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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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