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

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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

Abstract
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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 and 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 $\mathbb{R}^N2$, Birkhäuser Verlag, 2006.
[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 $\mathbb{R}^N2$ and related topics, J. Funct. Anal, 165 (1999), 117-149. doi: 10.1006/jfan.1999.3390.
[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.
[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.
[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.
[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.
[7]	K. J. Brown and N. Stavrakakis, Global bifurcation results for a semilinear elliptic equation on all $\mathbb{R}^N2$, Duke Math. J., 85 (1996), 77-94. doi: 10.1215/S0012-7094-96-08503-8.
[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.
[9]	S. Cingolani, Positive solutions to perturbed elliptic problems in $\mathbb{R}^N2$ involving critical Sobolev exponent, Nonlinear Analysis, 48 (2002), 1165-1178. doi: 10.1016/S0362-546X(00)00245-5.
[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.
[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.

show all references

References:

[1]	A. Ambrosetti and Andrea Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb{R}^N2$, Birkhäuser Verlag, 2006.
[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 $\mathbb{R}^N2$ and related topics, J. Funct. Anal, 165 (1999), 117-149. doi: 10.1006/jfan.1999.3390.
[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.
[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.
[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.
[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.
[7]	K. J. Brown and N. Stavrakakis, Global bifurcation results for a semilinear elliptic equation on all $\mathbb{R}^N2$, Duke Math. J., 85 (1996), 77-94. doi: 10.1215/S0012-7094-96-08503-8.
[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.
[9]	S. Cingolani, Positive solutions to perturbed elliptic problems in $\mathbb{R}^N2$ involving critical Sobolev exponent, Nonlinear Analysis, 48 (2002), 1165-1178. doi: 10.1016/S0362-546X(00)00245-5.
[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.
[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.

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