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A minimization problem involving a fractional Hardy-Sobolev type inequality. (English) Zbl 1445.35018

Summary: In this work, we obtain existence of nontrivial solutions to a minimization problem involving a fractional Hardy-Sobolev type inequality in the case of inner singularity. Precisely, for \(\lambda>0\), we analyze the attainability of the optimal constant \[\mu_{\alpha,\lambda}(\Omega):=\inf \bigg\{[u]^2_{s,\Omega}+\lambda\int_{\Omega}\vert u\vert^2\,dx:u\in H^s(\Omega),\int_{\Omega}\frac{\vert u(x)\vert^{2_{s,\alpha}}}{\vert x\vert^{\alpha}}\,dx=1\bigg\},\] where \(0< s< 1\), \(n> 4s\), \(0\le \alpha < 2s\), \(2_{s,\alpha}=\frac{2(n-\alpha)}{n-2s}\), and \(\Omega \subset \mathbb{R}^n\) is a bounded domain such that \(0\in \Omega\).

MSC:

35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
35R11 Fractional partial differential equations
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References:

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