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In this article, we investigate the existence of multiple positive solutions to the following multi-critical Schrödinger equation: (0.1) − Δ u + λ V ( x ) u = μ ∣ u ∣ p − 2 u + ∑ i = 1 k ( ∣ x ∣ − ( N − α i ) ∗ ∣ u ∣ 2 i ∗ ) ∣ u ∣ 2 i ∗ − 2 u in R N , u ∈ H 1 ( R N ) , \left\{\begin{array}{l}-\Delta u+\lambda V\left(x)u=\mu | u{| }^{p-2}u+\mathop{\displaystyle \sum }\limits_{i=1}^{k}\left(| x{| }^{-\left(N-{\alpha }_{i})}\ast | u{| }^{{2}_{i}^{\ast }})| u{| }^{{2}_{i}^{\ast }-2}u\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\hspace{0.33em}\in {H}^{1}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}\end{array}\right. where λ , μ ∈ R + , N ≥ 4 \lambda ,\mu \in {{\mathbb{R}}}^{+},N\ge 4 , and 2 i ∗ = N + α i N − 2 {2}_{i}^{\ast }=\frac{N+{\alpha }_{i}}{N-2} with N − 4 < α i < N N-4\lt {\alpha }_{i}\lt N , i = 1 , 2 , … , k i=1,2,\ldots ,k are critical exponents and 2 < p < 2 min ∗ = min { 2 i ∗ : i = 1 , 2 , … , k } 2\lt p\lt {2}_{\min }^{\ast }={\rm{\min }}\left\{{2}_{i}^{\ast }:i=1,2,\ldots ,k\right\} . Suppose that Ω = int V − 1 ( 0 ) ⊂ R N \Omega ={\rm{int}}\hspace{0.33em}{V}^{-1}\left(0)\subset {{\mathbb{R}}}^{N} is a bounded domain, we show that for λ \lambda large, problem (0.1) possesses at least cat Ω ( Ω ) {{\rm{cat}}}_{\Omega }\left(\Omega ) positive solutions.

In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schrodinger equation $$\left\{ {\matrix{ { - {\rm{\Delta }}u + V\left( x \right)u = {\mu _q}u + a{{\left| u \right|}^q}u\,\,\,{\rm{in}}\,{\mathbb{R}^2},} \hfill \cr {\int_{{\mathbb{R}^2}} {{{\left| u \right|}^2}dx = 1,} } \hfill \cr } } \right.$$ where μq is the Lagrange multiplier. We show that for q > 2 close to 2, the problem admits two solutions: one is the local minimal solution uq and the other one is the mountain pass solution υq. Furthermore, we study the limiting behavior of uq and υq when q → 2+. Particularly, we describe precisely the blow-up formation of the excited state υq.

Background Our previous work shows Autophagy enhanced resistance to cisplatin in seminoma. The expression of the N6-methyladenosine (m6A) methyltransferases METTL3 was significantly increased in the cisplatin-resistant TCam-2 cell line of seminoma. We aimed to investigate the role of m6A methylation in autophagy and the chemosensitivity of seminoma cells. Methods Plasmid and siRNA were used to overexpress and knockdown METTL3. Autophagy was detected by western blot and immunofluorescence, respectively. The expression of downstream targets of METTL3 was detected by quantitative real-time PCR (qRT-PCR) and western blot, and the m6A level of them was detected by MeRIP-qPCR. Chemosensitivity of the TCam-2 cell line was identified through MTT assay. Results Upon METTL3 overexpression, autophagy of TCam-2 cell line was enhanced and its sensitivity to cisplatin was decreased. The use of autophagy inhibitors 3-methyladenine (3-MA) could reverse the protective effect of METTL3 on TCam-2 cells. We found that the up-regulation of METTL3 could increase the m6A modification level of ATG5 transcript, thus increased expression of ATG5. Moreover, knockdown of ATG5 reduced METTL3-induced autophagy, suggesting that ATG5 was a potential target for METTL3 to promote autophagy. Conclusions In summary, our research unveiled the unique mechanism by which m6A methylation regulates autophagy and chemosensitivity of the TCam-2 cell line and METTL3 was a potential target to overcome the cisplatin resistance of seminoma.

