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Objective To assess serum levels of zinc in children patients with pityriasis alba (PA) and study mRNA expression profile of oxidative stress pathway by bioinformatics technology to investigate the relationship between oxidative stress and PA. Methods Healthy controls (167 cases) and children patients with PA (196 cases) were choosed from the Department of Child Health Care and Dermatology of Panyu Maternal and Child Care Service Centre of Guangzhou. Zinc was assayed by using inductively coupled plasma mass spectrometry; zinc levels in the PA and control groups were compared according to different age groups. Total RNA from peripheral blood of patients with PA and healthy children (5 each) were extracted and purified. High-throughput sequencing was used to detect mRNA expression levels and identify differentially expressed superoxide dismutase (SOD) mRNAs between patients with PA and healthy controls. The correlation between differential mRNAs was analyzed. Results The mean zinc level of patients with PA of different ages was 3.99±0.81, 4.37±1.04, 4.48±0.92, 5.07±0.99, 5.34±0.88, respectively, while that of health controls was 4.77±0.83, 5.61±1.01, 5.42±0.43, 5.91±0.66, 5.69±0.54 (all P

In this paper, we discuss the generalized quasilinear Schrödinger equation with nonlocal term: $ \begin{align} -\mathrm{div}(g^{2}(u)\nabla u)+g(u)g'(u)|\nabla u|^{2}+V(x)u = \left(|x|^{-\mu}\ast F(u)\right)f( u),\; \; x\in \mathbb{R}^{N}, \;\;\;\;\;\;\;\;({{\rm{P}}})\end{align} $ where $ N\geq 3 $, $ \mu\in(0, N) $, $ g\in \mathbb{C}^{1}(\mathbb{R}, \mathbb{R}^{+}) $, $ V\in \mathbb{C}^{1}(\mathbb{R}^N, \mathbb{R}) $ and $ f\in \mathbb{C}(\mathbb{R}, \mathbb{R}) $. Under some "Berestycki-Lions type conditions" on the nonlinearity $ f $ which are almost necessary, we prove that problem $ (\rm P) $ has a nontrivial solution $ \bar{u}\in H^{1}(\mathbb{R}^{N}) $ such that $ \bar{v} = G(\bar{u}) $ is a ground state solution of the following problem $ \begin{align} - \Delta v+V(x)\frac{G^{-1}(v)}{g(G^{-1}(v))} = \left(|x|^{-\mu}\ast F(G^{-1}(v))\right)f( G^{-1}(v)),\; \; x\in \mathbb{R}^{N}, \;\;\;\;\;\;\;\;({{\rm{\bar P}}})\end{align} $ where $ G(t): = \int_{0}^{t} g(s) ds $. We also give a minimax characterization for the ground state solution $ \bar{v} $.

In this paper, we study Kirchhoff equations with logarithmic nonlinearity: \begin{equation*} \begin{cases} -(a+b\int_\Omega|\nabla u|^2)\Delta u+ V(x)u=|u|^{p-2}u\ln u^2, & \mbox{in}\ \Omega,\\ u=0,& \mbox{on}\ \partial\Omega, \end{cases} \end{equation*} where $a,b>0$ are constants, $4

Coronary heart disease is one of the most serious health problems in the world nowadays. By segmenting and examining the coronary arteries in medical images, we can find artery stenosis and plaque, which are the main causes of this certain disease. Segmenting coronary arteries manually is time-consuming and subjective, and the traditional segmentation method requires a good initial point, which is also difficult to apply for the 3D coronary computed tomography angiography (CTA) data. In this paper, we propose a joint framework for coronary CTA segmentation based on deep learning and traditional level set method. A 3D fully convolutional network (FCN) is used to learn the 3D semantic features of coronary arteries, which provides an excellent initial point for the traditional level set. Moreover, an attention gate is added to the entire network, aiming to enhance the vessels and suppress irrelevant regions. The output of 3D FCN with attention gate is optimized by the level set to smooth the boundary to better fit the ground truth segmentation. The proposed algorithm is evaluated by the Jaccard index (JI) and the Dice similarity coefficient (DSC) scores. The experimental results show that the proposed framework provides significant better segmentation results than the vanilla 3D FCN intuitively and quantitively.

This paper is dedicated to studying the following fractional Choquard equation \begin{equation*} (-\triangle)^s u+V(x)u=\left(\int_{\mathbb{R}^N}\frac{Q(y)F(u(y))}{|x-y|^\mu}\mathrm{d}y\right)Q(x)f(u), \qquad u\in H^s(\mathbb{R}^{N}), \end{equation*} where $s\in(0,1)$, $N\ge 3$, $\mu\in(0,N)$, $V(x)$ and $Q(x)$ are periodic or asymptotically periodic, and $F(t)=\int_0^tf(s)\mathrm{d}s$. By combining the non-Nehari manifold approach with some new inequalities, we establish the existence of Nehari type ground state solutions for the above problem in the periodic and asymptotically periodic cases under mild assumptions on $f$. Our results generalize and improve the ones in [Y. H. Chen, C. G. Liu, Nonlinearity 29(2016), 1827–1842] and some related literature.

This article concerns the planar Schrodinger-Poisson system $$\displaylines{ -\Delta u+V(x)u+\phi u=f(x,u), \quad x\in \mathbb{R}^2,\cr \Delta \phi= u^2, \quad x\in \mathbb{R}^2, } $$ where V(x) and f(x, u) are periodic or asymptotically periodic in x. By combining the variational approach, the non-Nehari manifold approach and new analytic techniques, we establish the existence of ground state solutions for the above problem in the periodic and asymptotically periodic cases. In particular, in our study, f is not required to satisfy the Ambrosetti-Rabinowitz type condition or the Nehari-type monotonic condition.

In this article we prove the existence of ground state solutions for the quasilinear Schrodinger equation $$ -\Delta u+V(x)u-\Delta (u^2)u= g(u), \quad x\in \mathbb{R}^N, $$ where $N\ge 3$, $V\in \mathcal{C}^1(\mathbb{R}^N, [0, \infty))$ satisfies mild decay conditions and $g\in \mathcal{C}(\mathbb{R}, \mathbb{R})$ satisfies Berestycki-Lions conditions which are almost necessary. In particular, we introduce some new inequalities and techniques to overcome the lack of compactness.

We consider the nonlinear fractional Kirchhoff equation $$ \Big(a+b\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x\Big) (-\Delta)^\alpha u+V(x)u=f(u) \quad \text{in } \mathbb R^3, u\in H^{\alpha}(\mathbb R^3), $$ where a>0, $b\ge 0$, $\alpha\in(3/4, 1)$ are three constants, V(x) is differentiable and $f\in C^1(\mathbb R, \mathbb R)$. Our main results show the existence of ground state solutions of Nehari-Pohozaev type, and the existence of the least energy solutions to the above problem with general superlinear and subcritical nonlinearity. These results are proved by applying variational methods and some techniques from [27].

Abstract In the present paper, we consider the existence of ground state sign-changing solutions for the semilinear Dirichlet problem 0.1 {−△u+λu=f(x,u),x∈Ω;u=0,x∈∂Ω, $$ \left \{ \textstyle\begin{array}{l@{\quad}l} -\triangle u+\lambda u=f(x, u), & \hbox{$x\in\Omega$;} \\ u=0, & \hbox{$x\in\partial\Omega$,} \end{array}\displaystyle \right . $$ where Ω⊂RN $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with a smooth boundary ∂Ω, λ>−λ1 $\lambda>-\lambda_{1}$ is a constant, λ1 $\lambda_{1}$ is the first eigenvalue of (−△,H01(Ω)) $(-\triangle, H_{0}^{1}(\Omega))$, and f∈C(Ω×R,R) $f\in C(\Omega\times\mathbb{R}, \mathbb{R})$. Under some standard growth assumptions on f and a weak version of Nehari type monotonicity condition that the function t↦f(x,t)/|t| $t\mapsto f(x, t)/|t|$ is non-decreasing on (−∞,0)∪(0,∞) $(-\infty, 0)\cup(0, \infty)$ for every x∈Ω $x\in\Omega$, we prove that (0.1) possesses one ground state sign-changing solution, which has precisely two nodal domains. Our results improve and generalize some existing ones.

Abstract In this paper, we studied the following fractional Kirchhoff-type equation: ( a + b ∫ R N | ( − △ ) α 2 u | 2 d x ) ( − △ ) α u + V ( x ) u = f ( x , u ) , x ∈ R N , $$\biggl(a+b \int_{\mathbb{R}^{N}} \bigl\vert (-\triangle)^{\frac{\alpha }{2}}u \bigr\vert ^{2}\,\mathrm{d}x \biggr) (-\triangle)^{\alpha }u+V(x)u=f(x,u), \quad x\in{\mathbb{R}}^{N}, $$ where a, b are positive constants, α ∈ ( 0 , 1 ) $\alpha\in(0,1)$ , N ∈ ( 2 α , 4 α ) $N\in (2\alpha,4\alpha)$ , ( − △ ) α $(-\triangle)^{\alpha}$ is the fractional Laplacian operator, V ( x ) $V(x)$ and f ( x , u ) $f(x,u)$ are periodic or asymptotically periodic in x. Under some weaker conditions on the nonlinearity, we obtain the existence of ground state solutions for the above problem in periodic case and asymptotically periodic case, respectively. In particular, our results unify both asymptotically cubic and super-cubic nonlinearities, which are new even for α = 1 $\alpha=1$ .

This article concerns the fourth-order elliptic equation $$\displaylines{ \Delta^2u-\Delta u+\lambda V(x)u=f(x, u)+\mu \xi(x)|u|^{p-2}u, \quad x\in \mathbb{R}^{N},\cr u\in H^2(\mathbb{R}^{N}), }$$ where $\lambda >0$ is a parameter, $V\in C(\mathbb{R}^{N},\mathbb{R})$ and $V^{-1}(0)$ has nonempty interior. Under some mild assumptions, we establish the existence of two nontrivial solutions. Moreover, the concentration of these solutions is explored on the set $V^{-1}(0)$ as $\lambda\to\infty$. As an application, we give the similar results and concentration phenomenona for the above problem with concave and convex nonlinearities.

In this article, we prove the existence of infinitely many solutions for the fractional $p$-Laplacian equation $$ (-\Delta)^s_p u+V(x)|u|^{p-2}u=f(x,u),\quad x\in \mathbb{R}^N $$ where $s\in(0,1)$, $2\leq p

Abstract In this paper we study the existence of sign-changing solutions for nonlinear problems involving the fractional Laplacian 0.1 { ( − Δ ) s u − λ u = f ( x , u ) , x ∈ Ω , u = 0 , x ∈ R n ∖ Ω , $$ \left \{ \textstyle\begin{array}{l@{\quad}l} (-\Delta)^{s}u-\lambda u=f(x,u),&x\in\Omega, \\ u=0,&x\in\mathbb{R}^{n}\setminus\Omega, \end{array}\displaystyle \right . $$ where Ω ⊂ R n $\Omega\subset\mathbb{R}^{n}$ ( n ≥ 2 $n\geq2$ ) is a bounded smooth domain, s ∈ ( 0 , 1 ) $s\in(0,1)$ , ( − Δ ) s $(-\Delta)^{s}$ denotes the fractional Laplacian, λ is a real parameter, the nonlinear term f satisfies superlinear and subcritical growth conditions at zero and at infinity. When λ ≤ 0 $\lambda\leq0$ , we prove the existence of a positive solution, a negative solution and a sign-changing solution by combing minimax method with invariant sets of descending flow. When λ ≥ λ 1 s $\lambda\geq \lambda_{1}^{s}$ (where λ 1 s $\lambda_{1}^{s}$ denotes the first eigenvalue of the operator ( − Δ ) s $(-\Delta)^{s}$ in Ω with homogeneous Dirichlet boundary data), we prove the existence of a sign-changing solution by using a variation of linking type theorems.

In this paper, we prove the existence of many solutions for the following quasilinear Schrödinger equation \begin{equation*} -\Delta u - u\Delta(|u|^2) + V(|x|)u = f(|x|,u),\qquad x \in \mathbb{R}^N. \end{equation*} Under some generalized assumptions on $f$, we obtain infinitely many radial solutions for $N\geq 2$, many non-radial solutions for $N=4$ and $N \geq 6$, and a non radial solution for $N=5$. Our results generalize and extend some existing results.

In this article, we prove the existence of multiple solutions for following fractional Schrodinger-Poisson system with sign-changing potential $$\displaylines{ (-\Delta)^s u+V(x)u+\lambda\phi u=f(x,u),\quad x\in\mathbb{R}^3,\cr (-\Delta)^t\phi=u^2,\quad x\in\mathbb{R}^3, }$$ where $(-\Delta)^\alpha$ denotes the fractional Laplacian of order $\alpha\in(0,1)$, and the potential V is allowed to be sign-changing. Under certain assumptions on f, we obtain infinitely many solutions for this system.

This article concerns the nonlinear Dirac-Poisson system $$\displaylines{ -i\sum^3_{k=1}\alpha_{k}\partial_{k}u + (V(x)+a)\beta u + \omega u-\phi u =F_u(x,u),\cr -\Delta \phi=4\pi|u|^2, }$$ in $\mathbb{R}^3$, where $V(x)$ is a potential function and $F(x,u)$ is an asymptotically quadratic nonlinearity modeling various types of interaction. Since the effects of the nonlocal term, we use some special techniques to deal with the nonlocal term. Moreover, the existence of infinitely many stationary solutions is obtained for system with periodicity assumption via variational methods.

We consider the perturbed biharmonic equations $$ \varepsilon^4 \Delta^2 u+V(x)u=f(x,u),\quad x\in\mathbb{R}^N $$ and $$ \varepsilon^4 \Delta^2 u+V(x)u=Q(x)|u|^{2^{\ast\ast}-2}u+f(x,u), \quad x\in\mathbb{R}^N $$ where $\Delta^2$ is the biharmonic operator, $N\geq 5$, $2^{\ast\ast}=\frac{2N}{N-4}$ is the Sobolev critical exponent, Q(x) is a bounded positive function. Under some mild conditions on V and f, we show that the above equations have at least one nontrivial solution provided that $\varepsilon \leq \varepsilon_0$, where the bound $\varepsilon_0$ is formulated in terms of N, V, Q and f.

In this article we study the Schrodinger equation $$ -\Delta u=f(x,u),\quad x\in\Omega, \quad u\in H_0^1(\Omega), $$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ and f(x,u) is asymptotically linear at infinity with respect to u. Inspired by the works of Salvatore [14] on sign-changing solutions, in which $f(x,u)$ is asymptotically linear at zero with respect to $u$, we prove, via the constraint variational method and the quantitative deformation lemma, that the equation possesses one sign-changing solution with exactly two nodal domains.

In this article, we study the existence of least energy sign-changing solutions for nonlinear problems involving fractional Laplacian. By introducing some new ideas and combining constraint variational method with the quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution.

This article concerns the periodic solutions for a nonlinear Dirac equation. Under suitable assumptions on the nonlinearity, we show the existence of nontrivial and ground state solutions.

In this article, we consider the existence of infinitely many nontrivial solutions for the following semilinear Schr\"odinger equation with electromagnetic potential $$ \big(-i\nabla+A(x)\big)^2u+V(x)u=f(x,|u|)u,\quad\text{in } \mathbb{R}^N $$ where i is the imaginary unit, V is the scalar (or electric) potential, A is the vector (or magnetic) potential. We establish the existence of infinitely many solutions via variational methods.

In this article, we sutdy the multiplicity of homoclinic solutions to the perturbed second-order discrete Hamiltonian system $$ \Delta[p(n)\Delta u(n-1)]-L(n)u(n)+\nabla W(n,u(n))+\theta\nabla F(n,u(n))=0, $$ where L(n) and W(n,x) are neither autonomous nor periodic in n. Under the assumption that W(n,x) is only locally superquardic as $|x|\to \infty$ and even in x and F(n,x) is a perturbation term, we establish some existence criteria to guarantee that the above system has multiple homoclinic solutions by minimax method in critical point theory.

This article concerns the Schr\"odinger equation $$\displaylines{ -\Delta u+V(x)u=f(x, u), \quad \text{for } x\in\mathbb{R}^N,\cr u(x)\to 0, \quad \text{as } |x| \to \infty, }$$ where V and f are periodic in x, and 0 is a boundary point of the spectrum $\sigma(-\Delta+V)$. Assuming that f(x,u) is asymptotically linear as $|u|\to\infty$, existence of a ground state solution is established using some new techniques.

This article concerns the Schrodinger equation $$\displaylines{ -\Delta u+V(x)u=f(x, u), \quad \text{for } x\in\mathbb{R}^N,\\ u(x)\to 0, \quad \text{as } |x| \to \infty\,. }$$ Assuming that V and f are periodic in x, and 0 is a boundary point of the spectrum $\sigma(-\Delta+V)$, two types of ground state solutions are obtained with some super-quadratic conditions.

This article concerns the elliptic system $$\displaylines{ -\Delta u+V(x)u=W_{v}(x, u, v), \quad x\in \mathbb{R}^{N},\cr -\Delta v+V(x)v=W_{u}(x, u, v), \quad x\in \mathbb{R}^{N},\cr u, v\in H^{1}(\mathbb{R}^{N}), }$$ where V and W are periodic in x, and W(x,z) is super-linear in z=(u,v). We use a new technique to show that the above system has a nontrivial solution under concise super-quadratic conditions. These conditions show that the existence of a nontrivial solution depends mainly on the behavior of W(x,u,v) as $|u+v| \to 0$ and $|au+bv| \to \infty$ for some positive constants a,b.

