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November  2013, 12(6): 2565-2575. doi: 10.3934/cpaa.2013.12.2565

Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation

Jun Yang 1, and  Yaotian Shen 2,

1. 

Department of Mathematics, South China University of Technology, Guangzhou, 510640, China
2. 

Department of Mathematics, South China University of Technology, Guangzhou 510640

Received  July 2012 Revised  January 2013 Published  May 2013

Some critical Sobolev-Hardy inequalities with weight of distance function $d^{\frac{\alpha}{p}p^*}$ are established in a bounded domain $\Omega$, where $d$ is the distance to the boundary $\partial\Omega$. Using these inequalities we get the result that the embedding $\mathcal{D}^{1, 2}(\Omega, d^\alpha)\hookrightarrow L^q(\Omega, d^{\beta})$ is compact if $1\leq q<2^*$ and $\beta >\frac{\alpha}{2}q+\frac{q}{2^*}-1$. By the compactness result and critical-point theory about sign-changing solutions, we obtain infinitely many sign-changing solutions to a degenerate Dirichlet elliptic equation $-\hbox{div}(d^\alpha \nabla u)- \frac{(1-\alpha )^2}{4} d^{\alpha-2} u=f(x,u)$ provided that $f(x,u)$ satisfies suitable conditions.
Keywords: Sobolev-Hardy inequalities, degenerate elliptic equation., critical Hardy potential, Sign-changing solution.
Mathematics Subject Classification: Primary: 35J57, 35J66; Secondary: 35J7.
Citation: Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure &amp; Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565
References:
[1]

G. H. Hardy, Note on a theorem of Hilbert, Mathematische Zeitschrift, 6 (1920), 314-317.  Google Scholar

[2]

H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Revista Matemática de la Universidad Complutense de madrid, 10 (1997), 443-469.  Google Scholar

[3]

F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Transactions of the American Mathematical Society, 356 (2004), 2149-2168.  Google Scholar

[4]

Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its application, Proceedings of the American Mathematical Society, 130 (2002), 489-505.  Google Scholar

[5]

Adimurthi and M. J. Esteban, An improved Hardy-Sobolev inequality in $W^{1,p}$ and its application to Schrödinger operators, Nonlinear Differential Equatons and Applications, 12 (2005), 243-263.  Google Scholar

[6]

B. Abdellaoui, E. Colorado and I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities, Calculus of Variations and Partial Differential Equations, 23 (2005), 327-345.  Google Scholar

[7]

Y. T. Shen, The Dirichlet problem for degenerate or singular elliptic equation of high order, Journal of China University of Science and Technology, 10 (1980), 1-11. Google Scholar

[8]

Y. T. Shen and X. K. Guo, Weighted Poincaré inequalities on unbounded domains and nonlinear elliptic boundary value problems, Acta Mathematica Scientia, 4 (1984), 277-286.  Google Scholar

[9]

G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants, Trans. Amer. Math. Soc., 356 (2004), 2169-2196. Google Scholar

[10]

H. Brezis and M. Marcus, Hardy's inequalities revisited, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, Ser. IV 25 (1997), 217-237.  Google Scholar

[11]

S. Filippas, V. G. Maz'ya and A. Tertikas, On a question of Brezis and marcus, Calc. of Variations and P.D.E., 25 (2006), 491-501.  Google Scholar

[12]

S. Filippas, V. G. Maz'ya and A. Tertikas, Critical Hardy-Sobolev Inequalities, Journal de Mathématiques Pures et Appliquées, 87 (2007), 37-56.  Google Scholar

[13]

J. Dávila and L. Dupaigne, Hardy-type inequalities, J. Eur. Math. Soc., 6 (2004), 335-365.  Google Scholar

[14]

M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators, Annali Mat. Pura Appl., 80 (1968), 1-122.  Google Scholar

[15]