In this paper, we investigate the existence of multiple solutions to the following multi-critical elliptic problem \begin{equation}\label{eq:0.1} \left\{\begin{aligned} -\Delta u & =\lambda |u|^{p-2}u +\sum_{i=1}^k(|x|^{-(N-\alpha_i)}*|u|^{2^*_i})|u|^{2^*_i-2}u\quad {\rm in}\quad \Omega,\\ &u\in H^1_0(\Omega)\\ \end{aligned}\right. \end{equation} in connection with the topology of the bounded domain $\Omega\subset \mathbb{R}^N, \,N\geq 4$, where $\lambda>0$, $2^*_i=\frac{N+\alpha_i}{N-2}$ with $N-40$ such that if $0

Testicular germ cell tumours (TGCTs) rank as the most common malignancy in men aged 20-34 years, and seminomas are the most type of TGCTs. As a crucial anti-tumour agent with explicit toxicity, cisplatin may render resistance through intertwined mechanisms, even in disease entities with high curative ratio, such as seminoma. Previously, we established cisplatin-resistant seminoma TCam-2 (TCam-2/CDDP) cells and showed that epigenetic regulations, such as non-coding RNA (ncRNA) interactions, might orchestrate cell fate decisions in the cisplatin treatment context in seminoma. N6-methyladenosine (m6A) is the most prevalent internal modification in mRNA. In the present study, we assessed cisplatin resistance in seminoma from the perspective of m6 A, another manner of epigenetic modification. The global m6 A enrichment of TCam-2 and TCam-2/CDDP was depicted. Then, we elucidated whether transcription factor-activating enhancer-binding protein 2C (TFAP2C) was functionally m6 A-modified by methyltransferase-like protein 3 (METTL3), which acted as an m6 A 'writer', and insulin-like growth factor 2 mRNA-binding protein 1 (IGF2BP1), which acted as an m6 A 'reader'. Enhanced stability of TFAP2C mRNA promoted seminoma cell survival under cisplatin treatment burden probably through up-regulation of DNA repair-related genes. Hopefully, this study will help improve our understanding of the subtleties of the tumour cellular coping strategy in response to chemotherapy. Targeting factors that are involved in m6 A methylation may be an effective strategy for circumventing cisplatin resistance in seminoma.

In this paper, we establish two refinement of Sobolev-Hardy inequalities in terms of Morrey spaces. Then, with help of these inequalities, we show the existence of nontrivial solutions for doubly critical problems in \begin{document}$ \mathbb{R}^N $\end{document} involving \begin{document}$ p $\end{document} -Laplacian \begin{document}$ \begin{equation*} \label{eq:1} -\Delta_pu = \frac{|u|^{p^\ast_{\alpha}-2}u}{|y|^{\alpha}}+\frac{|u|^{p^\ast_{\beta}-2}u}{|y|^{\beta}}, \end{equation*} $\end{document} where \begin{document}$ x = (y,z)\in\mathbb{R}^K\times\mathbb{R}^{N-K},1\leq K\leq N,1 and \begin{document}$ p^\ast_{\alpha} = \frac{p(N-\alpha)}{N-p} $\end{document} is the critical Hardy-Sobolev exponent, and critical problems involving fractional Laplacian \begin{document}$ \begin{equation*} (-\Delta)^{s}u = \frac{|u|^{2^\ast_{s,\alpha}-2}u}{|y|^{\alpha}}+\frac{|u|^{2^\ast_{s,\beta}-2}u}{|y|^{\beta}}, \end{equation*} $\end{document} where \begin{document}$ 0 and \begin{document}$ 2^\ast_{s,\alpha} = \frac{2(N-\alpha)}{N-2s} $\end{document} .

In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation \[ \left\{\begin{array}{@{}ll} -\Delta u + V(x) u = \mu_q u + a \vert u \vert ^q u & {\rm in}\ \mathbb{R}^2,\\ \int_{\mathbb{R}^2} \vert u \vert ^2\,{\rm d}x =1, & \end{array} \right.\]where μq is the Lagrange multiplier. For ellipse-shaped potentials V(x), we show that for q > 2 close to 2, the equation admits an excited solution uq, and furthermore, we study the limiting behaviour of uq when q → 2+. Particularly, we describe precisely the blow-up formation of the excited state uq.

