic problem \begin{eqnarray*} (-\Delta)^m u=\lambda u+Q(x)|u|^{2^*-2}u & in \Omega, \\ (\frac{\partial}{\partial\nu })^j u |_{\partial\Omega}=0, j=0, 1, 2, \cdots, m-1,& \end{eqnarray*} where $\Omega$ is an open bounded domain with smooth boundary in $R^N$, $m\geq 1,N>2m, 2^*=\frac{2N}{N-2m}$ is the critical Sobolev exponent, $Q(x)$ is positive and continuous in $\overline{\Omega}$. We prove the existence of nontrivial and sign-changing solutions when $\Omega$, $Q(x)$ are invariant under a group of orthogonal transformations and $0 < \lambda < \lambda_1$, where $\lambda_1$ is the first Dirichlet eigenvalue of $(-\Delta)^m$ in $\Omega$.
Keywords: Polyharmonic elliptic problem, symmetric domain., critical Sobolev exponent.
Mathematics Subject Classification: Primary: 35B33, 35J35; Secondary: 35G3.
Citation: Jingbo Dou, Qianqiao Guo. Solutions for polyharmonic elliptic problems with critical nonlinearities in symmetric domains. Communications on Pure & Applied Analysis, 2012, 11 (2) : 453-464. doi: 10.3934/cpaa.2012.11.453
References:
[1]

H. Brézis 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

[2]

O. Rey, A multiplicity result for a variational problem with lack of compactness,, Nonl. Anal., 133 (1989), 1241.  doi: 10.1016/0362-546X(89)90009-6.  Google Scholar

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M. Lazzo, Solutions positives multiples pour une équation elliptique non linéaire avec l'exposant critique de Sobolev,, C. R. Acad. Sci. Paris S\'erie. I, 314 (1992), 61.   Google Scholar

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G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents,, J. Funct. Anal., 69 (1986), 289.  doi: 10.1016/0022-1236(86)90094-7.  Google Scholar

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A. Castro and M. Clapp, The effect of the domain topology on the number of minimal nodal solutions of an elliptic equation at critical growth in a symmetric domain,, Nonlinearity, 16 (2003), 579.  doi: 10.1088/0951-7715/16/2/313.  Google Scholar

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A. Cano and M. Clapp, Multiple positive and 2-nodal symmetric solutions of elliptic problems with critical nonlinearity,, J. Diff. Eqs., 237 (2007), 133.  doi: 10.1016/j.jde.2007.03.002.  Google Scholar

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P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators,, J. Math. Pures Appl., 69 (1990), 55.   Google Scholar

[8]

P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[9]

F. Gazzola, Critical growth problems for polyharmonic operators,, Proc. Roy. Soc. Edinburgh, 128A (1998), 251.   Google Scholar

[10]

H. Grunau, The Dirichlet problem for some semilinear elliptic differential equations of arbitrary order,, Analysis, 11 (1991), 83.  doi: 10.1080/03605309508821090.  Google Scholar

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H. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents,, Calc. Var. PDE., 3 (1995), 243.  doi: 10.1007/BF01205006.  Google Scholar

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C. A. Swanson, The best Sobolev constant,, Applicable Anal., 47 (1992), 227.  doi: 10.1080/00036819208840142.  Google Scholar

[13]

M. Clapp and M. Squassina, Nonhomogeneous polyharmonic elliptic problems at critical growth with symmetric data,, Comm. Pure Appl.Anal., 2 (2003), 171.  doi: 10.3934/cpaa.2003.2.171.  Google Scholar

[14]

Q. Guo and P. Niu, Nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains,, J. Diff. Eqs., 245 (2008), 3974.  doi: 10.1016/j.jde.2008.08.002.  Google Scholar

[15]

M. Clapp, A global compactness result for elliptic problems with critical nonlinearity on symmetric domains,, Nonlinear Equations: Methods, (2003), 117.   Google Scholar

[16]

