American Institute of Mathematical Sciences

In this paper we prove the existence of at least one positive solution for nonlocal semipositone problem of the type

\begin{document}$ (P_\lambda^\mu)\left\{ \begin{array}{rcl} (-\Delta)^s u& = & \lambda(u^{q}-1)+\mu u^r \text{ in } \Omega\\ u&>&0 \text{ in } \Omega\\ u&\equiv &0 \text{ on }{\mathbb R^N\setminus\Omega}. \end{array}\right. $\end{document}

when the positive parameters \begin{document}$ \lambda $\end{document} and \begin{document}$ \mu $\end{document} belong to certain range. Here \begin{document}$ \Omega\subset \mathbb{R}^N $\end{document} is assumed to be a bounded open set with smooth boundary, \begin{document}$ s\in (0, 1), N> 2s $\end{document} and \begin{document}$ 0<q<1<r\leq \frac{N+2s}{N- 2s}. $\end{document} First we consider \begin{document}$ (P_ \lambda^\mu) $\end{document} when \begin{document}$ \mu = 0 $\end{document} and prove that there exists \begin{document}$ \lambda_0\in(0, \infty) $\end{document} such that for all \begin{document}$ \lambda> \lambda_0 $\end{document} the problem \begin{document}$ (P_ \lambda^0) $\end{document} admits at least one positive solution. In fact we will show the existence of a continuous branch of maximal solutions of \begin{document}$ (P_\lambda^0) $\end{document} emanating from infinity. Next for each \begin{document}$ \lambda>\lambda_0 $\end{document} and for all \begin{document}$ 0<\mu<\mu_{\lambda} $\end{document} we establish the existence of at least one positive solution of \begin{document}$ (P_\lambda^\mu) $\end{document} using variational method. Also in the sub critical case, i.e., for \begin{document}$ 1<r<\frac{N+2s}{N-2s} $\end{document}, we show the existence of second positive solution via mountain pass argument.

We study the quasilinear Dirichlet boundary problem

\begin{document}$ \begin{equation} \nonumber \begin{cases} -Qu = \lambda e^{u}, \text{in}~~ \Omega, \\ u = 0, \qquad \;~~\text{on}~~~~ \partial\Omega, \end{cases} \end{equation} $\end{document}

where \begin{document}$ \lambda>0 $\end{document} is a parameter, \begin{document}$ \Omega\subset\mathbb{R}^{N} $\end{document} (\begin{document}$ N\geq2 $\end{document}) is a bounded domain, and the operator \begin{document}$ Q $\end{document}, known as Finsler-Laplacian or anisotropic Laplacian, is defined by

\begin{document}$ Qu: = \sum\limits_{i = 1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)). $\end{document}

Here, \begin{document}$ F_{\xi_{i}} = \frac{\partial F}{\partial\xi_{i}}(\xi) $\end{document} and \begin{document}$ F: \mathbb{R}^{N}\rightarrow [0, +\infty) $\end{document} is a convex function of \begin{document}$ C^{2}(\mathbb{R}^{N}\setminus\{0\}) $\end{document}, and satisfies certain assumptions. We derive the existence of extremal solution and obtain that it is regular, if \begin{document}$ N\leq9 $\end{document}.

We also concern the Hénon type anisotropic Liouville equation,

\begin{document}$ -Qu = (F^{0}(x))^{\alpha}e^{u} ~~\text{in} ~~\mathbb{R}^{N}, $\end{document}

where \begin{document}$ \alpha>-2 $\end{document}, \begin{document}$ N\geq2 $\end{document} and \begin{document}$ F^{0} $\end{document} is the support function of \begin{document}$ K: = \{x\in\mathbb{R}^{N}:F(x)<1\} $\end{document}. We obtain the Liouville theorem for stable solutions and finite Morse index solutions for \begin{document}$ 2\leq N<10+4\alpha $\end{document} and \begin{document}$ 3\leq N<10+4\alpha^{-} $\end{document} respectively, where \begin{document}$ \alpha^{-} = \min\{\alpha, 0\} $\end{document}.