Let Ω e be a bounded domain in R N with small holes. In this paper, we study the existence of solutions for the following problem { ( − Δ ) s u = u 2 s ⁎ − 1 , u > 0 , in Ω e , u = 0 , in R N ∖ Ω e , where 0 s 1 and 2 s ⁎ = 2 N N − 2 s . We construct solutions which blow up like a volcano near the center of each hole in Ω e

The technology to sinter copper inks in air is beneficial for the applications in flexible printed electronics. However, copper based ink could hardly produce conductive copper patterns when sintered in air, because copper is easily oxidized. In this study, a convenient fabrication process for conductive copper films with a simple heat treatment on hotplate in air was successfully developed. The ink was prepared by mixing copper submicron particles with formic acid and an amino-alcohol. The copper oxide on the surface of untreated copper submicron particles was converted to decomposable copper formate. More importantly, the copper submicron particles could be activated by the chemisorbed HCOOH to sinter at low temperatures. 3-Dimethylamino-1,2-propanediol (DMAPD) was introduced to protect copper from oxidation when sintered in air. In addition, DMAPD could promote the decomposition of copper formate by forming copper–amine complex. A resistivity of 54 ± 2 μΩ cm was obtained after sintered at 200 °C for 50 s (63 ± 4 μΩ cm for only 10 s). This simple, convenient and rapid sintering of submicron copper inks in air provides an alternative fabrication method of copper patterns in printed electronics.

In this paper, we consider the following elliptic problem: (0.1) − Δ u + V ( x ) u + λ ( κ α ⁎ | u | p ) | u | p − 2 u = ( κ β ⁎ | u | q ) | u | q − 2 u , x ∈ R 3 , where λ > 0 , 0 α 3 , 0 β 3 , 1 + α 3 p 3 + α , q > 1 and κ α ( x ) = | x | α − 3 , κ β ( x ) = | x | β − 3 . If V = 1 , we show the existence of a ground state solution and multiple solutions, as well as the nonexistence result for (0.1) . Furthermore, we investigate the limit behavior of ground state solutions of (0.1) as λ → 0 + . Ground state solutions are also obtained if the potential V is not a constant.

In this paper, we consider the existence of non-trivial solutions for the following equation \begin{document}$\mathcal{H}_{0J}u = |u|^{p-2}u+λ u\;\;\;\;{\rm in}\,\,\mathbb{R}^3,\;\;\;\;\;\;\;\;(1)$ \end{document} where \begin{document} $\mathcal{H}_{0J}$ \end{document} is the higher-order Schrodinger operator with \begin{document} $J∈\mathbb{N}$ \end{document} , \begin{document} $2 , and \begin{document} $λ∈\mathbb{R}$ \end{document} is a parameter. Let \begin{document} $E(u)$ \end{document} be the corresponding variational functional of problem (1). We look for solutions of equation (1) by finding minimizers of the minimization problem \begin{document}$E_ρ = \inf\{E(u)|u∈ H^{J}(\mathbb{R}^3):\,\,\|u\|_{L^2(\mathbb{R}^3)} = ρ\}.$ \end{document} We show that problem (1) admits at least a solution provided that in the case \begin{document} $J$ \end{document} being odd, \begin{document} $2 and \begin{document} $ρ>0$ \end{document} small or \begin{document} $2+J and \begin{document} $ρ>0$ \end{document} large; and for the case \begin{document} $J$ \end{document} being even, \begin{document} $3 and \begin{document} $ρ>0$ \end{document} small.

The role of TDRG1 in tumorigenesis and the progression of seminoma, as well as its role in regulating chemosensitivity of seminoma to cisplatin through the PI3K/Akt/mTOR signaling pathway, has been previously defined. However, the detailed mechanism underlying TDRG1 expression and concomitant chemoresistance conditions are unknown. Furthermore, it has been reported that non‐protein‐coding RNAs play an important role in a variety of vital processes including cellular chemosensitivity. However, the role of non‐protein‐coding RNAs in regulating the chemosensitivity of seminoma remains unknown. In this study, using microarray analysis, we found that long non‐coding RNA H19 was upregulated while miRNA‐106b‐5p was downregulated in an established cisplatin‐resistant TCam‐2 cell line. Moreover, H19 acts as a miRNA‐106b‐5p sponge and thus impairs the function of miRNA‐106b‐5p on its target gene, TDRG1. Based on these findings, we propose that H19 promotes the expression of TDRG1 by sequestering miRNA‐106b‐5p and uses this mechanism to facilitate cell survival in cisplatin‐based chemotherapeutic conditions. These findings elucidate the mechanisms, at least partially, applied to deregulate TDRG1 and cisplatin sensitivity, and may provide new therapeutic possibilities for chemoresistant seminoma.