A. Kristály and C. Varga, Multiple solutions for a degenerate elliptic equation involving sublinear terms at infinity, J. Math. Anal. Appl., 352 (2009), 139-148.  Google Scholar

[16]

Y. M. Chen, Regularity of solutions to the Dirichlet problem for degenerate elliptic equation, Chin. Ann. Math., Ser. B, 24 (2003), 529-540.  Google Scholar

[17]

Y. T. Shen and Y. X. Yao, Nonlinear elliptic equations with critical potential and critical parameter, Proceedings of the Royal Society of Edinburgh, Sect. A, 136 (2006), 1041-1051.  Google Scholar

[18]

M. M. Zou, "Sign-Changing Critical Point Theory," Springer-Verlag, New York, 2008. Google Scholar

[19]

E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes in Mathematics, 5 (1999), A.M.S.  Google Scholar

show all references

References:
[1]

G. H. Hardy, Note on a theorem of Hilbert, Mathematische Zeitschrift, 6 (1920), 314-317.  Google Scholar

[2]

H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Revista Matemática de la Universidad Complutense de madrid, 10 (1997), 443-469.  Google Scholar

[3]

F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Transactions of the American Mathematical Society, 356 (2004), 2149-2168.  Google Scholar

[4]

Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its application, Proceedings of the American Mathematical Society, 130 (2002), 489-505.  Google Scholar

[5]

Adimurthi and M. J. Esteban, An improved Hardy-Sobolev inequality in $W^{1,p}$ and its application to Schrödinger operators, Nonlinear Differential Equatons and Applications, 12 (2005), 243-263.  Google Scholar

[6]

B. Abdellaoui, E. Colorado and I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities, Calculus of Variations and Partial Differential Equations, 23 (2005), 327-345.  Google Scholar

[7]

Y. T. Shen, The Dirichlet problem for degenerate or singular elliptic equation of high order, Journal of China University of Science and Technology, 10 (1980), 1-11. Google Scholar

[8]

Y. T. Shen and X. K. Guo, Weighted Poincaré inequalities on unbounded domains and nonlinear elliptic boundary value problems, Acta Mathematica Scientia, 4 (1984), 277-286.  Google Scholar

[9]

G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants, Trans. Amer. Math. Soc., 356 (2004), 2169-2196. Google Scholar

[10]

H. Brezis and M. Marcus, Hardy's inequalities revisited, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, Ser. IV 25 (1997), 217-237.  Google Scholar

[11]

S. Filippas, V. G. Maz'ya and A. Tertikas, On a question of Brezis and marcus, Calc. of Variations and P.D.E., 25 (2006), 491-501.  Google Scholar

[12]

S. Filippas, V. G. Maz'ya and A. Tertikas, Critical Hardy-Sobolev Inequalities, Journal de Mathématiques Pures et Appliquées, 87 (2007), 37-56.  Google Scholar

[13]

J. Dávila and L. Dupaigne, Hardy-type inequalities, J. Eur. Math. Soc., 6 (2004), 335-365.  Google Scholar

[14]

M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators, Annali Mat. Pura Appl., 80 (1968), 1-122.  Google Scholar

[15]

A. Kristály and C. Varga, Multiple solutions for a degenerate elliptic equation involving sublinear terms at infinity, J. Math. Anal. Appl., 352 (2009), 139-148.  Google Scholar

[16]

Y. M. Chen, Regularity of solutions to the Dirichlet problem for degenerate elliptic equation, Chin. Ann. Math., Ser. B, 24 (2003), 529-540.  Google Scholar

[17]

Y. T. Shen and Y. X. Yao, Nonlinear elliptic equations with critical potential and critical parameter, Proceedings of the Royal Society of Edinburgh, Sect. A, 136 (2006), 1041-1051.  Google Scholar

[18]

M. M. Zou, "Sign-Changing Critical Point Theory," Springer-Verlag, New York, 2008. Google Scholar

[19]

E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes in Mathematics, 5 (1999), A.M.S.  Google Scholar

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