D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities,, Trans. Amer. Math. Soc., 357 (2005), 2909.  doi: 10.1090/S0002-9947-04-03769-9.  Google Scholar

[17]

M. Struwe, "Variational Methods,", Springer-Verlag, (1996).  doi: 10.1007/978-3-540-74013-1.  Google Scholar

[18]

H. Grunau and G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions,, Math. Ann., 307 (1997), 589.  doi: 10.1007/s002080050052.  Google Scholar

[19]

F. Gazzola, H. Grunau and G. Sweers, "Polyharmonic Boundary Value Problems,", Springer, (2009), 305.  doi: 10.1007/978-3-642-12245-3.  Google Scholar

show all references

References:
[1]

H. Brézis 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

[2]

O. Rey, A multiplicity result for a variational problem with lack of compactness,, Nonl. Anal., 133 (1989), 1241.  doi: 10.1016/0362-546X(89)90009-6.  Google Scholar

[3]

M. Lazzo, Solutions positives multiples pour une équation elliptique non linéaire avec l'exposant critique de Sobolev,, C. R. Acad. Sci. Paris S\'erie. I, 314 (1992), 61.   Google Scholar

[4]

G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents,, J. Funct. Anal., 69 (1986), 289.  doi: 10.1016/0022-1236(86)90094-7.  Google Scholar

[5]

A. Castro and M. Clapp, The effect of the domain topology on the number of minimal nodal solutions of an elliptic equation at critical growth in a symmetric domain,, Nonlinearity, 16 (2003), 579.  doi: 10.1088/0951-7715/16/2/313.  Google Scholar

[6]

A. Cano and M. Clapp, Multiple positive and 2-nodal symmetric solutions of elliptic problems with critical nonlinearity,, J. Diff. Eqs., 237 (2007), 133.  doi: 10.1016/j.jde.2007.03.002.  Google Scholar

[7]

P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators,, J. Math. Pures Appl., 69 (1990), 55.   Google Scholar

[8]

P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[9]

F. Gazzola, Critical growth problems for polyharmonic operators,, Proc. Roy. Soc. Edinburgh, 128A (1998), 251.   Google Scholar

[10]

H. Grunau, The Dirichlet problem for some semilinear elliptic differential equations of arbitrary order,, Analysis, 11 (1991), 83.  doi: 10.1080/03605309508821090.  Google Scholar

[11]

H. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents,, Calc. Var. PDE., 3 (1995), 243.  doi: 10.1007/BF01205006.  Google Scholar

[12]

C. A. Swanson, The best Sobolev constant,, Applicable Anal., 47 (1992), 227.  doi: 10.1080/00036819208840142.  Google Scholar

[13]

M. Clapp and M. Squassina, Nonhomogeneous polyharmonic elliptic problems at critical growth with symmetric data,, Comm. Pure Appl.Anal., 2 (2003), 171.  doi: 10.3934/cpaa.2003.2.171.  Google Scholar

[14]

Q. Guo and P. Niu, Nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains,, J. Diff. Eqs., 245 (2008), 3974.  doi: 10.1016/j.jde.2008.08.002.  Google Scholar

[15]

M. Clapp, A global compactness result for elliptic problems with critical nonlinearity on symmetric domains,, Nonlinear Equations: Methods, (2003), 117.   Google Scholar

[16]

D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities,, Trans. Amer. Math. Soc., 357 (2005), 2909.  doi: 10.1090/S0002-9947-04-03769-9.  Google Scholar

[17]

M. Struwe, "Variational Methods,", Springer-Verlag, (1996).  doi: 10.1007/978-3-540-74013-1.  Google Scholar

[18]

H. Grunau and G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions,, Math. Ann., 307 (1997), 589.  doi: 10.1007/s002080050052.  Google Scholar

[19]

F. Gazzola, H. Grunau and G. Sweers, "Polyharmonic Boundary Value Problems,", Springer, (2009), 305.  doi: 10.1007/978-3-642-12245-3.  Google Scholar

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