High-performance actuators are crucial to enable mechanical versatility of wearable robots, which are required to be lightweight, highly backdrivable, and with high bandwidth. State-of-the-art actuators, e.g., series elastic actuators, have to compromise bandwidth to improve compliance (i.e., backdrivability). In this article, we describe the design and human–robot interaction modeling of a portable hip exoskeleton based on our custom quasi-direct drive actuation (i.e., a high torque density motor with low ratio gear). We also present a model-based performance benchmark comparison of representative actuators in terms of torque capability, control bandwidth, backdrivability, and force tracking accuracy. This article aims to corroborate the underlying philosophy of “design for control,” namely meticulous robot design can simplify control algorithms while ensuring high performance. Following this idea, we create a lightweight bilateral hip exoskeleton to reduce joint loadings during normal activities, including walking and squatting. Experiments indicate that the exoskeleton is able to produce high nominal torque (17.5 Nm), high backdrivability (0.4 Nm backdrive torque), high bandwidth (62.4 Hz), and high control accuracy (1.09 Nm root mean square tracking error, 5.4% of the desired peak torque). Its controller is versatile to assist walking at different speeds and squatting. This article demonstrates performance improvement compared with state-of-the-art exoskeletons.

Background: Inorganic arsenic (iAs) is an environmental toxicant associated with an increased risk of prostate cancer in chronically exposed populations worldwide. However, the biological mechanisms underlying iAs-induced prostate carcinogenesis remain unclear. Objectives: We studied how iAs affects normal human prostate stem-progenitor cells (PrSPCs) and drives transformation and interrogated the molecular mechanisms involved. Methods: PrSPCs were enriched by spheroid culture from normal human primary or immortalized prostate epithelial cells, and their differentiation capability was evaluated by organoid culture. Microarray analysis was conducted to identify iAs-dysregulated genes, and lentiviral infection was used for stable manipulation of identified genes. Soft agar colony growth assays were applied to examine iAs-induced transformation. For in vivo study, PrSPCs mixed with rat urogenital sinus mesenchyme were grafted under the renal capsule of nude mice to generate prostatelike tissues, and mice were exposed to 5 ppm (∼65μM) iAs in drinking water for 3 months. Results: Low-dose iAs (1μM) disturbed PrSPC homeostasis in vitro, leading to increased self-renewal and suppressed differentiation. Transcriptomic analysis indicated that iAs activated oncogenic pathways in PrSPCs, including the KEAP1-NRF2 pathway. Further, iAs-exposed proliferative progenitor cells exhibited NRF2 pathway activation that was sustained in their progeny cells. Knockdown of NRF2 inhibited spheroid formation by driving PrSPC differentiation, whereas its activation enhanced spheroid growth. Importantly, iAs-induced transformation was suppressed by NRF2 knockdown. Mechanistically, iAs suppressed Vacuolar ATPase subunit VMA5 expression, impairing lysosome acidification and inhibiting autophagic protein degradation including p62, which further activated NRF2. In vivo, chronic iAs exposure activated NRF2 in both epithelial and stroma cells of chimeric human prostate grafts and induced premalignant events. Conclusions: Low-dose iAs increased self-renewal and decreased differentiation of human PrSPCs by activating the p62-NRF2 axis, resulting in epithelial cell transformation. NRF2 is activated by iAs through specific autophagic flux blockade in progenitor cells, which may have potential therapeutic implications. https://doi.org/10.1289/EHP6471

MicroRNA (miRNA)–gene interactions are well-recognized as involved in the progression of almost all cancer types including prostate cancer, which is one of the most common cancers in men. This study explored the significantly dysregulated genes and miRNAs and elucidated the potential miRNA–gene regulatory network in prostate cancer. Integrative analysis of prostate cancer and normal prostate transcriptomic data in The Cancer Genome Atlas dataset was conducted using both differential expression analysis and weighted correlation network analysis (WGCNA). Thirteen genes (RRM2, ORC6, CDC45, CDKN2A, E2F2, MYBL2, CCNB2, PLK1, FOXM1, CDC25C, PKMYT1, GTSE1, and CDC20) were potentially correlated with prostate cancer based on functional enrichment analyses. MiRNAs targeting these genes were predicted and eight miRNAs were intersections between those miRNAs and the hub miRNAs obtained from miRNA WGCNA analysis. Three genes (E2F2, RRM2, and PKMYT1) and four miRNAs (hsa-mir-17-5p, hsa-mir-20a-5p, hsa-mir-92a-3p, and hsa-mir-93-5p) were key factors according to the interaction network. RRM2 and PKMYT1 were significantly related to survival. These findings partially elucidated the dysregulation of gene expressions in prostate cancer. Efficient manipulations of the miRNA–gene interactions in prostate cancer may be exploited as promising therapeutics.

Introduction There are many Western reports on factors influencing coital frequency among men. However, no articles could be found about the factors influencing sexual activity among Chinese men. Aim The aim of this study was to identify the factors that influence the coital frequency of Chinese men. Main Outcome Measures The main outcome measures included self-reported monthly coital frequency, age, occupation, education level, andrology-related scales and dietary habits. Methods Data for 1,407 men aged 18–79 years were collected in the Health Management Center of the Third Xiangya Hospital of Central South University from January 2019 to May 2019. The respondents completed the questionnaires independently or with the help of an interviewer (who read or explained the questionnaires to them) to analyse the factors that influence coital frequency. Results In the previous 6 months, the sample had a mean monthly coital frequency (±SD) of 4.34 ± 3.18. Univariate logistic regression results indicated that the number of children (P = 0.004), IIEF-5 scores (P Conclusion The coital frequency of Chinese men is associated with erectile function, anthropometric parameters, age, occupation, and dietary habits.

In this paper, we show the existence of nontrivial solutions for doubly critical nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians. Our approach is elementary without the use of neither the concentration-compactness principle nor the extension argument developed in [4].

We previously identified testis developmental related gene 1 (TDRG1), a gene implicated in proliferation of TCam‐2 seminoma cells. Recent evidence has revealed that autophagy influences the chemosensitivity of cancer cells to chemotherapy. However, whether TDRG1 protein regulates autophagy in seminoma cells and influences their sensitivity to cis‐dichlorodiammine platinum (CDDP) remains unknown. In this study, we used TCam‐2 cells and male athymic BALB/c nude mice with xenografts of TCam‐2 cells to investigate autophagy, cell viability, apoptosis and the p110β/Rab5/Vps34 (PI3‐kinase Class III) pathway under the conditions of TDRG1 overexpression or knockdown and with or without CDDP treatment. We found that TDRG1 upregulation promoted autophagy in both TCam‐2 cells and seminoma xenografts via p110β/Rab5/Vps34 activation. Inhibition of autophagy reduced cell viability and promoted apoptosis during CDDP treatment of TCam‐2 cells. Similarly, TDRG1 knockdown inhibited autophagy, reduced cell viability and promoted apoptosis during CDDP treatment of TCam‐2 cells. TDRG1 knockdown inhibited tumour growth and promoted apoptosis in TCam‐2 cell xenografts, whereas TDRG1 overexpression had the opposite effect. According to these results, we propose that high expression of TDRG1 promotes autophagy through the p110β/Rab5/Vps34 pathway in TCam‐2 cells. TDRG1 overexpression promotes autophagy and leads to CDDP resistance, whereas TDRG1 knockdown inhibits autophagy and promotes chemosensitivity to CDDP both in vivo and in vitro. This study has uncovered a novel role of TDRG1 in reducing chemoresistance during CDDP treatment and provides potential therapeutic strategies for the treatment of human seminoma.