This paper is concerned with the propagation dynamics of a nonlocal dispersal predator-prey model with two predators and one prey. Precisely, our main concern is the invasion process of the two predators into the habitat of one prey, when the two predators are weak competitors in the absence of prey. This invasion process is characterized by the spreading speed of the predators as well as the minimal wave speed of traveling waves connecting the predator-free state to the co-existence state. Particularly, the right-hand tail limit of wave profile is derived by the idea of contracting rectangle.

We consider positive solutions of semi-linear elliptic equations

\begin{document}$ - \epsilon^2 u'' +u = u^p $\end{document}

on compact metric graphs, where \begin{document}$ p \in (1,\infty) $\end{document} is a given constant and \begin{document}$ \epsilon $\end{document} is a positive parameter. We focus on the multiplicity of positive solutions for sufficiently small \begin{document}$ \epsilon $\end{document}. For each edge of the graph, we construct a positive solution which concentrates some point on the edge if \begin{document}$ \epsilon $\end{document} is sufficiently small. Moreover, we give the existence result of solutions which concentrate inner vertices of the graph.

In this paper, we prove sharp gradient estimates for positive solutions to the weighted heat equation on smooth metric measure spaces with compact boundary. As an application, we prove Liouville theorems for ancient solutions satisfying the Dirichlet boundary condition and some sharp growth restriction near infinity. Our results can be regarded as a refinement of recent results due to Kunikawa and Sakurai.

In this paper, we consider the weighted fourth order equation

\begin{document}$ \Delta(|x|^{-\alpha}\Delta u)+\lambda \text{div}(|x|^{-\alpha-2}\nabla u)+\mu|x|^{-\alpha-4}u = |x|^\beta u^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\}, $\end{document}

where \begin{document}$ n\geq 5 $\end{document}, \begin{document}$ -n<\alpha<n-4 $\end{document}, \begin{document}$ p>1 $\end{document} and \begin{document}$ (p,\alpha,\beta,n) $\end{document} belongs to the critical hyperbola

\begin{document}$ \frac{n+\alpha}{2}+\frac{n+\beta}{p+1} = n-2. $\end{document}

We prove the existence of radial solutions to the equation for some \begin{document}$ \lambda $\end{document} and \begin{document}$ \mu $\end{document}. On the other hand, let \begin{document}$ v(t): = |x|^{\frac{n-4-\alpha}{2}}u(|x|) $\end{document}, \begin{document}$ t = -\ln |x| $\end{document}, then for the radial solution \begin{document}$ u $\end{document} with non-removable singularity at origin, \begin{document}$ v(t) $\end{document} is a periodic function if \begin{document}$ \alpha \in (-2,n-4) $\end{document} and \begin{document}$ \lambda $\end{document}, \begin{document}$ \mu $\end{document} satisfy some conditions; while for \begin{document}$ \alpha \in (-n,-2] $\end{document}, there exists a radial solution with non-removable singularity and the corresponding function \begin{document}$ v(t) $\end{document} is not periodic. We also get some results about the best constant and symmetry breaking, which is closely related to the Caffarelli-Kohn-Nirenberg type inequality.

In this paper, we consider the global existence of the Cauchy problem for a version of one velocity Baer-Nunziato model with dissipation for the mixture of two compressible fluids in \begin{document}$ \mathbb{R}^3 $\end{document}. We get the existence theory of global strong solutions by using the decaying properties of the solutions. The energy method combined with the low-high-frequency decomposition is used to derive such properties and hence the global existence. As a byproduct, the optimal time decay estimates of all-order spatial derivatives of the pressure and the velocity are obtained.

We consider fully coupled cooperative systems on \begin{document}$ \mathbb{R}^n $\end{document} with coefficients that decay exponentially at infinity. Expanding some results obtained previously on bounded domain, we prove that the existence of a strictly positive supersolution ensures the first eigenvalue to exist and to be nonzero. This result is applied to show that the topological solutions for a Chern-Simons model, described by a semilinear system on \begin{document}$ \mathbb{R}^2 $\end{document} with exponential nonlinearity, are nondegenerate.