This article concerns the fourth-order elliptic equations $$\displaylines{ \Delta^{2}u-\Delta u+\lambda V(x)u=f(x, u), \quad x\in \mathbb{R}^N,\cr u\in H^{2}(\mathbb{R}^N), }$$ where $\lambda >0$ is a parameter, $V\in C(\mathbb{R}^N)$ and $V^{-1}(0)$ has nonempty interior. Under some mild assumptions, we establish the existence of nontrivial solutions. Moreover, the concentration of solutions is also explored on the set $V^{-1}(0)$ as $\lambda\to\infty$.

We consider the system of coupled nonlinear Schrodinger equations $$\displaylines{ -\varepsilon^2\Delta u+a(x) u=H_{u}(x, u, v)+\mu(x) v, \quad x\in \mathbb{R}^N,\cr -\varepsilon^2\Delta v+b(x) v=H_{v}(x, u, v)+\mu(x) u, \quad x\in \mathbb{R}^N,\cr u,v\in H^1(\mathbb{R}^N), }$$ where $N\geq 3$, $a, b, \mu \in C(\mathbb{R}^N)$ and $H_{u}, H_{v}\in C(\mathbb{R}^N\times \mathbb{R}^2, \mathbb{R})$. Under conditions that $a_0=\inf a=0$ or $b_0=\inf b=0$ and $|\mu(x)|^2\le \theta a(x)b(x)$ with $\theta\in(0, 1)$ and some mild assumptions on $H$, we show that the system has at least one nontrivial solution provided that $0

We study the existence of nontrivial solution of the following equation without compactness: (-Δ)pαu+up-2u=f(x,u), x∈RN, where N,p≥2, α∈(0,1), (-Δ)pα is the fractional p-Laplacian, and the subcritical p-superlinear term f∈C(RN×R) is 1-periodic in xi for i=1,2,…,N. Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of fractional p-Laplacian type equation. To overcome this difficulty, by adding coercive potential term and using mountain pass theorem, we get the weak solution uλ of perturbation equations. And we prove that uλ→u as λ→0. Finally, by using vanishing lemma and periodic condition, we get that u is a nontrivial solution of fractional p-Laplacian equation.

In this article, we study the elliptic boundary value problem $$\displaylines{ -\Delta u+a(x)u=g(x, u) \quad \text{in } \Omega,\cr u = 0\quad \text{on } \partial \Omega, }$$ where $ \Omega\subset \mathbb{R}^N$ $(N\geq3)$ is a bounded domain with smooth boundary $ \partial\Omega$ and the potential $a(x)$ is allowed to be sign-changing. We establish the existence of infinitely many nontrivial solutions by variant fountain theorem developed by Zou for sublinear nonlinearity.

In this article, we study the multiplicity of solutions for the semilinear elliptic equation $$\displaylines{ -\Delta u+a(x)u=f(x, u), \quad x\in \Omega,\cr u=0, \quad x \in \partial\Omega, }$$ where $ \Omega\subset \mathbb{R}^N$ $(N\geq3)$, the potential a(x) satisfies suitable integrability conditions, and the primitive of the nonlinearity f is of super-quadratic growth near infinity and is allowed to change sign. Our super-quadratic conditions are weaker the usual super-quadratic conditions.

This paper studies the global asymptotic stability for positive solutions to the higher order rational difference equation xn=(∏j=1m(xn−kj+1)+∏j=1m(xn−kj−1))/(∏j=1m(xn−kj+1)−∏j=1m(xn−kj−1)), n=0,1,2,…, where m is odd and x−km,x−km+1,…,x−1∈(0,∞). Our main result generalizes several others in the recent literature and confirms a conjecture by Berenhaut et al., 2007.

For any a > 0, we study the existence of normalized solutions and ground state solutions to the following Schrödinger equation with L²-constraint: where A G R is a Lagrange multiplier, the potential b G C(R2, (0, oo)) satisfies 0 < limb(y) < infæeK2 b(x) and appears as a converse direction of the Rabinowitz-type trapping potential, and the reaction f G C(R, R) enjoys critical exponential growth of Trudinger--Moser type. Under two different kinds of assumptions on /, we prove several new existence results, which, in the context of normalized solutions, can be considered as both counterparts of planar unconstrained critical problems and extensions of unconstrained Schrôdinger problems with Rabinowitz-type trapping potential. Especially, in this scenario, we develop some sharp estimates of energy levels and ingenious analysis techniques to restore the compactness which are novel even for 6(x) = constant. We believe that these techniques will allow not only treating other L²-constrained problems in the Trudinger--Moser critical setting but also generalizing previous results to the case of variable potentials. [ABSTRACT FROM AUTHOR]