In this paper, we consider the existence of static solutions to the nonlinear Chern-Simons-Schrodinger system 1 \begin{document}$ \begin{equation} \left\{\begin{array}{ll} -iD_0\Psi-(D_1D_1+D_2D_2)\Psi+V\Psi = |\Psi|^{p-2}\Psi,\\ \partial_0A_1-\partial_1A_0 = -\frac 12i\lambda[\overline{\Psi}D_2\Psi-\Psi\overline{D_2\Psi}],\\ \partial_0A_2-\partial_2A_0 = \frac 12i\lambda[\overline{\Psi}D_1\Psi-\Psi\overline{D_1\Psi}],\\ \partial_1A_2-\partial_2A_1 = -\frac12\lambda|\Psi|^2.\\ \end{array} \right. \end{equation} $\end{document} with an external potential \begin{document}$ V(x) $\end{document} , where \begin{document}$ D_{0} = \partial_{t}+i\lambda A_{0} $\end{document} and \begin{document}$ D_{k} = \partial_{x_k}-i\lambda A_{k}, \, k = 1,2, $\end{document} for \begin{document}$ (x_1,x_2,t)\in \mathbb{R}^{2,1} $\end{document} are covariant derivatives, \begin{document}$ \lambda $\end{document} is the coupling number. Suppose that \begin{document}$ V(x) $\end{document} satisfies \begin{document}$ \lim_{|x|\to\infty}V(x) = +\infty $\end{document} , we show for \begin{document}$ 2 that there exists \begin{document}$ \lambda^*>0 $\end{document} such that if \begin{document}$ 0 , problem (1) has two nontrivial static solutions \begin{document}$ (\Psi_\lambda, A_0^\lambda, A_1^\lambda,A_2^\lambda) $\end{document} . Moreover, there also exists \begin{document}$ \tilde\lambda>0 $\end{document} such that if \begin{document}$ \lambda>\tilde\lambda $\end{document} , problem (1) has no nontrivial solutions. While for \begin{document}$ p>4 $\end{document} we assume in addition that \begin{document}$ x\cdot \nabla V(x)\geq 0 $\end{document} , then problem (1) admits a mountain pass solution for all \begin{document}$ \lambda>0 $\end{document} .

We previously identified TDRG1 (testis developmental related gene 1), a novel gene with exclusive expression in testis, promoted the proliferation and progression of cultured human seminoma cells through PI3K/Akt/mTOR signaling. As increasing evidence reveal that aberrant activation of this signaling is involved in cisplatin resistance. Then, in this study, we further explored whether TDRG1 regulated the chemosensitivity of seminoma TCam-2 cells to cisplatin. Our researches showed TDRG1 could regulate the viability of TCam-2 cells following cisplatin treatment in vitro through control of both cell apoptosis and cell cycle. Mechanistically, we observed TDRG1 positively regulated the expression levels of the key elements in PI3K/Akt/mTOR pathway including p-PI3K, p-Akt and p-mTOR and also affected the translocation of nuclear p-Akt in TCam-2 cells during cisplatin treatment. Meanwhile, the levels of Bad, cytochrome c, caspase-9 ratio (activated/total), caspase-3 ratio (activated/total) and cleaved-PARP were negatively modulated by TDRG1, which meant the involvement of mitochondria-mediated apoptotic pathway. Furthermore, we found the effect of TDRG1 knockdown or TDRG1 overexpression could be reversed by IGF-1, a PI3K signaling activator, or LY294002, a inhibitor of this pathway, respectively. Similar effects of TDRG1 on cisplatin chemosensitivity and associated molecular mechanism were also confirmed in vivo by employing xenograft assays. In addition, the positive correlation between TDRG1 and p-PI3K, or p-Akt, was found in tumor tissues from seminoma patients. In conclusion, we uncover that TDRG1 regulates chemosensitivity of TCam-2 cells to cisplatin through PI3K/Akt/mTOR signaling and mitochondria-mediated apoptotic pathway both in vitro and in vivo.

In this paper, we show that the minimizing problem Λ s , N , k , α = inf u ∈ H ˙ s ⁢ ( ℝ N ) , u ≢ 0 ⁡ ∫ ℝ N | ( - Δ ) s 2 ⁢ u ⁢ ( x ) | 2 ⁢ 𝑑 x ( ∫ ℝ N | u ⁢ ( x ) | 2 s , α * | y | α ⁢ 𝑑 x ) 2 2 s , α * $\Lambda_{s,N,k,\alpha}=\inf_{u\in\dot{H}^{s}(\mathbb{R}^{N}),u\not\equiv 0}% \frac{\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u(x)|^{2}\,dx}{\bigl{(}\int% _{\mathbb{R}^{N}}\frac{|u(x)|^{2^{*}_{s,\alpha}}}{|y|^{\alpha}}\,dx\bigr{)}^{% \frac{2}{2^{*}_{s,\alpha}}}}$ is achieved by a positive, cylindrically symmetric and strictly decreasing function u ⁢ ( x ) ${u(x)}$ provided 0 < s < N 2 ${0 , 0 < α < 2 ⁢ s ${0 , where x = ( y , z ) ∈ ℝ k × ℝ N - k ${x=(y,z)\in\mathbb{R}^{k}\times\mathbb{R}^{N-k}}$ and 2 s , α * = 2 ⁢ ( N - α ) N - 2 ⁢ s ${2^{*}_{s,\alpha}=\frac{2(N-\alpha)}{N-2s}}$ . Decaying laws for the minimizer u are also established.

