Solutions for polyharmonic elliptic problems with critical nonlinearities in symmetric domains

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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

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

Abstract
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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:

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[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.
[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.
[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.
[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.
[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.
[7]	P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators,, J. Math. Pures Appl., 69 (1990), 55.
[8]	P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681. doi: 10.1512/iumj.1986.35.35036.
[9]	F. Gazzola, Critical growth problems for polyharmonic operators,, Proc. Roy. Soc. Edinburgh, 128A (1998), 251.
[10]	H. Grunau, The Dirichlet problem for some semilinear elliptic differential equations of arbitrary order,, Analysis, 11 (1991), 83. doi: 10.1080/03605309508821090.
[11]	H. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents,, Calc. Var. PDE., 3 (1995), 243. doi: 10.1007/BF01205006.
[12]	C. A. Swanson, The best Sobolev constant,, Applicable Anal., 47 (1992), 227. doi: 10.1080/00036819208840142.
[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.
[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.
[15]	M. Clapp, A global compactness result for elliptic problems with critical nonlinearity on symmetric domains,, Nonlinear Equations: Methods, (2003), 117.
[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.
[17]	M. Struwe, "Variational Methods,", Springer-Verlag, (1996). doi: 10.1007/978-3-540-74013-1.
[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.
[19]	F. Gazzola, H. Grunau and G. Sweers, "Polyharmonic Boundary Value Problems,", Springer, (2009), 305. doi: 10.1007/978-3-642-12245-3.

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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.
[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.
[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.
[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.
[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.
[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.
[7]	P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators,, J. Math. Pures Appl., 69 (1990), 55.
[8]	P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681. doi: 10.1512/iumj.1986.35.35036.
[9]	F. Gazzola, Critical growth problems for polyharmonic operators,, Proc. Roy. Soc. Edinburgh, 128A (1998), 251.
[10]	H. Grunau, The Dirichlet problem for some semilinear elliptic differential equations of arbitrary order,, Analysis, 11 (1991), 83. doi: 10.1080/03605309508821090.
[11]	H. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents,, Calc. Var. PDE., 3 (1995), 243. doi: 10.1007/BF01205006.
[12]	C. A. Swanson, The best Sobolev constant,, Applicable Anal., 47 (1992), 227. doi: 10.1080/00036819208840142.
[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.
[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.
[15]	M. Clapp, A global compactness result for elliptic problems with critical nonlinearity on symmetric domains,, Nonlinear Equations: Methods, (2003), 117.
[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.
[17]	M. Struwe, "Variational Methods,", Springer-Verlag, (1996). doi: 10.1007/978-3-540-74013-1.
[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.
[19]	F. Gazzola, H. Grunau and G. Sweers, "Polyharmonic Boundary Value Problems,", Springer, (2009), 305. doi: 10.1007/978-3-642-12245-3.

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