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

show all references

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

[1]

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

[2]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[3]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363

[4]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[5]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[6]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511

[7]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[8]

Jean Dolbeault, Maria J. Esteban, Michał Kowalczyk, Michael Loss. Improved interpolation inequalities on the sphere. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 695-724. doi: 10.3934/dcdss.2014.7.695

[9]

Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151

[10]

V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153

[11]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597

[12]

Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053

[13]

Gioconda Moscariello, Antonia Passarelli di Napoli, Carlo Sbordone. Planar ACL-homeomorphisms : Critical points of their components. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1391-1397. doi: 10.3934/cpaa.2010.9.1391

[14]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[15]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[16]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995

[17]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243

[18]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

[19]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[20]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]