In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-\Delta)^{\frac \alpha 2} u=\lambda |u|^{r-2}u+\mu\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in }\quad \Omega, \\ \\ u=0 &{\rm on }\quad \partial\Omega, \end{array} \right. \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb R^N$ with $0\in \Omega$, $\lambda,\mu>0,0, Comment: 21 pages

In this paper, we consider the following problem involving fractional Laplacian operator: \begin{equation}\label{eq:0.1} (-\Delta)^{\alpha} u= |u|^{2^*_\alpha-2-\varepsilon}u + \lambda u\,\, {\rm in}\,\, \Omega,\quad u=0 \,\, {\rm on}\, \, \partial\Omega, \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $\varepsilon\in [0, 2^*_\alpha-2)$, $06\alpha$ and $\lambda>0$. An application of this compactness result is that problem \eqref{eq:0.1} possesses infinitely many solutions under the same assumptions., Comment: 34 pages

In this paper, we investigate the constrained minimization problem \begin{equation}\label{eq:0.1} e(a):=\inf_{\{u\in \mathcal{H},\|u\|_2^2=1\}}E_a(u), \end{equation} where the energy functional \begin{equation} \label{eq:0.2} E_a(u)=\int_{\mathbb{R}^3}(u\sqrt{-\Delta+m^2}\,u+Vu^2)\,dx -\frac{a}{2}\int_{\mathbb{R}^3}(|x|^{-1}*u^2)u^2\,dx \end{equation} with $m\in \mathbb{R}$, $a>0$, is defined on a Sobolev space $\mathcal{H}$. We show that there exists a threshold $a^*>0$ so that $e(a)$ is achieved if $0

This paper is concerned with the following H’enon type problem where B1(0) is the unit ball in ℝN centered at the origin, 1 < p, q and if N ≥ 3. We show that there exists α∗ > 0 such that the ground state solution of the problem is non-radial if α > α∗. We also consider the limiting behavior of the ground state solution (u, v) as p + q → 2∗. We prove that the maximum points of two components u and v concentrate at the same point on the boundary ∂B1(0).

In this paper, we are concerned with the existence of positive solutions of the following nonlinear elliptic system involving critical Hardy-Sobolev exponent \begin{equation*}\label{eq:1}(*) \left\{ \begin{array}{lll} -\Delta u= \frac{2\alpha}{\alpha+\beta}\frac{u^{\alpha-1}v^\beta}{|x|^s}-\lambda u^p, & \quad {\rm in}\quad \Omega,\\[2mm] -\Delta v= \frac{2\beta}{\alpha+\beta}\frac{u^\alpha v^{\beta-1}}{|x|^s}-\lambda v^p, & \quad {\rm in}\quad \Omega,\\[2mm] u\gt 0, v\gt 0, &\quad {\rm in}\quad \Omega,\\[2mm] u=v=0, &\quad {\rm on}\quad \partial\Omega, \end{array} \right. \end{equation*} where \(N\geq 4\) and \(\Omega\) is a \(C^1\) bounded domain in \(\mathbb{R}^N\) with \(0\in\partial\Omega\). \(0\lt s \lt 2\), \(\alpha+\beta=2^*(s)=\frac{2(N-s)}{N-2}\), \(\alpha,\beta\gt 1\), \(\lambda\gt 0\) and \(1 \lt p\lt \frac{N+2}{N-2}\). The case when 0 belongs to the boundary of \(\Omega\) is closely related to the mean curvature at the origin on the boundary. We show in this paper that problem \((*)\) possesses at least a positive solution.