Doubly nonlocal system with Hardy-Littlewood-Sobolev critical nonlinearity

This article concerns about the existence and multiplicity of weak solutions for the following nonlinear doubly nonlocal problem with critical nonlinearity in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{ \begin{split} (-\Delta)^su &= \lambda |u|^{q-2}u + \left(\int_{\Omega}\frac{|v(y)|^{2^*_\mu}}{|x-y|^\mu}~\mathrm{d}y\right) |u|^{2^*_\mu-2}u\; \text{in}\; \Omega (-\Delta)^sv &= \delta |v|^{q-2}v + \left(\int_{\Om}\frac{|u(y)|^{2^*_\mu}}{|x-y|^\mu}~\mathrm{d}y \right) |v|^{2^*_\mu-2}v \; \text{in}\; \Omega u &=v=0\; \text{in}\; \mb R^n\setminus\Omega, \end{split} \right. \end{equation*} where $\Omega$ is a smooth bounded domain in \mb R^n, $n >2s$, $s \in (0,1)$, $(-\Delta)^s$ is the well known fractional Laplacian, $\mu \in (0,n)$, $2^*_\mu = \displaystyle\frac{2n-\mu}{n-2s}$ is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality, $1<q<2$ and $\lambda,\delta >0$ are real parameters. We study the fibering maps corresponding to the functional associated with $(P_{\lambda,\delta})$ and show that minimization over suitable subsets of Nehari manifold renders the existence of atleast two non trivial solutions of (P_{\la,\delta}) for suitable range of \la and $\delta$.Comment: 37 page

A Global multiplicity result for a very singular critical nonlocal equation

In this article, we show the global multiplicity result for the following nonlocal singular problem \begin{equation*} (P_\la):\;\quad (-\De)^s u = u^{-q} + \la u^{{2^*_s}-1}, \quad u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om, \end{equation*} where \Om is a bounded domain in \mb{R}^n with smooth boundary \partial \Om, n > 2s,\; s \in (0,1),\; \la >0,\; q>0 satisfies $q(2s-1)<(2s+1)$ and $2^*_s=\frac{2n}{n-2s}$. Employing the variational method, we show the existence of at least two distinct weak positive solutions for (P_\la) in $X_0$ when \la \in (0,\La) and no solution when \la>\La, where \La>0 is appropriately chosen. We also prove a result of independent interest that any weak solution to $(P_\lambda)$ is in $C^\alpha(\R^n)$ with $\alpha=\alpha(s,q)\in (0,1)$. The asymptotic behaviour of weak solutions reveals that this result is sharp.Comment: 23 page

Pregnancy in porphyria and its complications: a case report

As with many rare diseases, little is known about the porphyrias and reproduction. Knowledge pertaining to complications in pregnancy in a patient with porphyria is even more scant. The information which we have is from attending on individual cases. The following case report demonstrates a rare case of porphyria cutanea tarda in a 30 year old female patient, diagnosed around pubertal age, who presented to the Emergency department as a case of severe anemia who had delivered a still born baby the same day. Still births along with spontaneous abortions, preeclampsia and low birth weights are few of the known complications of pregnancy in patients with porphyria

The Association of Obesity with Walking Independent of Knee Pain: The Multicenter Osteoarthritis Study

Practice guidelines recommend addressing obesity for people with knee OA, however, the association of obesity with walking independent of pain is not known. We investigated this association within the Multicenter Osteoarthritis Study, a cohort of older adults who have or are at high risk of knee OA. Subjects wore a StepWatch to record steps taken over 7 days. We measured knee pain from a visual analogue scale and obesity by BMI. We examined the association of obesity with walking using linear regression adjusting for pain and covariates. Of 1788 subjects, the mean steps/day taken was 8872.9 ± 3543.4. Subjects with a BMI ≥35 took 3355 fewer steps per day independent of knee pain compared with those with a BMI ≤25 (95% CI −3899, −2811). BMI accounted for 9.7% of the variability of walking while knee pain accounted for 2.9%. BMI was associated with walking independent of knee pain

Relation of Temporal Asymmetry During Walking to Two-Year Knee Pain Outcomes in Those With Mild-to-Moderate Unilateral Knee Pain: An Exploratory Analysis From the Multicenter Osteoarthritis Study.

OBJECTIVE: We aimed to explore the cross-sectional relation of unilateral knee pain severity and temporal asymmetry during walking and to determine relations of temporal asymmetry during walking to 2-year changes in ipsilateral and contralateral knee pain in those with mild-to-moderate unilateral knee pain. METHODS: The Multicenter Osteoarthritis Study is a prospective cohort study of adults with or at risk for knee osteoarthritis. The current study included participants with unilateral knee pain. Gait was assessed during self-selected and fast walking at baseline. Knee pain was assessed at baseline and 2 years. We calculated limb symmetry indices (LSIs; nonpainful limb/painful limb × 100) for stance, single-limb support time, and double-limb support time, then examined their relations to unilateral knee pain severity, incident contralateral knee pain, and persistent ipsilateral knee pain. RESULTS: Unilateral knee pain severity was not associated with temporal asymmetry during self-selected or fast walking. At 2 years, 17.1% of participants had incident contralateral knee pain and 51.4% had persistent ipsilateral knee pain. For self-selected walking, greater LSIs (i.e., longer time on the nonpainful limb) for stance and single-limb support time were associated with decreased odds of incident contralateral knee pain. Measures of temporal asymmetry were not associated with persistent ipsilateral knee pain, except for single-limb support time during fast walking. CONCLUSION: For those with unilateral knee pain, temporal asymmetry during walking is not associated with pain severity. However, select measures of stance and single-limb support time during self-selected and fast walking relate to longitudinal knee pain outcomes