American Institute of Mathematical Sciences

Let \begin{document}$ \Omega\subset\mathbb{R}^2 $\end{document} be a bounded smooth domain, we study the following anisotropic elliptic problem

\begin{document}$ \begin{cases} -\nabla(a(x)\nabla \upsilon)+a(x)\upsilon = 0& \text{in}\, \, \, \, \, \Omega, \\ \dfrac{\partial \upsilon}{\partial\nu} = e^\upsilon-s\phi_1-h(x) & \text{on}\, \ \, \partial\Omega, \end{cases} $\end{document}

where \begin{document}$ \nu $\end{document} denotes the outer unit normal vector to \begin{document}$ \partial\Omega $\end{document}, \begin{document}$ h\in C^{0, \alpha}( \partial\Omega) $\end{document}, \begin{document}$ s>0 $\end{document} is a large parameter, \begin{document}$ a(x) $\end{document} is a positive smooth function and \begin{document}$ \phi_1 $\end{document} is a positive first Steklov eigenfunction. We show that this problem has an unbounded number of solutions for all sufficiently large \begin{document}$ s $\end{document}, which give a positive answer to a generalization of the Lazer-McKenna conjecture for this case. Moreover, the solutions found exhibit multiple concentration behavior around boundary maxima of \begin{document}$ a(x)\phi_1 $\end{document} as \begin{document}$ s\rightarrow+\infty $\end{document}.

We have in a previous study introduced a novel elliptic operator \begin{document}$ \Delta_{(p, q)} u = |\nabla u|^q\Delta_1 u +(p-1)|\nabla u|^{p-2} \Delta_{\infty} u $\end{document}, \begin{document}$ p \ge 1 $\end{document}, \begin{document}$ q\ge 0, $\end{document} as a generalization of the \begin{document}$ p $\end{document}-Laplace operator. In this paper, we establish the well-posedness of the parabolic equation \begin{document}$ u_{t} = |\nabla u|^{1-q}\Delta_{(1+q, q)}, $\end{document} where \begin{document}$ q = q(|\nabla u|) $\end{document} is continuous and has range in \begin{document}$ [0, 1], $\end{document} in the framework of viscosity solutions. We prove the consistency and convergence of the numerical scheme of finite differences of this parabolic equation. Numerical simulations shows the advantage of this operator applied to image enhancement.

In this paper, we study the existence of radial and nonradial solutions to the scalar field equations with fractional operators. For radial solutions, we prove the existence of infinitely many solutions under \begin{document}$ N \geq 2 $\end{document}. We also show the existence of least energy solution (with the Pohozaev identity) and its mountain pass characterization. For nonradial solutions, we prove the existence of at least one nonradial solution under \begin{document}$ N \geq 4 $\end{document} and infinitely many nonradial solutions under either \begin{document}$ N = 4 $\end{document} or \begin{document}$ N \geq 6 $\end{document}. We treat both of the zero mass and the positive mass cases.

In this paper, we consider a viscoelastic rotating Euler-Bernoulli beam that has one end fixed to a rotated motor in a horizontal plane and to a tip mass at the other end. For a large class relaxation function \begin{document}$ q $\end{document}, namely, \begin{document}$ q^{\prime}(t) \leq -\zeta(t)H(q(t)) $\end{document}, where \begin{document}$ H $\end{document} is an increasing and convex function near the origin and \begin{document}$ \zeta $\end{document} is a nonincreasing function, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial decay.

We provide a new proof of the Hardy–Moser–Trudinger inequality and the existence of its extremals which are established by Wang and Ye ("G. Wang, and D. Ye, A Hardy–Moser–Trudinger inequality, Adv. Math, 230 (2012) 294–230.") without using the blow-up analysis method. Our proof is based on the transformation of functions via the transplantation of Green functions. This method enables us to compute explicitly the concentrating level of the Hardy–Moser–Trudinger functional over the normalizing concentrating sequences which is crucial to prove the existence of extremals for the Hardy–Moser–Trudinger inequality. Some comments on the applications of this approach to some other Moser–Trudinger type inequalities are given.

