American Institute of Mathematical Sciences

In this paper, we consider the following Schrödinger-Poisson system

\begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} -\triangle u+u+K(x)\phi(x)u = a(x)|u|^{p-2}u, \ \ \ \ x\in { \mathbb{R}}^{3},\\ -\triangle \phi = K(x)u^2, \ \ \ \ x\in { \mathbb{R}}^{3}, \end{array} \right. \end{equation*} $\end{document}

where \begin{document}$ p\in [4,6) $\end{document}, \begin{document}$ a(x)\ge \lim_{|x|\to\infty}a(x) = a_{\infty}>0 $\end{document} and \begin{document}$ \lim_{|x|\to\infty}K(x) = 0 $\end{document}. Lack of any symmetry property of \begin{document}$ a $\end{document} and \begin{document}$ K $\end{document}, we present some new sufficient conditions to guarantee the existence of a positive ground state solution of above system. Our results extend and complement the ones of [G. Cerami, G. Vaira, J. Differential Equations 248 (2010)] in which \begin{document}$ p\in (4,6) $\end{document}, \begin{document}$ a $\end{document} and \begin{document}$ K $\end{document} need to satisfy additional integrability conditions.

We consider the existence and multiplicity of nontrivial solutions for a semilinear biharmonic equation with the concave-convex nonlinearities \begin{document}$ f(x) |u|^{q-1} u $\end{document} and \begin{document}$ h(x) |u|^{p-1} u $\end{document} under certain conditions on \begin{document}$ f(x), \, h(x) $\end{document}, \begin{document}$ p $\end{document} and \begin{document}$ q $\end{document}. Applying the Nehari manifold method along with the fibering maps and the minimization method, we study the effect of \begin{document}$ f(x) $\end{document} and \begin{document}$ h(x) $\end{document} on the existence and multiplicity of nontrivial solutions for the semilinear biharmonic equation. When \begin{document}$ h(x)^+ \neq 0 $\end{document}, we prove that the equation has at least one nontrivial solution if \begin{document}$ f(x)^+ = 0 $\end{document} and that the equation has at least two nontrivial solutions if \begin{document}$ \int_\Omega |f^+|^r\, \text{d}x \in (0, \varLambda^r) $\end{document}, where \begin{document}$ r $\end{document} and \begin{document}$ \varLambda $\end{document} are explicit numbers. These results are novel, which improve and extend the existing results in the literature.

In this paper, we deal with a system of fractional Hartree equations. By means of a direct method of moving planes, the radial symmetry and monotonicity of positive solutions are presented.

In this paper, we consider a two-species competitive and diffusive system with nonlocal delays. We investigate the existence of traveling wave fronts of the system by employing linear chain techniques and geometric singular perturbation theory. The existence of the traveling wave fronts analogous to a bistable wavefront for a single species is proved by transforming the system with nonlocal delays to a six-dimensional system without delay.

This paper is concerned with the traveling wave solutions for a class of predator-prey model with nonlocal dispersal. By adopting the truncation method, we use Schauder's fixed-point theorem to obtain the existence of traveling waves connecting the semi-trivial equilibrium to non-trivial leftover concentrations for \begin{document}$ c>c_{*} $\end{document}, in which \begin{document}$ c_* $\end{document} is the minimal wave speed. Meanwhile, through the limiting approach and the delicate analysis, we establish the existence of traveling wave solutions with the critical speed. Finally, we show the nonexistence of traveling waves for \begin{document}$ 0<c<c_{*} $\end{document} by the characteristic equation corresponding to the linearization of original system at the semi-trivial equilibrium. Throughout the whole paper, we need to overcome the difficulties brought by the nonlocal dispersal and the non-preserving of system itself.

We are concerned with the polynomial stability and the integrability of the energy for second order integro-differential equations in Hilbert spaces with positive definite kernels, where the memory can be oscillating or sign-varying or not locally absolutely continuous (without any control conditions on the derivative of the kernel). For this stability problem, tools from the theory of existing positive definite kernels can not be applied. In order to solve the problem, we introduce and study a new mathematical concept – generalized positive definite kernel (GPDK). With the help of GPDK and its properties, we obtain an efficient criterion of the polynomial stability for evolution equations with such a general but more complicated and useful memory. Moreover, in contrast to existing positive definite kernels, GPDK allows us to directly express the decay rate of the related kernel.

In this paper, we consider a quartic polynomial differential system with multiple parameters, and obtain the existence and number of limit cycles with the help of the Melnikov function under perturbation of polynomials of degree \begin{document}$ n = 4 $\end{document}.

