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In this article, we consider the existence of multiple nodal solutions of the nonlinear Choquard equation -Δu + u = (∣x∣-1 * ∣u∣ p )∣u∣p-2u in R 3 , u ∈ H1 (R3 ), where p ∈ (5/2, 5). We show that for any positive integer k, the above problem has at least one radially symmetrical solution changing sign exactly k-times. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the constrained minimization problem e(a):=inf{u∊H ∣∣u∣∣22=1}Ea(u), where the energy functional Ea(u) = ∫ℝ3(u√-Δ + m2 u + Vu2)dx-a/2∫ℝ3(∣x∣-1 *u2)u2dx with m∊ℝ, a>0, is defined on a Sobolev space H. We show that there exists a threshold (a* > 0 so that e(a) is achieved if 0

In this article, we consider the existence of nodal solutions for the Schrodinger-Poisson type problem ... where a; b are positive constants, p ∊ (4; 6) and V (x) is a radial smooth function. For each k ∊ N+, we show the existence of to nodal solution changing sign exactly k times. [ABSTRACT FROM AUTHOR]

Testis developmental related gene 1 (TDRG1) is a novel human testis-specific gene. TDRG1 is differentially expressed in cancerous tissue compared with normal testicular tissue and demonstrates a unique expression pattern in normal testes; therefore, this gene may be involved in the occurrence and development of testicular germ cell tumors (TGCT). In the present study, the expression level of TDRG1 was downregulated in human TGCT NTERA-2 cells by RNA interference (RNAi) in order to investigate the association between TDRG1 and TGCT. The TDRG1 mRNA and protein expression levels in NTERA-2 cells were significantly inhibited following transfection with specific RNAi plasmids. The ability to proliferate (inhibited by 15.4% at day 3 and 26.1% at day 5; P<0.001) and invade (reduced by 49.1%; P=0.01) in vitro was suppressed in cells in which the expression level of TDRG1 was reduced, and a corresponding increase in the apoptotic potential was observed (the early apoptotic potential and total apoptotic potential were increased by 75%; P=0.019 and 54.8%; P=0.009, respectively). The results of the present study indicated that the biological behavior of NTERA-2 cells is associated with TDRG1 expression levels, and that this gene may be a novel target candidate in the treatment of TGCT. [ABSTRACT FROM AUTHOR]

In this paper, we prove that any Hs (Ω) solution u of the problem s (0.1) (-Δ)su = f(u) in Ω,u = 0 on ∂ύ, belongs to L∞ (Ω) for the nonlinearity of f(t) being subcritical and critical. This implies that the solution u is classical if f(t) is C1, γ for some 0 < γ < 1. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of positive solutions of the following nonlinear elliptic system involving critical Hardy-Sobolev exponent ... where N¹ ≥ 4 and Ω is a C¹ bounded domain in ℝN with 0 ∈ ∂Ω. 0 < s < 2, α+β = 2* (s) = 2(N-2)/N-2,α, β > 1, λ > 0 and 1 < p < N+2/N-2 . The case when 0 belongs to the boundary of Ω 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. [ABSTRACT FROM AUTHOR]

This aticle concerns the existence of positive solutions to the nonlinear elliptic system involving critical Hardy-Sobolev exponent -Δu = 2λα / α + β uα-1vβ / |π(x)|s - up, in Ω, -Δu = 2λα / α + β uα-1vβ / |π(x)|s - vp, in Ω, u > 0, v > 0, in Ω, u = v = 0, on ∂Ω, where N ≥ 4 and Ω is a C¹ bounded domain in ℝN, 0 < s < 2, α + β = 2*(s) = 2(N-s)/N-2, α, β > 1, λ > 0 and 1 ≤ p < N/N-2. Let P be a linear subspace of ℝN such that k = dimℝ P ≥ 2, and π be the orthogonal projection on P with respect to the Euclidean structure. We consider mainly the case when P⊥ ∩ Ω = ∅ and P⊥ ∩ ∂Ω ≠ ∅. We show that there exists λ* > 0 such that the system above possesses at least one positive solution for 0 < λ < λ* provided that at each point x ∈ P⊥ ∩ ∂Ω the principal curvatures of ∂Ω at x are non-positive, but not all vanish. [ABSTRACT FROM AUTHOR]

In this article, we investigate the asymptotic behavior of positive ground state solutions, as α → ∞, for the following Hénon type system -Δu = 2p/p+q|x|αup-1vq, -Δv = 2q/p+q|x|αupvq-1, in B1(0) with zero boundary condition, where B1(0) ⊂ ℝN (N ≥ 3) is the unit ball centered at the origin, p, q > 1, p + q < 2* = 2N/(N - 2). We show that both components of the ground solution pair (u, v) concentrate on the same point on the boundary ∂B1(0) as α → ∞. [ABSTRACT FROM AUTHOR]

Let (M, g) be a compact, connected, orientable, Riemannian n-manifold of class C∞ with Riemannian metric g (n ≥ 3). We study the existence of solutions to the equation -ε²Δgu + V (x)u = K(x)|u|p-2u on this Riemannian manifold. Here 2 < p < 2* = 2n/(n - 2), V (x) and K(x) are continuous functions. We show that the shape of V (x) and K(x) affects the number of solutions, and then prove the existence of multiple solutions. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of finding standing waves - solutions to nonlinear Schrödinger equations with vanishing potential and sign-changing nonlinearities. This involves searching for solutions of the problem -ϵ2Δu+V(x)u=Q(x)∣u∣p-1u in RN(0.1) We show that the problem has a solution, and the maximum point of the solution is concentrated on a minimum point of some function as ϵ→o. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence of solutions for the strongly indefinite elliptic system -Δu = λu + f(x, v) in Ω, -Δv = λv + g(x, u) in Ω, u = v = 0, on ∂Ω, where Ω is a bounded domain in ℝN (N ≥ 3) with smooth boundary, λk0 < λ < λk0+1, where λk is the kth eigenvalue of -Δ in Ω with zero Dirichlet boundary condition. Both cases when f, g being superlinear and asymptotically linear at infinity are considered. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions for the nonlinear elliptic system -Δu + u = |u|p-1u + λv in Ω, -Δv + v = |v|p-1v + λu in Ω, u = v = 0 on ∂Ω, where is a exterior domain in ∇N, N ≥ 3. We show that the system possesses at least one nontrivial positive solution. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Neumann problem for a critical elliptic system in exterior domains. Assuming that the coefficient Q(x) is a positive smooth function and λ, μ ≥ 0 are parameters, we examine the common effect of the mean curvature of the boundary ∂Ω and the shape of the graph of the coefficient Q(x) on the existence of the least energy solutions. [ABSTRACT FROM AUTHOR]

In this paper, we prove, under natural assumptions, that the semilinear elliptic system -Δu+u=g(x,v), x ∈ RN, -Δv+v=f(x,u), x∈RN (0.1) possesses at least one positive solution pair (u, v) ∈ H¹(RN) × H¹(RN), where f(x, t) and g(x, t) are continuous functions on RN × R and asymptotically linear as t → + ∞. [ABSTRACT FROM AUTHOR]

In this corrigendum we give a correct proof for equation (3.17), as well as the assertion on line 15, page 1364, of "Existence of solutions for semilinear elliptic problems without (PS) condition", Proc. Amer. Math. Soc. 132(5) (2004), 1355-1366. [ABSTRACT FROM AUTHOR]

We consider the existence of rotating solitary waves (vortices) for a coupled Schrodinger equations by finding solutions to the singular system. where λ1; λ2; μ1; μ2 are positive parameters, β ≠ 0. We show that this system has a positive least energy solution for the cases when either β is negative or α is positive and small or large. Moreover, if λ1 = λ2, then the solution is unique. We also study the limiting behavior of the least energy solutions in the repulsive case for β → ∞1, and phase separation. [ABSTRACT FROM AUTHOR]

In this article, we consider the regularity and symmetry of positive solutions to the nonlinear integral system u(x) = ∫ℝn Kα(x - y) v(y)q/|y|β dy, v(x) = ∫ℝn Kα(x - y) u(y)p/|y|β dy for x ∈ ℝn, + where Kα(x) is the kernel of the operator (-Δ) + id of order α, with 0 ≤ β < 2α < n, 1 < p, q < (n - β)/β and 1/p + 1 + 1/q + 1 > n - 2α+β/n . We show that positive solution pairs (u, v) ∈ Lp+1 (ℝn)xLq+1(ℝn) are locally Hölder continuous ö ℝN\{0}, radially symmetric about the origin, and strictly decreasing. [ABSTRACT FROM AUTHOR]

In this paper we present a novel variational approach to the question of existence and multiplicity of positive solutions to semipositone problems in a bounded smooth domain of ℝN. We consider both the sublinear and superlinear cases. [ABSTRACT FROM AUTHOR]

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 (~65 μM) iAs in drinking water for 3 months. RESULTS: Low-dose iAs (1 lM) 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. [ABSTRACT FROM AUTHOR]

Human testis development-related gene 1 (TDRG1) is a recently identified gene that is expressed exclusively in the testes and promotes the development of testicular germ cell tumors. In this study, the role of TDRG1 in the development of testicular seminoma, which is the most common testicular germ cell tumor, was further investigated. Based on polymerase chain reaction, Western blotting, and immunohistochemistry tests, both gene and protein expression levels of TDRG1 were significantly upregulated in testicular seminoma tissues compared with normal testicular tissues. Additionally, the levels of phosphoinositide-3 kinase (PI3K)/p110 and Akt phosphorylation were dramatically upregulated in testicular seminoma tissues. Accordingly, in our cell experiment, seminoma TCam-2 cells were subjected to different treatments: the TDRG1 knockout, TDRG1 overexpression, PI3K inhibition (LY294002 administration), or PI3K activation (insulin-like growth factor-1 administration). Cell proliferation, the proliferation index, apoptosis rate, cell adhesive capacity, and cell invasion capability were assessed. Cells with both TDRG1 knockout and PI3K inhibition exhibited decreased cell proliferation, proliferation indexes, cell adhesion capacity, and cell invasion capability and increased apoptosis rates. Most of these effects were reversed by TDRG1 overexpression or PI3K activation, indicating that both TDRG1- and PI3K-mediated signaling promote proliferation and invasion of testicular seminoma cells. The knockout of TDRG1 significantly decreased the phosphorylation levels of PI3K/p85, PI3K/p110, Akt, and mammalian target of rapamycin (mTOR; Ser2448). Except for PI3K/p110, TDRG1 overexpression had the opposite effects on phosphorylation levels. Phosphorylated mTOR at Ser2481 and Thr2446 was not affected by TDRG1 or PI3K in our tests. Thus, these results indicate that TDRG1 promotes the development and migration of seminoma cells via the regulation of the PI3K/Akt/mTOR signaling pathway; this contributes to an understanding of the precise mechanisms underlying the development and migration of seminomas and lays a theoretical foundation for the development of appropriate therapies. [ABSTRACT FROM AUTHOR]

Objective: MicroRNA-100 (miR-100) has been demonstrated to be downregulated in bladder cancer tissues, and enforced expression of this miRNA may inhibit cell growth and colony formation of human bladder cancer 5637 cells in vitro. However, the clinical significance of miR-100 in human bladder cancer has not yet been elucidated. Thus, the aim of this study was to investigate the diagnostic and prognostic values of miR-100 in this disease. Methods: Expression levels of miR-100 in 126 pairs of bladder cancer and adjacent normal tissues were detected by TaqMan real-time quantitative RT-PCR assay. In order to determine its prognostic value, overall survival (OS) and progression-free survival (PFS) were evaluated using the Kaplan-Meier method, and multivariate analysis was performed using the Cox proportional hazard analysis. Results: Expression levels of miR-100 in bladder cancer tissues were significantly lower than those in adjacent normal tissues (mean expression level: 2.6 ± 1.2 vs. 3.9 ± 1.5, P < 0.001). When categorized into low vs. high expression, low miR-100 expression was negatively associated with the stage (P = 0.01), the recurrence (P = 0.008), the progression (P = 0.01), and the death (P < 0.001) of patients with bladder cancer. Moreover, low miR-100 expression clearly predicted poorer PFS (P = 0.001) and OS (P < 0.001). In the multivariate analysis, low miR-100 expression was an independent prognostic factor for both PFS (P = 0.01) and OS (P = 0.008). Conclusion: Our data offer the convincing evidence that miR-100 may play an important role in the progression of bladder cancer and that the reduced expression of this miRNA may be independently associated with shorter PFS and OS of patients, suggesting that miR-100 might be a potential marker for further risk stratification in the treatment of this cancer. [ABSTRACT FROM AUTHOR]