In this paper, we give a new and more general sufficient condition for exponential stability of thermoelastic Bresse systems with heat flux given by Cattaneo's law acting in shear and longitudinal motion equations. This condition, which we also prove to be necessary in some special cases, is given by a relation between the constants of the system and generalizes the well-known equal wave speed condition.

In this paper, we consider the following equation with the higher-order fractional Laplacian \begin{document}$ (-\Delta)^s $\end{document} for \begin{document}$ s = m+\frac{\alpha}{2} $\end{document}:

\begin{document}$ \begin{equation*} (-\Delta)^{s} u(x) = f(u(x)), \qquad x\in\mathbb{R}^n_+, \end{equation*} $\end{document}

where \begin{document}$ m\in \mathbb{N}^* $\end{document}, \begin{document}$ 0<\alpha<2 $\end{document}. By developing a narrow region principle in unbounded domain and establishing a equivalence of differential equation and integral equation, together with the method of moving planes, we deduce the monotonicity property of positive solutions and the Liouville theorem of nonnegative solutions.

In this article we present a new technique to obtain a lower bound for the principal Dirichlet eigenvalue of a fully nonlinear elliptic operator. We illustrate the construction of an appropriate radial function required to obtain the bound in several examples. In particular we use our results to prove that

\begin{document}$ \lim_{p\to \infty}\lambda_{1,p}(\Omega) = \lambda_{1,\infty}(\Omega) = \left(\frac{\pi}{2R}\right)^2 $\end{document}

where \begin{document}$ R $\end{document} is the largest radius of a ball included in the domain \begin{document}$ \Omega\subset {\mathbb R}^n $\end{document}, and \begin{document}$ \lambda_{1,p}(\Omega) $\end{document} and \begin{document}$ \lambda_{1,\infty}(\Omega) $\end{document} are the principal eigenvalue for the homogeneous \begin{document}$ p $\end{document}-laplacian and the homogeneous infinity laplacian respectively.

We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We obtain decay estimates for large time in Lebesgue spaces.

This paper is concerned with a class of free boundary models with "nonlocal diffusions'' and different free boundaries, which are natural extensions of free boundary problems of reaction diffusion systems with different free boundaries in [M.X.Wang and Y.Zhang, J. Differ. Equ., 264 (2018), 3527-3558] and references therein. These different free boundaries, which may intersect each other as time evolves, are used to describe the spreading front of the species. We prove that such kind of nonlocal diffusion problems has a unique global solution. Moreover, we investigate the long time behavior of global solution and criteria of spreading and vanishing for the classical Lotka-Volterra competition, prey-predator and mutualist models.

In this paper, we establish two refinement of Sobolev-Hardy inequalities in terms of Morrey spaces. Then, with help of these inequalities, we show the existence of nontrivial solutions for doubly critical problems in \begin{document}$ \mathbb{R}^N $\end{document} involving \begin{document}$ p $\end{document}-Laplacian

\begin{document}$ \begin{equation*} \label{eq:1} -\Delta_pu = \frac{|u|^{p^\ast_{\alpha}-2}u}{|y|^{\alpha}}+\frac{|u|^{p^\ast_{\beta}-2}u}{|y|^{\beta}}, \end{equation*} $\end{document}

where \begin{document}$ x = (y,z)\in\mathbb{R}^K\times\mathbb{R}^{N-K},1\leq K\leq N,1<p<N,0<\alpha,\beta<\min\{K,\frac{NKp}{N^2-(N-K)p}\} $\end{document} and \begin{document}$ p^\ast_{\alpha} = \frac{p(N-\alpha)}{N-p} $\end{document} is the critical Hardy-Sobolev exponent, and critical problems involving fractional Laplacian

\begin{document}$ \begin{equation*} (-\Delta)^{s}u = \frac{|u|^{2^\ast_{s,\alpha}-2}u}{|y|^{\alpha}}+\frac{|u|^{2^\ast_{s,\beta}-2}u}{|y|^{\beta}}, \end{equation*} $\end{document}

where \begin{document}$ 0<s<\frac{N}2,0<\alpha,\beta<\min\{K,\frac{2NKs}{N^2-2s(N-K)}\} $\end{document} and \begin{document}$ 2^\ast_{s,\alpha} = \frac{2(N-\alpha)}{N-2s} $\end{document}.

