American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Tug-of-war games and the infinity Laplacian with spatial dependence
  • CPAA Home
  • This Issue
  • Next Article
    On qualitative analysis for a two competing fish species model with a combined non-selective harvesting effort in the presence of toxicity
September  2013, 12(5): 1943-1957. doi: 10.3934/cpaa.2013.12.1943

Elliptic equations with cylindrical potential and multiple critical exponents

Xiaomei Sun 1, and  Yimin Zhang 2,

1. 

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
2. 

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, PO Box 71010, Wuhan 430071E01103

Received  November 2011 Revised  August 2012 Published  January 2013

In this paper, we deal with the following problem: \begin{eqnarray*} -\Delta u-\lambda |y|^{-2}u=|y|^{-s}u^{2^{*}(s)-1}+u^{2^{*}-1}\ \ \ in \ \ R^N , y\neq 0\\ u\geq 0 \end{eqnarray*} where $u(x)=u(y,z): R^m\times R^{N-m}\longrightarrow R$, $N\geq 4$, $2 < m < N$, $\lambda < (\frac{m-2}{2})^2$ and $0 < s < 2$, $2^*(s)=\frac{2(N-s)}{N-2}$, $2^*=\frac{2N}{N-2}$. Using the Variational method, we proved the existence of a ground state solution for the case $0 < \lambda < (\frac{m-2}{2})^2$ and the existence of a cylindrical weak solution under the case $\lambda<0$.
Keywords: Variational method, cylindrical weight, multiple critical exponents..
Mathematics Subject Classification: Primary: 35J15, 35J61; Secondary: 35J2.
Citation: Xiaomei Sun, Yimin Zhang. Elliptic equations with cylindrical potential and multiple critical exponents. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1943-1957. doi: 10.3934/cpaa.2013.12.1943
References:
[1]

J. Bellazzini and C. Bonanno, Nonlinear Schrödinger equations with strongly singular potentials,, Proceedings of the Royal Society of Edinburgh, 140A (2010), 707.  doi: 10.1017/S0308210509001401.  Google Scholar

[2]

M. Badiale, V. Bergio and S. Rolando, A nonlinear elliptic equation with singular potential and applications to nonlinear field equations,, J. Eur. Math. Soc., 9 (2007), 355.   Google Scholar

[3]

M. Badiale, M. Guida and S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities,, Adv. Diff. Equ., 12 (2007), 1321.  doi: 10.1007/s00009-005-0055-5.  Google Scholar

[4]

M. Badiale and S. Rolando, Nonlinear elliptic equations with subhomogeneous potentials,, Nonlinear Analysis, 72 (2010), 602.  doi: 10.1016/j.na.2009.06.111.  Google Scholar

[5]

M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics,, Arch. Ration. Mech. Anal., 163 (2002), 252.  doi: 10.1007/s002050200201.  Google Scholar

[6]

M. Bhakta and K. Sandeep, Hardy-Sobolev-Maz'ya type equations in bounded domains,, J. Differential Equations, 247 (2009), 119.  doi: 10.1016/j.jde.2008.12.011.  Google Scholar

[7]

H. Brezis and E. Lieb, A relation between pointwise convergence of functionals and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486.  doi: 10.2307/2044999.  Google Scholar

[8]

D. Castorina, I. Fabbri, G. Mancini and K. Sandeep, Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations,, J. Differential Equations, 246 (2009), 1187.  doi: 10.1016/j.jde.2008.09.006.  Google Scholar

[9]

D. M. Cao and P. G. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials,, J. Differential Equations, 224 (2006), 332.  doi: 10.1016/j.jde.2005.07.010.  Google Scholar

[10]

D. M. Cao and Y. Y. Li, Results on positive solutions of elliptic equations with a critical Hardy-Sobolev operator,, Methods and Applications of Analysis, 15 (2008), 081.  doi: 10.1.1.140.417.  Google Scholar

[11]

L. D'Ambrosio, Hardy type inequalities related to degenerate elliptic differential operators,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 4 (2005), 451.   Google Scholar

[12]

R. Filippucci, P. Pucci and F. Robert, On a p-Laplace equation with multiple critical nonlinearities,, J. Math. Pures Appl., 91 (2009), 156.  doi: 10.1016/j.matpur.2008.09.008.  Google Scholar

[13]

F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms,, Trans. Amer. Math. Soc., 356 (2004), 2149.  doi: 10.1090/S0002-9947-03-03395-6.  Google Scholar

[14]

N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities,, Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire, 21 (2004), 767.  doi: 10.1016/j.anihpc.2003.07.002.  Google Scholar

[15]

Y. Y. Li and C. S. Lin, A nonlinear Elliptic PDE with two Sobolev-Hardy critical exponents,, Arch. Rational Mech. Anal., 203 (2012), 943.  doi: 10.1007/s00205-011-0467-2.  Google Scholar

[16]

G. Mancini, I. Fabbri and K. Sandeep, Classification of solutions of a critical Hardy-Sobolev operator,, J. Differential Equations, 224 (2006), 258.  doi: 10.1016/j.jde.2005.07.001.  Google Scholar

[17]

G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\Bbb H^n$,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (2008), 635.   Google Scholar

[18]

R. Musina, Ground state solutions of a critical problem involving cylindrical weights,, Nonlinear Anal., 68 (2008), 3972.  doi: 10.1016/j.na.2007.04.034.  Google Scholar

[19]

R. S. Palais, The principle of symmetric criticality,, Commun. Math. Phys., 69 (1979), 19.  doi: 10.1007/BF01941322.  Google Scholar

[20]

J. B. Su and Z. Q. Wang, Sobolev type embedding and quasilinear elliptic equations with radial potentials,, J. Differential Equa., 250 (2011), 223.  doi: 10.1016/j.jde.2010.08.025.  Google Scholar

[21]

J. B. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials,, J. Differential Equa., 238 (2007), 201.  doi: 10.1016/j.jde.2007.03.018.  Google Scholar

[22]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353.  doi: 10.1007/BF02418013.  Google Scholar

[23]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent,, Adv. Differential Equations, 2 (1996), 241.   Google Scholar

[24]

A. Tertikas and K. Tintarev, On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality,, Ann. Mat. Pura e Appl., 186 (2007), 645.  doi: 10.1007/s10231-006-0024-z.  Google Scholar

[25]

S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space,, Ann. Inst. Henry Poincar\'e-Analyse Nonlin\'eaire, 12 (1995), 319.   Google Scholar

[26]

M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Third edition, (2000).   Google Scholar

show all references

References:
[1]

J. Bellazzini and C. Bonanno, Nonlinear Schrödinger equations with strongly singular potentials,, Proceedings of the Royal Society of Edinburgh, 140A (2010), 707.  doi: 10.1017/S0308210509001401.  Google Scholar

[2]

M. Badiale, V. Bergio and S. Rolando, A nonlinear elliptic equation with singular potential and applications to nonlinear field equations,, J. Eur. Math. Soc., 9 (2007), 355.   Google Scholar

[3]

M. Badiale, M. Guida and S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities,, Adv. Diff. Equ., 12 (2007), 1321.  doi: 10.1007/s00009-005-0055-5.  Google Scholar

[4]

M. Badiale and S. Rolando, Nonlinear elliptic equations with subhomogeneous potentials,, Nonlinear Analysis, 72 (2010), 602.  doi: 10.1016/j.na.2009.06.111.  Google Scholar

[5]

M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics,, Arch. Ration. Mech. Anal., 163 (2002), 252.  doi: 10.1007/s002050200201.  Google Scholar

[6]

M. Bhakta and K. Sandeep, Hardy-Sobolev-Maz'ya type equations in bounded domains,, J. Differential Equations, 247 (2009), 119.  doi: 10.1016/j.jde.2008.12.011.  Google Scholar

[7]

H. Brezis and E. Lieb, A relation between pointwise convergence of functionals and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486.  doi: 10.2307/2044999.  Google Scholar

[8]

D. Castorina, I. Fabbri, G. Mancini and K. Sandeep, Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations,, J. Differential Equations, 246 (2009), 1187.  doi: 10.1016/j.jde.2008.09.006.  Google Scholar

[9]

D. M. Cao and P. G. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials,, J. Differential Equations, 224 (2006), 332.  doi: 10.1016/j.jde.2005.07.010.  Google Scholar

[10]

D. M. Cao and Y. Y. Li, Results on positive solutions of elliptic equations with a critical Hardy-Sobolev operator,, Methods and Applications of Analysis, 15 (2008), 081.  doi: 10.1.1.140.417.  Google Scholar

[11]

L. D'Ambrosio, Hardy type inequalities related to degenerate elliptic differential operators,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 4 (2005), 451.   Google Scholar

[12]

R. Filippucci, P. Pucci and F. Robert, On a p-Laplace equation with multiple critical nonlinearities,, J. Math. Pures Appl., 91 (2009), 156.  doi: 10.1016/j.matpur.2008.09.008.  Google Scholar

[13]

F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms,, Trans. Amer. Math. Soc., 356 (2004), 2149.  doi: 10.1090/S0002-9947-03-03395-6.  Google Scholar

[14]

N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities,, Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire, 21 (2004), 767.  doi: 10.1016/j.anihpc.2003.07.002.  Google Scholar

[15]

Y. Y. Li and C. S. Lin, A nonlinear Elliptic PDE with two Sobolev-Hardy critical exponents,, Arch. Rational Mech. Anal., 203 (2012), 943.  doi: 10.1007/s00205-011-0467-2.  Google Scholar

