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

Previous Article
Dynamics of vacuum states for one-dimensional full compressible Navier-Stokes equations
CPAA Home
This Issue
Next Article
Four positive solutions of a quasilinear elliptic equation in $ R^N$

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

Abstract
Related Papers

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 & 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.
[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.
[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.
[4]	Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its application,, Proceedings of the American Mathematical Society, 130 (2002), 489.
[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.
[6]	B. Abdellaoui, E. Colorado and I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities,, Calculus of Variations and Partial Differential Equations, 23 (2005), 327.
[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.
[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.
[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.
[10]	H. Brezis and M. Marcus, Hardy's inequalities revisited,, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 25 (1997), 217.
[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.
[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.
[13]	J. Dávila and L. Dupaigne, Hardy-type inequalities,, J. Eur. Math. Soc., 6 (2004), 335.
[14]	M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators,, Annali Mat. Pura Appl., 80 (1968), 1.
[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.
[16]	Y. M. Chen, Regularity of solutions to the Dirichlet problem for degenerate elliptic equation,, Chin. Ann. Math., 24 (2003), 529.
[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.
[18]	M. M. Zou, "Sign-Changing Critical Point Theory,", Springer-Verlag, (2008).
[19]	E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities,, Courant Lecture Notes in Mathematics, 5 (1999).

show all references

References:

[1]	G. H. Hardy, Note on a theorem of Hilbert,, Mathematische Zeitschrift, 6 (1920), 314.
[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.
[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.
[4]	Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its application,, Proceedings of the American Mathematical Society, 130 (2002), 489.
[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.
[6]	B. Abdellaoui, E. Colorado and I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities,, Calculus of Variations and Partial Differential Equations, 23 (2005), 327.
[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.
[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.
[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.
[10]	H. Brezis and M. Marcus, Hardy's inequalities revisited,, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 25 (1997), 217.
[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.
[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.
[13]	J. Dávila and L. Dupaigne, Hardy-type inequalities,, J. Eur. Math. Soc., 6 (2004), 335.
[14]	M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators,, Annali Mat. Pura Appl., 80 (1968), 1.
[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.
[16]	Y. M. Chen, Regularity of solutions to the Dirichlet problem for degenerate elliptic equation,, Chin. Ann. Math., 24 (2003), 529.
[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.
[18]	M. M. Zou, "Sign-Changing Critical Point Theory,", Springer-Verlag, (2008).
[19]	E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities,, Courant Lecture Notes in Mathematics, 5 (1999).

[1]	Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439
[2]	Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
[3]	Tsung-Fang Wu. On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function. Communications on Pure & Applied Analysis, 2008, 7 (2) : 383-405. doi: 10.3934/cpaa.2008.7.383
[4]	Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033
[5]	Fengshuang Gao, Yuxia Guo. Multiple solutions for a critical quasilinear equation with Hardy potential. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1977-2003. doi: 10.3934/dcdss.2019128
[6]	Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499
[7]	Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[8]	Jinhui Chen, Haitao Yang. A result on Hardy-Sobolev critical elliptic equations with boundary singularities. Communications on Pure & Applied Analysis, 2007, 6 (1) : 191-201. doi: 10.3934/cpaa.2007.6.191
[9]	Mateus Balbino Guimarães, Rodrigo da Silva Rodrigues. Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2697-2713. doi: 10.3934/cpaa.2013.12.2697
[10]	Craig Cowan. Optimal Hardy inequalities for general elliptic operators with improvements. Communications on Pure & Applied Analysis, 2010, 9 (1) : 109-140. doi: 10.3934/cpaa.2010.9.109
[11]	Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151
[12]	Aixia Qian, Shujie Li. Multiple sign-changing solutions of an elliptic eigenvalue problem. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 737-746. doi: 10.3934/dcds.2005.12.737
[13]	Chunhua Wang, Jing Yang. Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1603-1628. doi: 10.3934/dcds.2016.36.1603
[14]	Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363
[15]	Jann-Long Chern, Yong-Li Tang, Chuan-Jen Chyan, Yi-Jung Chen. On the uniqueness of singular solutions for a Hardy-Sobolev equation. Conference Publications, 2013, 2013 (special) : 123-128. doi: 10.3934/proc.2013.2013.123
[16]	Guirong Liu, Yuanwei Qi. Sign-changing solutions of a quasilinear heat equation with a source term. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1389-1414. doi: 10.3934/dcdsb.2013.18.1389
[17]	Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure & Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015
[18]	Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164
[19]	Stathis Filippas, Luisa Moschini, Achilles Tertikas. Trace Hardy--Sobolev--Maz'ya inequalities for the half fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (2) : 373-382. doi: 10.3934/cpaa.2015.14.373
[20]	Masato Hashizume, Chun-Hsiung Hsia, Gyeongha Hwang. On the Neumann problem of Hardy-Sobolev critical equations with the multiple singularities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 301-322. doi: 10.3934/cpaa.2019016

2017 Impact Factor: 0.884

American Institute of Mathematical Sciences