In this paper, we are concerned with the following two-component system of Schrödinger equations with Hartree nonlinearity:

\begin{document}$ \begin{equation*} \begin{cases} -\varepsilon ^{2}\Delta u+V_{1}\left( x\right) u+\lambda _{1}\left(I_{\varepsilon }\ast |u|^{p+1}\right)|u|^{p-1}u\\ \qquad\quad\; = \left(\mu _{1}|u|^{2p}+\beta(x) |u|^{q-1}|v|^{q+1}\right)u, & \text{in }\mathbb{R}^{N}, \\ -\varepsilon ^{2}\Delta v+V_{2}\left( x\right) v+\lambda _{2}\left(I_{\varepsilon }\ast |v|^{p+1}\right)|v|^{p-1}v\\ \qquad\quad\; = \left(\mu _{2}|v|^{2p}+\beta(x) |v|^{q-1}|u|^{q+1}\right)v, & \text{in }\mathbb{R}^{N}\,, \\ u,v\in H^{1}(\mathbb{R}^{N}),\quad u,v>0, \end{cases} \end{equation*} $\end{document}

where \begin{document}$ 0<\varepsilon \ll 1 $\end{document} is a small parameter, \begin{document}$ 0<q\leq p $\end{document}, \begin{document}$ I_{\varepsilon}(x) = \frac{\Gamma((N-\alpha)/2)} {\Gamma(\alpha/2)\pi^{\frac{N}{2}}2^{\alpha}\varepsilon^{\alpha}}\frac{1}{|x|^{N-\alpha}}, \; x\in\mathbb{R}^{N}\setminus\{0\} $\end{document}, \begin{document}$ \alpha\in(0,N),\; N = 3,4,5 $\end{document} and \begin{document}$ \lambda _{l}\geq0,\; \mu _{l}>0,\; l = 1,2, $\end{document} are constants. Under some suitable assumptions on the potentials \begin{document}$ V_{l}(x),l = 1,2, $\end{document} and the coupled function \begin{document}$ \beta(x) $\end{document}, we prove the existence and multiplicity of positive solutions for the above system by using energy estimates, the Nehari manifold technique and the Lusternik-Schnirelmann theory. Furthermore, the existence and nonexistence of the least energy positive solutions are also explored.

The aim of this paper is to study symmetry and monotonicity for positive solutions to fractional equations. We first consider the following problems in bounded domains in the sense of distributions

\begin{document}$ \begin{equation*} \begin{cases} (-\Delta)^su = \frac{g(u)}{|x|^{2s}}+f(x,u)\ \ \ &\mbox{in}\ \Omega,\\ u>0\ \ \ \ \ \ \ \ \ \ \ &\mbox{in}\ \Omega,\\ u = 0\ \ \ \ \ \ \ \ \ \ \ &\mbox{in}\ \mathbb R^n\setminus\Omega, \end{cases} \end{equation*} $\end{document}

where \begin{document}$ n>2s $\end{document}, \begin{document}$ 0<s<1 $\end{document}. We prove that all positive solutions are radically symmetric about the origin. Compare to results in [1], we use a completely different method under the weaker conditions in \begin{document}$ f $\end{document}. Next we consider a problem in infinite cylinders. We establish the symmetry and monotonicity of positive solutions by using the method of moving planes. This result can be seen as the nonlocal counterparts of [3].

In this paper, we consider the following fractional critical Schrödinger-Poisson system without perturbation terms

\begin{document}$ \begin{equation*} \begin{cases} (-\Delta)^su+V(x)u+K(x)\phi u = |u|^{2_s^{\ast}-2}u & \hbox{in $\mathbb{R}^3$,} \\ (-\Delta)^t\phi = K(x)u^2& \hbox{in $\mathbb{R}^3$,} \end{cases} \end{equation*} $\end{document}

where \begin{document}$ s\in(\frac{3}{4},1) $\end{document}, \begin{document}$ t\in(0,1) $\end{document}. Under some suitable assumptions on potentials \begin{document}$ V(x) $\end{document} and \begin{document}$ K(x) $\end{document}, we show that the ground state positive solutions do not exist, by using the topological argument, we establish the existence of positive bound state solutions in the range \begin{document}$ (\frac{s}{3}\mathcal{S}_s^{\frac{3}{2s}},\frac{2s}{3}\mathcal{S}_s^{\frac{3}{2s}}) $\end{document}.

