American Institute of Mathematical Sciences

Advanced
August  2016, 21(6): 1999-2009. doi: 10.3934/dcdsb.2016033

Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent

Jing Zhang 1, and  Shiwang Ma 2,

1. 

Mathematics Science College, Inner Mongolia Normal University, Hohhot 010022, China
2. 

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071

Received  January 2014 Revised  April 2016 Published  June 2016

In this paper, we are concerned with the following nonlinear Schrödinger equations with hardy potential and critical Sobolev exponent \begin{equation}\label{eq0.1} \left\{\begin{array}{ll} -\Delta u+\lambda a(x)u^q=\mu\frac{u}{|x|^{2}}+|u|^{2^{*}-2}u,& \textrm{in}\, \mathbb{R}^N, \\ u>0, & \textrm{in}\,\mathcal{D}^{1,2}(\mathbb{R}^N), （1） \end{array} \right. \end{equation} where $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent, $0\leq \mu<\overline{\mu}=\frac{(N-2)^2}{4}$, $a(x)\in C(\mathbb{R}^N)$. We first use an abstract perturbation method in critical point theory to obtain the existence of positive solutions of （1） for small value of $|\lambda|$. Secondly, we focus on an anisotropic elliptic equation of the form \begin{equation}\label{eq0.2} -{\rm div}(B_\lambda(x)\nabla u)+\lambda a(x)u^q=\mu\frac{u}{|x|^{2}}+|u|^{2^*-2}u, x\in\mathbb{R}^N. （2） \end{equation} The same abstract method is used to yield existence result of positive solutions of （2） for small value of $|\lambda|$.
Keywords: Positive solutions, anisotropic problem., critical exponent, pertubation method.
Mathematics Subject Classification: Primary: 35B33, 35B20; Secondary: 35B0.
Citation: 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
References:
[1]

A. Ambrosetti and Andrea Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbbR^N$, Birkhäuser Verlag, 2006.  Google Scholar

[2]

A. Ambrosetti, J. Garcia Azorero and I. Peral, Perturbation of $\Delta u+u^{\frac{N+2}{N-2}}=0$, The scalar curvature problems in $\mathbbR^N$ and related topics, J. Funct. Anal, 165 (1999), 117-149. doi: 10.1006/jfan.1999.3390.  Google Scholar

[3]

A. Ambrosetti and M. Badiale, Variational perturbative methods and bifurcation of bounds states from the the essential spectrum, Proc. Roy. Soc. Edinburgh A, 128 (1998), 1131-1161. doi: 10.1017/S0308210500027268.  Google Scholar

[4]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.  Google Scholar

[5]

M. Badiale, J. Garcia Azorero and I. Peral, Perturbation results for an anisotropic SchrHodinger equation via a variational form, NoDEA, 7 (2000), 201-230. doi: 10.1007/s000300050005.  Google Scholar

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations I - existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar

[7]

K. J. Brown and N. Stavrakakis, Global bifurcation results for a semilinear elliptic equation on all $\mathbbR^N$, Duke Math. J., 85 (1996), 77-94. doi: 10.1215/S0012-7094-96-08503-8.  Google Scholar

[8]

F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extermal functions, Comm. Pure Appl. Math., 54 (2001), 229-258. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[9]

S. Cingolani, Positive solutions to perturbed elliptic problems in $\mathbbR^N$ involving critical Sobolev exponent, Nonlinear Analysis, 48 (2002), 1165-1178. doi: 10.1016/S0362-546X(00)00245-5.  Google Scholar

[10]

O. Rey, The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3.  Google Scholar

[11]

N. S. Trudinger, On Harnack type inequalities and theri application to quasilinear elliptic equations, Comm. Pure Appl. Math., 20 (1967), 721-747. doi: 10.1002/cpa.3160200406.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and Andrea Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbbR^N$, Birkhäuser Verlag, 2006.  Google Scholar

[2]

A. Ambrosetti, J. Garcia Azorero and I. Peral, Perturbation of $\Delta u+u^{\frac{N+2}{N-2}}=0$, The scalar curvature problems in $\mathbbR^N$ and related topics, J. Funct. Anal, 165 (1999), 117-149. doi: 10.1006/jfan.1999.3390.  Google Scholar

[3]

A. Ambrosetti and M. Badiale, Variational perturbative methods and bifurcation of bounds states from the the essential spectrum, Proc. Roy. Soc. Edinburgh A, 128 (1998), 1131-1161. doi: 10.1017/S0308210500027268.  Google Scholar

[4]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.  Google Scholar

[5]

M. Badiale, J. Garcia Azorero and I. Peral, Perturbation results for an anisotropic SchrHodinger equation via a variational form, NoDEA, 7 (2000), 201-230. doi: 10.1007/s000300050005.  Google Scholar

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations I - existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar

[7]

K. J. Brown and N. Stavrakakis, Global bifurcation results for a semilinear elliptic equation on all $\mathbbR^N$, Duke Math. J., 85 (1996), 77-94. doi: 10.1215/S0012-7094-96-08503-8.  Google Scholar

[8]

F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extermal functions, Comm. Pure Appl. Math., 54 (2001), 229-258. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[9]

S. Cingolani, Positive solutions to perturbed elliptic problems in $\mathbbR^N$ involving critical Sobolev exponent, Nonlinear Analysis, 48 (2002), 1165-1178. doi: 10.1016/S0362-546X(00)00245-5.  Google Scholar

[10]

O. Rey, The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3.  Google Scholar

[11]

N. S. Trudinger, On Harnack type inequalities and theri application to quasilinear elliptic equations, Comm. Pure Appl. Math., 20 (1967), 721-747. doi: 10.1002/cpa.3160200406.  Google Scholar

[1]

Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure & Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567
[2]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[3]

Liping Wang, Dong Ye. Concentrating solutions for an anisotropic elliptic problem with large exponent. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3771-3797. doi: 10.3934/dcds.2015.35.3771
[4]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
[5]

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
[6]

Maoding Zhen, Jinchun He, Haoyuan Xu, Meihua Yang. Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6523-6539. doi: 10.3934/dcds.2019283
[7]

Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773
[8]

Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557
[9]

Zhongliang Wang. Nonradial positive solutions for a biharmonic critical growth problem. Communications on Pure & Applied Analysis, 2012, 11 (2) : 517-545. doi: 10.3934/cpaa.2012.11.517
[10]

Rui-Qi Liu, Chun-Lei Tang, Jia-Feng Liao, Xing-Ping Wu. Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1841-1856. doi: 10.3934/cpaa.2016006
[11]

Elvise Berchio, Filippo Gazzola. Positive solutions to a linearly perturbed critical growth biharmonic problem. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 809-823. doi: 10.3934/dcdss.2011.4.809
[12]

Antonio Capella. Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1645-1662. doi: 10.3934/cpaa.2011.10.1645
[13]

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
[14]

Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907
[15]

Sanjiban Santra. On the positive solutions for a perturbed negative exponent problem on $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1441-1460. doi: 10.3934/dcds.2018059
[16]

Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076
[17]

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
[18]

Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025
[19]

Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure & Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921
[20]

Ting Guo, Xianhua Tang, Qi Zhang, Zu Gao. Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1563-1579. doi: 10.3934/cpaa.2020078

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (163)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy