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

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

 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 $|\lambda|$.
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:
  A. Ambrosetti and Andrea Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbbR^N$,, Birkhäuser Verlag, (2006). Google Scholar  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.  doi: 10.1006/jfan.1999.3390.  Google Scholar  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.  doi: 10.1017/S0308210500027268.  Google Scholar  A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations,, Arch. Rational Mech. Anal., 140 (1997), 285.  doi: 10.1007/s002050050067.  Google Scholar  M. Badiale, J. Garcia Azorero and I. Peral, Perturbation results for an anisotropic SchrHodinger equation via a variational form,, NoDEA, 7 (2000), 201.  doi: 10.1007/s000300050005.  Google Scholar  H. Berestycki and P. L. Lions, Nonlinear scalar field equations I - existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar  K. J. Brown and N. Stavrakakis, Global bifurcation results for a semilinear elliptic equation on all $\mathbbR^N$,, Duke Math. J., 85 (1996), 77.  doi: 10.1215/S0012-7094-96-08503-8.  Google Scholar  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.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar  S. Cingolani, Positive solutions to perturbed elliptic problems in $\mathbbR^N$ involving critical Sobolev exponent,, Nonlinear Analysis, 48 (2002), 1165.  doi: 10.1016/S0362-546X(00)00245-5.  Google Scholar  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.  doi: 10.1016/0022-1236(90)90002-3.  Google Scholar  N. S. Trudinger, On Harnack type inequalities and theri application to quasilinear elliptic equations,, Comm. Pure Appl. Math., 20 (1967), 721.  doi: 10.1002/cpa.3160200406.  Google Scholar

show all references

##### References:
  A. Ambrosetti and Andrea Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbbR^N$,, Birkhäuser Verlag, (2006). Google Scholar  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.  doi: 10.1006/jfan.1999.3390.  Google Scholar  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.  doi: 10.1017/S0308210500027268.  Google Scholar  A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations,, Arch. Rational Mech. Anal., 140 (1997), 285.  doi: 10.1007/s002050050067.  Google Scholar  M. Badiale, J. Garcia Azorero and I. Peral, Perturbation results for an anisotropic SchrHodinger equation via a variational form,, NoDEA, 7 (2000), 201.  doi: 10.1007/s000300050005.  Google Scholar  H. Berestycki and P. L. Lions, Nonlinear scalar field equations I - existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar  K. J. Brown and N. Stavrakakis, Global bifurcation results for a semilinear elliptic equation on all $\mathbbR^N$,, Duke Math. J., 85 (1996), 77.  doi: 10.1215/S0012-7094-96-08503-8.  Google Scholar  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.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar  S. Cingolani, Positive solutions to perturbed elliptic problems in $\mathbbR^N$ involving critical Sobolev exponent,, Nonlinear Analysis, 48 (2002), 1165.  doi: 10.1016/S0362-546X(00)00245-5.  Google Scholar  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.  doi: 10.1016/0022-1236(90)90002-3.  Google Scholar  N. S. Trudinger, On Harnack type inequalities and theri application to quasilinear elliptic equations,, Comm. Pure Appl. Math., 20 (1967), 721.  doi: 10.1002/cpa.3160200406.  Google Scholar
  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  Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461  Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436  Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469  Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056  Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351  Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052  Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117  Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013  Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251  Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216  Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019  Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384  Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076  Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078  Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462  Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250  Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120  Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453  Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

2019 Impact Factor: 1.27