This paper is concerned with blow-up solution for the Cauchy problem of two-component Camassa-Holm equation with generalized weak dissipation. By Kato's theorem and monotonicity, we investigate the local well-posedness of Cauchy problem and establish the blow-up criteria and the blow-up rate. Moreover, the property of blow-up points set is characterized.

Communications on Pure and Applied Analysis, 19 (2020), 3785–3803

This article was accidentally posted online but only to be discovered that the same article had been published (see [1]) in the previous issue of the same journal. Thus this publication is retracted. The Editorial Office offers apologies for the confusion and inconvenience it might have caused.

In this paper, we first establish the existence of trajectory attractors for the 3D smectic-A liquid crystal flow system and 3D smectic-A liquid crystal flow-\begin{document}$ \alpha $\end{document} model, and then prove that the latter trajectory attractor converges to the former one as the parameter \begin{document}$ \alpha\rightarrow 0^{+} $\end{document}.

In this paper we study algebraic structures of the classes of the \begin{document}$ L_2 $\end{document} analytic Fourier–Feynman transforms on Wiener space. To do this we first develop several rotation properties of the generalized Wiener integral associated with Gaussian paths. We then proceed to analyze the \begin{document}$ L_2 $\end{document} analytic Fourier–Feynman transforms associated with Gaussian paths. Our results show that these \begin{document}$ L_2 $\end{document} analytic Fourier–Feynman transforms are actually linear operator isomorphisms from a Hilbert space into itself. We finally investigate the algebraic structures of these classes of the transforms on Wiener space, and show that they indeed are group isomorphic.

In this paper, we investigate the chemotaxis-fluid system with singular sensitivity and logistic source in bounded convex domain with smooth boundary. We present the global existence of very weak solutions under appropriate regularity assumptions on the initial data. Then, we show that system possesses a global bounded classical solution. Finally, we present a unique globally bounded classical solution for a fluid-free system. In addition, the asymptotic behavior of the solutions is studied, and our results generalize and improve some well-known results in the literature, and partially results are new.

We consider the semilinear wave equation with time-dependent damping

\begin{document}$ \partial_{tt}u-\Delta u +\mu (1+t)^{-\beta} \partial_t u = |u|^p, \quad (t, x)\in (0, \infty)\times D^c, $\end{document}

where \begin{document}$ D^c = \mathbb{R}^N\backslash D $\end{document}, \begin{document}$ D $\end{document} is the closed unit ball in \begin{document}$ \mathbb{R}^N $\end{document}, \begin{document}$ N\geq 2 $\end{document}, \begin{document}$ \mu>0 $\end{document}, \begin{document}$ p>1 $\end{document} and \begin{document}$ -1<\beta<1 $\end{document}. The considered equation is investigated under the boundary conditions:

\begin{document}$ u(t, x) \left(\mbox{or } \frac{\partial u}{\partial n^+}(t, x)\right) = b(t)f(x)\, \, \mbox{on}\, \, (0, \infty)\times \partial D, $\end{document}

where \begin{document}$ n^+ $\end{document} is the outward (relative to \begin{document}$ D^c $\end{document}) unit normal on \begin{document}$ \partial D $\end{document}. General blow-up results are established for the considered problems. Moreover, for a certain class of functions \begin{document}$ b $\end{document}, the critical exponent in the sense of Fujita is obtained.

We give a discrete characterization of the trace of a class of Sobolev spaces on the Sierpinski gasket to the bottom line. This includes the \begin{document}$ L^2 $\end{document} domain of the Laplacian as a special case. In addition, for Sobolev spaces of low orders, including the domain of the Dirichlet form, the trace spaces are Besov spaces on the line.