We study convergence of 3D lattice sums via expanding spheres. It is well-known that, in contrast to summation via expanding cubes, the expanding spheres method may lead to formally divergent series (this will be so e.g. for the classical NaCl-Madelung constant). In the present paper we prove that these series remain convergent in Cesaro sense. For the case of second order Cesaro summation, we present an elementary proof of convergence and the proof for first order Cesaro summation is more involved and is based on the Riemann localization for multi-dimensional Fourier series.

We present a uniform(-in-time) stability of the relativistic Cucker-Smale (RCS) model in a suitable framework and study its application to a uniform mean-field limit which lifts earlier classical results for the CS model in a relativistic setting. For this, we first provide a sufficient framework for an exponential flocking for the RCS model in terms of the diameters of state observables, coupling strength and communication weight function, and then we use the obtained exponential flocking estimate to derive a uniform \begin{document}$ \ell_{q,p} $\end{document}-stability of the RCS model under appropriate conditions on initial data and system parameters. As an application of the derived uniform \begin{document}$ \ell_{q,p} $\end{document}-stability estimate, we show that a uniform mean-field limit of the RCS model can be made for some admissible class of solutions uniformly in time. This justifies a formal derivation of the kinetic RCS equation [18] in a rigorous setting.

In this paper, we consider the asymptotic behavior of the ground state and its energy for the nonlinear Schrödinger system with three wave interaction on the parameter \begin{document}$ \gamma $\end{document} as \begin{document}$ \gamma \to \infty $\end{document}. In addition we prove the existence of the positive threshold \begin{document}$ \gamma^* $\end{document} such that the ground state is a scalar solution for \begin{document}$ 0 \le \gamma < \gamma^* $\end{document} and is a vector solution for \begin{document}$ \gamma > \gamma^* $\end{document}.

In this paper, we study the existence of periodic solutions of the following differential delay equations

\begin{document}$ \begin{equation} z^{\prime\prime}(t) = \sum\limits_{k = 1}^{M-1}(-1)^kf(z(t-k)), \notag \end{equation} $\end{document}

where \begin{document}$ f\in C(\mathbf{R}^N, \mathbf{R}^N) $\end{document}, \begin{document}$ M,N\in \mathbf{N} $\end{document} and \begin{document}$ M $\end{document} is odd. By making use of \begin{document}$ S^1 $\end{document}-geometrical index theory, we obtain an estimation about the number of periodic solutions in term of the difference between eigenvalues of asymptotically linear matrices at the origin and at infinity.

In this paper we prove a partial Hölder regularity result for weak solutions \begin{document}$ u:\Omega\to \mathbb{R}^N $\end{document}, \begin{document}$ N\geq 2 $\end{document}, to non-autonomous elliptic systems with general growth of the type:

\begin{document}$ \begin{equation*} -{\rm{div}} a(x, u, Du) = b(x, u, Du) \quad \;{\rm{ in }}\; \Omega. \end{equation*} $\end{document}

The crucial point is that the operator \begin{document}$ a $\end{document} satisfies very weak regularity properties and a general growth, while the inhomogeneity \begin{document}$ b $\end{document} has a controllable growth.

The Zakharov system in dimension \begin{document}$ d = 2,3 $\end{document} is shown to have a local unique solution for any initial values in the space \begin{document}$ H^{s} \times H^{l} \times H^{l-1} $\end{document}, where a new range of regularity \begin{document}$ (s, l) $\end{document} is given, especially at the line \begin{document}$ s-l = -1 $\end{document}. The result is obtained mainly by the normal form reduction and the Strichartz estimates.

In this paper, we consider a two-species chemotaxis-Stokes system with \begin{document}$ p $\end{document}-Laplacian diffusion in two-dimensional smooth bounded domains. It is proved that the existence of time periodic solution for any \begin{document}$ \frac{15}{7}\leq p<3 $\end{document} and any large periodic source \begin{document}$ g_1(x,t) $\end{document} and \begin{document}$ g_2(x,t) $\end{document}.

We consider the Cauchy problems for Schrödinger equations with an inverse-square potential and a harmonic one. Since the Mehler type formulas are completed, the pseudo-conformal transforms can be constructed. Thus we can convert the problems into the nonautonomous Schrödinger equations without a harmonic oscillator.