# American Institute of Mathematical Sciences

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

 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.
Citation: Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure & 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.   Google Scholar [2] H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems,, Revista Matem$\acutea$tica de la Universidad Complutense de madrid, 10 (1997), 443.   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.   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.   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.   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.   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.   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.   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.   Google Scholar [10] H. Brezis and M. Marcus, Hardy's inequalities revisited,, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 25 (1997), 217.   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.   Google Scholar [12] S. Filippas, V. G. Maz'ya and A. Tertikas, Critical Hardy-Sobolev Inequalities,, Journal de Math$\acutee$matiques Pures et Appliqu$\acutee$es, 87 (2007), 37.   Google Scholar [13] J. Dávila and L. Dupaigne, Hardy-type inequalities,, J. Eur. Math. Soc., 6 (2004), 335.   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.   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.   Google Scholar [16] Y. M. Chen, Regularity of solutions to the Dirichlet problem for degenerate elliptic equation,, Chin. Ann. Math., 24 (2003), 529.   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, 136 (2006), 1041.   Google Scholar [18] M. M. Zou, "Sign-Changing Critical Point Theory,", Springer-Verlag, (2008).   Google Scholar [19] E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities,, Courant Lecture Notes in Mathematics, 5 (1999).   Google Scholar

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##### References:
 [1] G. H. Hardy, Note on a theorem of Hilbert,, Mathematische Zeitschrift, 6 (1920), 314.   Google Scholar [2] H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems,, Revista Matem$\acutea$tica de la Universidad Complutense de madrid, 10 (1997), 443.   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.   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.   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.   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.   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.   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.   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.   Google Scholar [10] H. Brezis and M. Marcus, Hardy's inequalities revisited,, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 25 (1997), 217.   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.   Google Scholar [12] S. Filippas, V. G. Maz'ya and A. Tertikas, Critical Hardy-Sobolev Inequalities,, Journal de Math$\acutee$matiques Pures et Appliqu$\acutee$es, 87 (2007), 37.   Google Scholar [13] J. Dávila and L. Dupaigne, Hardy-type inequalities,, J. Eur. Math. Soc., 6 (2004), 335.   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.   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.   Google Scholar [16] Y. M. Chen, Regularity of solutions to the Dirichlet problem for degenerate elliptic equation,, Chin. Ann. Math., 24 (2003), 529.   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, 136 (2006), 1041.   Google Scholar [18] M. M. Zou, "Sign-Changing Critical Point Theory,", Springer-Verlag, (2008).   Google Scholar [19] E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities,, Courant Lecture Notes in Mathematics, 5 (1999).   Google Scholar
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