In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝ N {\mathbb{R}^{N}} .
NUS1 is responsible for encoding of the Nogo-B receptor (NgBR), which is a subunit of cis-prenyltransferase. Over 25 variants in NUS1 have been reported, and these variants have been found to be associated with various phenotypes, such as congenital disorders of glycosylation (CDG) and developmental and epileptic encephalopathy (DEE). We report on the case of a patient who presented with language and motor retardation, epilepsy, and electroencephalogram abnormalities. Upon conducting whole-exome sequencing, we discovered a novel pathogenic variant (chr6:118024873, NM_138459.5: c.791 + 6T>G) in NUS1, which was shown to cause Exon 4 to be skipped, resulting in a loss of 56 amino acids. Our findings strongly suggest that this novel variant of NUS1 is responsible for the development of neurological disorders, including epilepsy. It is believed that the truncation of Nogo-B receptor results in the loss of cis-prenyltransferase activity, which may be the underlying cause of the disease.
In this work we study the existence of positive solutions to the following fractional elliptic systems with Hardy-type singular potentials, and coupled by critical homogeneous nonlinearities \begin{equation*} \begin{cases} (-\Delta)^{s}u-\mu_{1}\frac{u}{|x|^{2s}}=|u|^{2^{\ast}_{s}-2}u+\frac{\eta\alpha}{2^{\ast}_{s}}|u|^{\alpha-2} |v|^{\beta}u+\frac{1}{2}Q_{u}(u,v) \ \ in \ \Omega, \\[2mm] (-\Delta)^{s}v-\mu_{2}\frac{v}{|x|^{2s}}=|v|^{2^{\ast}_{s}-2}v+\frac{\eta\beta}{2^{\ast}_{s}}|u|^{\alpha} |v|^{\beta-2}v+\frac{1}{2}Q_{v}(u,v) \ \ in \ \Omega, \\[2mm] \ \ u, \ v>0 \ \ \ \ \ in \ \ \Omega, \\[2mm] \ u=v=0 \ \ \ \ in \ \ \mathbb{R}^{N}\backslash\Omega, \end{cases} \end{equation*} where $(-\Delta)^{s}$ denotes the fractional Laplace operator, $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain such that $0\in\Omega$, $\mu_{1}, \mu_{2}\in [0,\Lambda_{N,s})$, $\Lambda_{N,s}=2^{2s}\frac{\Gamma^{2}(\frac{N+2s}{4})}{\Gamma^{2}(\frac{N-2s}{4})}$ is the best constant of the fractional Hardy inequality and $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent. In order to prove the main result, we establish some refined estimates on the extremal functions of the fractional Hardy-Sobolev type inequalities and we get the existence of positive solutions to the systems through variational methods.
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