[16]

G. Mancini, I. Fabbri and K. Sandeep, Classification of solutions of a critical Hardy-Sobolev operator,, J. Differential Equations, 224 (2006), 258.  doi: 10.1016/j.jde.2005.07.001.  Google Scholar

[17]

G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\Bbb H^n$,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (2008), 635.   Google Scholar

[18]

R. Musina, Ground state solutions of a critical problem involving cylindrical weights,, Nonlinear Anal., 68 (2008), 3972.  doi: 10.1016/j.na.2007.04.034.  Google Scholar

[19]

R. S. Palais, The principle of symmetric criticality,, Commun. Math. Phys., 69 (1979), 19.  doi: 10.1007/BF01941322.  Google Scholar

[20]

J. B. Su and Z. Q. Wang, Sobolev type embedding and quasilinear elliptic equations with radial potentials,, J. Differential Equa., 250 (2011), 223.  doi: 10.1016/j.jde.2010.08.025.  Google Scholar

[21]

J. B. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials,, J. Differential Equa., 238 (2007), 201.  doi: 10.1016/j.jde.2007.03.018.  Google Scholar

[22]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353.  doi: 10.1007/BF02418013.  Google Scholar

[23]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent,, Adv. Differential Equations, 2 (1996), 241.   Google Scholar

[24]

A. Tertikas and K. Tintarev, On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality,, Ann. Mat. Pura e Appl., 186 (2007), 645.  doi: 10.1007/s10231-006-0024-z.  Google Scholar

[25]

S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space,, Ann. Inst. Henry Poincar\'e-Analyse Nonlin\'eaire, 12 (1995), 319.   Google Scholar

[26]

M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Third edition, (2000).   Google Scholar

[1]

Dongsheng Kang. Quasilinear systems involving multiple critical exponents and potentials. Communications on Pure & Applied Analysis, 2013, 12 (2) : 695-710. doi: 10.3934/cpaa.2013.12.695
[2]

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
[3]

M. L. Miotto. Multiple solutions for elliptic problem in $\mathbb{R}^N$ with critical Sobolev exponent and weight function. Communications on Pure & Applied Analysis, 2010, 9 (1) : 233-248. doi: 10.3934/cpaa.2010.9.233
[4]

Dongsheng Kang, Fen Yang. Semilinear elliptic systems involving multiple critical exponents and singularities in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4247-4263. doi: 10.3934/dcds.2012.32.4247
[5]

F. R. Pereira. Multiple solutions for a class of Ambrosetti-Prodi type problems for systems involving critical Sobolev exponents. Communications on Pure & Applied Analysis, 2008, 7 (2) : 355-372. doi: 10.3934/cpaa.2008.7.355
[6]

Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315-335. doi: 10.3934/mbe.2008.5.315
[7]

Mousomi Bhakta, Debangana Mukherjee. Semilinear nonlocal elliptic equations with critical and supercritical exponents. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1741-1766. doi: 10.3934/cpaa.2017085
[8]

Ramzi Alsaedi. Perturbation effects for the minimal surface equation with multiple variable exponents. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 139-150. doi: 10.3934/dcdss.2019010
[9]

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
[10]

Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004
[11]

Seunghyeok Kim. On vector solutions for coupled nonlinear Schrödinger equations with critical exponents. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1259-1277. doi: 10.3934/cpaa.2013.12.1259
[12]

Alexandre Nolasco de Carvalho, Marcelo J. D. Nascimento. Singularly non-autonomous semilinear parabolic problems with critical exponents. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 449-471. doi: 10.3934/dcdss.2009.2.449
[13]

Shengbing Deng, Fethi Mahmoudi, Monica Musso. Bubbling on boundary submanifolds for a semilinear Neumann problem near high critical exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3035-3076. doi: 10.3934/dcds.2016.36.3035
[14]

Yinbin Deng, Shuangjie Peng, Li Wang. Infinitely many radial solutions to elliptic systems involving critical exponents. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 461-475. doi: 10.3934/dcds.2014.34.461
[15]

Filippo Gazzola. Critical exponents which relate embedding inequalities with quasilinear elliptic problems. Conference Publications, 2003, 2003 (Special) : 327-335. doi: 10.3934/proc.2003.2003.327
[16]

Yinbin Deng, Qi Gao, Dandan Zhang. Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 211-233. doi: 10.3934/dcds.2007.19.211
[17]

Zhilei Liang. On the critical exponents for porous medium equation with a localized reaction in high dimensions. Communications on Pure & Applied Analysis, 2012, 11 (2) : 649-658. doi: 10.3934/cpaa.2012.11.649
[18]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120
[19]

Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201
[20]

Monica Musso, Donato Passaseo. Multiple solutions of Neumann elliptic problems with critical nonlinearity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 301-320. doi: 10.3934/dcds.1999.5.301

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences