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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.

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.

In this paper, we consider the following fractional Schrodinger-Poisson problem, $$\begin{cases}\epsilon^{2s}(-\Delta)^{s}u+V(x)u+\phi{u}=\mid{u}\mid^{p-1}u, & x\in\mathbb{R}^N,\\(-\Delta)^t\phi=u^2, & x\in\mathbb{R}^N,\end{cases}$$ where e > 0 is a small parameter, N ⩾ 3 and V(x) is a potential function. We construct non-radial sign-changing solutions, whose components may have spikes clustering at the local minimum point of V(x).

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.

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} .

In this paper, we study the existence and concentration of normalized solutions to the nonlinear Schrodinger equation −Δu+k2|x|2u+Ωr2|x|2u−2g|u|2u=2ωu in R2 with ∫R2|u|2dx=1, where ω is the Lagrange multiplier, Ωr is the radial trapping frequency, and g > 0. We show that there is a gk*>0 such that the problem has a ground state solution ug if 0

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

Let (M ,g) be a smooth compact Riemannian manifold of dimension n 3 . We study the existence of positive weak solutions for the following quasilinear elliptic equation −(Δp)gu+up−1 = f (x,u,∇gu) in M , where (Δp)gu = divg(|∇u|p−2 g ∇u) is the p -Laplacian operator on Riemannian manifold (M ,g) with 1 < p < n .

The authors prove the existence of nontrivial solutions for the Schrodinger equation −Δu + V(x)u = λf(x, u) in ℝN, where f is superlinear, subcritical and critical at infinity, respectively, V is periodic.

We, prove the existence of nontrivial solutions for the Schrodinger equation -Delta u + V(x)u = a gamma (x) f (u) in R-N. where f is superlinear and suberitical at zero and infinity respectively, V is periodic and a(x) chances sign. (c) 2007 Elsevier Inc. All rights reserved.

In this paper, we study existence of solutions with boundary layer and peaks for an elliptic problem. We prove that the problem has a mountain-pass-type solution, which has a boundary layer and a single peak near the boundary. Moreover, we also study the existence of solutions with interior peak.

In this paper we establish the existence of a solution of the Schrodinger equation (1). The existence of a solution is obtained by a variational method based on the critical point theorem 2.4 from [3].

This paper analyses the existence and globally exponential stability of a class of cellular neural networks with distributed delays, with assuming global Lipschitz conditions on the activation functions, applying the idea of vector Lyapunov function, Young inequality and Halanay differential inequality with delay, some sufficient conditions are obtained to ensure the equilibrium point.

In this paper, the global exponential stability is studied for a class of cellular neural networks with distributed delays. With assuming global Lipschitz conditions on the activation functions, based on the vector Lyapunov function, using the technique by virtue of Young inequality and Halanay differential inequality with delay, some sufficient conditions are obtained to ensure the uniqueness equilibrium point and globally exponential stability.

By Lyapunov function and Halanay inequality etc., a class of the BAM type Cohen-Grossberg neural network with delays and impulsive is considered on time scales. It obtained some sufficient conditions about the existence of equilibrium point, globally exponential stability and globally exponentially robust stability. We can see the impulses do contribution to the exponential stability and robust stability. At last, some applications and examples can be demonstrate the results.

A class of impulsive neural networks with distributed delays is studied in this paper. With assuming global Lipschitz conditions on the activation functions, applying idea of vector Lyapunov function, combining Young inequality and Halanay differential inequality with delay, sufficient conditions are obtained for the existence of unique and global exponential stable equilibrium of the distributed delay neural networks.

In this paper, a class of impulsive neural networks with time-varying delays is considered to study the globally exponential stability New sufficient conditions for globally exponential stability are obtained by using the vector Lyapunov function, Young inequality and Halanay differential inequality with delay.

In this paper, with assuming global Lipschitz conditions on the activation functions, applying idea of vector Lyapunov function, Young inequality and Halanay differential inequality with delay, the global exponential stability of the equilibrium point for a class of Cohen-Grossberg neural networks with time-varying delays and impulses is investigated, the sufficient conditions for globally exponential stability of neural networks are obtained.

In this paper, a class of BAMtype Cohen-Grossberg neural networks with delays are studied. Some sufficient conditions ensuring global exponential stability of the system are obtained on time scales. The results extend and improve some recent works for neural networks.

In this paper, the main purpose is to study the global exponential stability of the equilibrium point for a class of cellular neural networks with impulses and distributed delays. With the assumption of global Lipschitz conditions on the activation function, applying idea of vector Lyapunov function, combining Halanay differential inequality with delay, a sufficient condition for globally exponential stability of neural networks is obtained.

In this paper, with assuming global Lipschitz conditions on the activation functions, applying idea of vector Lyapunov function, Young inequality and Halanay differential inequality with delay, the global exponential stability of the equilibrium point for a class of cellular neural networks with distributed delays and large impulses is investigated, the sufficient conditions for globally exponential stability of neural networks are obtained.