In this paper, by using critical point theory, we obtain some sufficient conditions on the existence of infinitely many positive solutions of the discrete Robin problem with \begin{document}$ \phi $\end{document}-Laplacian. We show that, an unbounded sequence of positive solutions and a sequence of positive solutions which converges to zero will emerge from the suitable oscillating behavior of the nonlinear term at infinity and at the zero, respectively. We also give two examples to illustrate our main results.

The present paper considers a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. The existence of the nontrivial positive equilibria is discussed, and some sufficient conditions for locally asymptotically stability of one of the positive equilibria are developed. Meanwhile, the existence of Hopf bifurcation is discussed by choosing time delays as the bifurcation parameters. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcated periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out to support the analytical results.

In this paper, we study the oscillatory behavior of solutions of a class of damped fractional partial differential equations subject to Robin and Dirichlet boundary value conditions. By using integral averaging technique and Riccati type transformations, we obtain some new sufficient conditions for oscillation of all solutions of this kind of fractional differential equations with damping term. Our results essentially enrich the ones in the existing literature. Finally, we also give two specific examples to illustrate our main results.

In this paper, we obtain some rapid convergence results for a class of set differential equations with initial conditions. By introducing the partial derivative of set valued function and the \begin{document}$ m $\end{document}-hyperconvex/hyperconcave functions (\begin{document}$ m\ge 1 $\end{document}), and using the comparison principle and quasilinearization, we derive two monotone iterative sequences of approximate solutions of such equations that involve the sum of two functions, and these approximate solutions converge uniformly to the unique solution with higher order.

In this paper, we consider the Dirichlet boundary value problem for a singularly perturbed reaction-diffusion equation with discontinuous reactive term. We use the asymptotic analysis to construct the formal asymptotic approximation of the solution with internal and boundary layers. The internal layer is located in the vicinity of a curve of the discontinuous reactive term. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution and estimate the accuracy of its asymptotic approximation.

In this paper, the chaotic oscillations of the initial-boundary value problem of linear hyperbolic partial differential equation (PDE) with variable coefficients are investigated, where both ends of boundary conditions are nonlinear implicit boundary conditions (IBCs). It separately considers that IBCs can be expressed by general nonlinear boundary conditions (NBCs) and cannot be expressed by explicit boundary conditions (EBCs). Finally, numerical examples verify the effectiveness of theoretical prediction.

This work concerns with the existence and multiplicity of positive solutions for the following quasilinear Schrödinger equation

\begin{document}$ -\Delta u+V(x)u-\Delta(u^2)u = a(\epsilon x)g(u), \; \; \; \; x\in\mathbb R^N, $\end{document}

where \begin{document}$ V(x)>0 $\end{document}, \begin{document}$ u>0 $\end{document}, \begin{document}$ a $\end{document} and \begin{document}$ g $\end{document} are continuous functions and \begin{document}$ a $\end{document} has \begin{document}$ m $\end{document} maximum points. With the change of variables we show that this equation has at least \begin{document}$ m $\end{document} nontrivial solutions by using variational methods, the Ekeland's variational principle, and some properties of the Nehari manifold. Some recent results are improved.

We prove a global \begin{document}$ W^{2, p} $\end{document}-estimate for the viscosity solution to fully nonlinear elliptic equations \begin{document}$ F(x, u, Du, D^{2}u) = f(x) $\end{document} with oblique boundary condition in a bounded \begin{document}$ C^{2, \alpha} $\end{document}-domain for every \begin{document}$ \alpha\in (0, 1) $\end{document}. Here, the nonlinearities \begin{document}$ F $\end{document} is assumed to be asymptotically \begin{document}$ \delta $\end{document}-regular to an operator \begin{document}$ G $\end{document} that is \begin{document}$ (\delta, R) $\end{document}-vanishing with respect to \begin{document}$ x $\end{document}. We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global \begin{document}$ W^{2, p} $\end{document}-estimate for the viscosity solution to fully nonlinear parabolic equations \begin{document}$ F(x, t, u, Du, D^{2}u)-u_{t} = f(x, t) $\end{document} with oblique boundary condition in a bounded \begin{document}$ C^{3} $\end{document}-domain.

We are concerned with dynamics of the weakly damped plate equation on a time-dependent domain. Under the assumption that the domain is time-like and expanding, we obtain the existence of time-dependent attractors, where the nonlinear term has a critical growth.