On the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity

We consider the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity $\varepsilon^{2s}M([u]_{s,A_\varepsilon}^2)(-\Delta)_{A_\varepsilon}^su + V(x)u =$ $|u|^{2_s^\ast-2}u + h(x,|u|^2)u,$ $\ \ x\in \mathbb{R}^N,$ where $u(x) \rightarrow 0$ as $|x| \rightarrow \infty,$ and $(-\Delta)_{A_\varepsilon}^s$ is the fractional magnetic operator with $0<s<1$, $2_s^\ast = 2N/(N-2s),$ $M : \mathbb{R}^{+}_{0} \rightarrow \mathbb{R}^{+}$ is a continuous nondecreasing function, $V:\mathbb{R}^N \rightarrow \mathbb{R}^+_0,$ and $A: \mathbb{R}^N \rightarrow \mathbb{R}^N$ are the electric and the magnetic potential, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that $\varepsilon < \mathcal {E}$; and (ii) for any $m^\ast \in \mathbb{N}$, has $m^\ast$ pairs of solutions if $\varepsilon < \mathcal {E}_{m^\ast}$, where $\mathcal {E}$ and $\mathcal {E}_{m^\ast}$ are sufficiently small positive numbers. Moreover, these solutions $u_\varepsilon \rightarrow 0$ as $\varepsilon \rightarrow 0$

Multiple solutions for critical Choquard-Kirchhoff type equations

AbstractIn this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents,−a+b∫RN|∇u|2dxΔu=αk(x)|u|q−2u+β∫RN|u(y)|2μ∗|x−y|μdy|u|2μ∗−2u,x∈RN,$$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$wherea> 0,b≥ 0, 0 <μ<N,N≥ 3,αandβare positive real parameters,2μ∗=(2N−μ)/(N−2)$\begin{array}{} 2^*_{\mu} = (2N-\mu)/(N-2) \end{array}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality,k∈Lr(ℝN), withr= 2∗/(2∗−q) if 1 <q< 2*andr= ∞ ifq≥ 2∗. According to the different range ofq, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting

Infinitely many solutions for the p-fractional Kirchhoff equations with electromagnetic fields and critical nonlinearity

In this paper, we consider the fractional Kirchhoff equations with electromagnetic fields and critical nonlinearity. By means of the concentration-compactness principle in fractional Sobolev space and the Kajikiya's new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions, which tend to zero for suitable positive parameters

The existence of countably many positive solutions for nonlinear singular m-point boundary value problems on the half-line

AbstractIn this paper, by introducing a new operator, improving and generating a p-Laplace operator for some p>1, we study the existence of countably many positive solutions for nonlinear boundary value problems on the half-line (φ(u′))′+a(t)f(u(t))=0,0<t<+∞,u(0)=∑i=1m−2αiu(ξi),u′(∞)=0, where φ:R→R is the increasing homeomorphism and positive homomorphism and φ(0)=0. We show the sufficient conditions for the existence of countably many positive solutions by using the fixed-point index theory and a new fixed-point theorem in cones

Solutions of stationary Kirchhoff equations involving nonlocal operators with critical nonlinearity in RN

In this paper, we consider the existence and multiplicity of solutions for fractional Schrödinger equations with critical nonlinearity in&nbsp;RN. We use the fractional version of Lions' second concentration-compactness principle and concentration-compactness principle at infinity to prove that (PSc) condition holds locally. Under suitable assumptions, we prove that it has at least one solution and, for any&nbsp;m ∈ N, it has at least&nbsp;m&nbsp;pairs of solutions. Moreover, these solutions can converge to zero in some Sobolev space as&nbsp;ε&nbsp;→ 0

Sign-changing solutions for Kirchhoff-type problems involving variable-order fractional Laplacian and critical exponents

In this paper, we are concerned with the Kirchhoff-type variable-order fractional Laplacian problem with critical variable exponent. By using constraint variational method and quantitative deformation lemma we show the existence of one least energy solution, which is strictly larger than twice of that of any ground state solution

Beneficial effect of fluid warming in elderly patients with bladder cancer undergoing Da Vinci roboticassisted laparoscopic radical cystectomy

OBJECTIVES: The enhanced recovery after surgery (ERAS) protocol recommends prevention of intraoperative hypothermia. However, the beneficial effect of maintaining normothermia after radical cystectomy has not been evaluated. This study aimed to investigate the efficacy of fluid warming nursing in elderly patients undergoing Da Vinci robotic-assisted laparoscopic radical cystectomy. METHODS: A total of 108 patients with bladder cancer scheduled to undergo DaVinci robotic-assisted laparoscopic radical cystectomy were recruited and randomly divided into the control group (n=55), which received a warming blanket (43o C) during the intraoperative period and the warming group (n=53), in which all intraoperative fluids were administered via a fluid warmer (41o C). The surgical data, body temperature, coagulation function indexes, and postoperative complications were compared between the two groups. RESULTS: Compared to the control group, the warming group had significantly less intraoperative transfusion (p=0.028) and shorter hospitalization days (po0.05). During the entire intraoperative period (from 1 to 6h), body temperature was significantly higher in the warming group than in the control group. There were significant differences in preoperative fibrinogen level, white blood cell count, total bilirubin level, intraoperative lactose level, postoperative thrombin time (TT), and platelet count between the control and warming groups. Multivariate linear regression analysis demonstrated that TT was the only significant factor, suggesting that the warming group had a lower TT than the control group. CONCLUSION: Fluid warming nursing can effectively reduce transfusion requirement and hospitalization days, maintain intraoperative normothermia, and promote postoperative coagulation function in elderly patients undergoing Da Vinci robotic-assisted laparoscopic radical cystectomy