The main purpose of this paper is to study the globally exponential stability of the equilibrium point for a class of cellular neural networks with distributed delays and large impulses. With assuming global Lipschitz conditions on the activation functions, applying idea of vector Lyapunov function, combining Halanay differential inequality with delay, the sufficient conditions for globally exponential stability of neural networks are obtained.

The main purpose of this paper is to study the globally exponential stability of the equilibrium point for a class of impulsive neural networks with time-varying delays. Without assuming global Lipschitz conditions on the activation functions, applying idea of vector Lyapunov function, combining Halanay differential inequality with delay, the sufficient conditions for globally exponential stability of neural networks are obtained.

The main purpose of this paper is to study the globally exponential stability of the equilibrium point for a class of impulsive neural networks with time-varying delays. Without assuming global Lipschitz conditions on the activation functions, applying idea of vector Lyapunov function, combining Young inequality and Halanay differential inequality with delay, the sufficient conditions for globally exponential stability of neural networks are obtained.

In this paper, a class of impulsive neural networks with time-varying delays is studied. Without assumption of global Lipschitz conditions on the activation functions, applying idea of vector Lyapunov function, combining Halanay differential inequality with delay, a sufficient condition for globally exponential stability of the equilibrium is obtained

In this paper , With assumption of global Lipschitz conditions on the activation functions, by Lyapunov function and Halanay differential inequality with delay, we study a class of cellular neural networks with distributed delays and large impulses. It obtained a sufficient condition for globally exponential stability of the equilibrium point.

In this paper we study the globally exponential stability of a class of neural networks with time-varying delays. Applying idea of vector Lyapunov function, Young inequality and Halanay differential inequality with delay, the sufficient conditions for the existence and globally exponential stability of the equilibrium point are obtained

By Lyapunov function and Halanay inequality, a class of the BAM type Cohen-Grossberg neural network with delays and impulsive is considered. It obtained some sufficient conditions about the existence of equilibrium point, globally exponential stability and globally exponentially robust stability. We can see the impulses do contribution to the exponential stability and robust stability.At last, an example can be demonstrate the results.

Employing general Halanay inequalities, Dinipsilas derivative and functional analysis techniques, several global exponential stability criteria of the equilibrium point are established for a class of delayed reaction-diffusion recurrent neural networks. These criteria are only dependent on the parameters of the system. The results correct some errors of the earlier publication.

Some sufficient conditions are obtained for the global exponential stability of a solution and existence of the periodic solution to the general bidirectional associative memory (BAM) neural networks with distributed delay and impulsive by using the Lyapunov functional and some inequality technique. These results are helpful for designing a globally exponential stable and periodic oscillatory BAM neural network, and the conditions can be easily applied in practice. An example is also given to illustrate our results.

By using the method of Lyapunov function and analysis technique, a model for bidirectional associative memory with impulsive is studied. Sufficient conditions are obtained for the existence and uniformly stability of a unique equilibrium solution . We can see that impulses do contribute to systempsilas uniformly stability. This stability property is independent of the stability of continuous-time BAM model.

This paper is concerned with the stability of the impulsive Cohen-Grossberg neural networks with variable delays. By means of matrices theory, Lyapunov function and analysis technique, sufficient conditions are obtained for the uniform stability of a unique equilibrium solution of this system. We can see that impulses do contribution to system's uniform stability, so the impulsive system is stable whether the corresponding continuous-time system is stable or not. As the applications of the main theorems, some uniform stability criteria are established for several well known impulsive ecological models with variable delays.

Derating of the working current level does not work for improving GaN-based light-emitting diode (LED) devices' reliability. The present work demonstrates that it is not the levels but the specific components of the applied electrical currents weighing more on LEDs' degradation. Existing defects are sources for tunneling currents and Shockley-Read-Hall (SRH) non-radiative recombination current, and the component of tunneling currents and SRH non-radiative recombination current in the applied electrical current will in turn induce fast increase of defect density. The current component from electron tunneling to deep levels in the vicinity of mixed/screw dislocations will affect more on LEDs' degradation than other components, such as heavy-hole tunneling via intermediate state. In a whole, the overflow leakage current from the active region and Auger recombination currents in the applied electrical current will generate positive effects to alleviate LEDs' degradation.

The main purpose of this paper is to establish the existence of a solution of the semilinear Schr¨odinger equation (1) −u + V (x)u = K(x)|u|2−2u + f(x, u) in RN, involving a critical Sobolev exponent 2 = 2N/(N − 2) with N 4 and a subcritical nonlinearity f : RN × R ! R. Throughout this paper it is assumed that (A) The coefficients V and K are continuous and 1-periodic functions in each variable xi, i = 1, . . . ,N. Moreover, we assume that K 0 on RN. In this case it is known that the operator − + V on L2(RN) has a purely continuous spectrum consisting of closed disjoint intervals. In this paper we consider the case: (B) 0 is in the spectral gap of the operator − + V . There are many existence results in the case K 0 on RN and we refer to the papers [4], [5], [9], [10], [13]. 1991 Mathematics Subject Classification. 35J65.