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

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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 $|\lambda|$.
    Mathematics Subject Classification: Primary: 35B33, 35B20; Secondary: 35B09.


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  • [1]

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


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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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