Symmetry and Nonexistence of Positive Solutions for a Fractional Laplacion System with Coupled Terms

In this paper, we study the problem for a nonlinear elliptic system involving fractional Laplacion: (equation 1.1) where 0 < α, β < 2, p, q > 0 and max{p, q} ≥ 1, α + γ > 0, β + τ > 0, n ≥ 2. First of all, while in the subcritical case, i.e. n + α + γ − p(n − α) − (q + 1)(n − β) > 0, n + β + τ − (p + 1)(n − α) − q(n − β) > 0, we prove the nonexistence of positive solution for the above system in R n . Moreover, though Doubling Lemma to obtain the singularity estimates of the positive solution on bounded domain Ω. In addition, while in the critical case, i.e. n+α+γ −p(n−α)−(q + 1)(n−β) = 0, n+β +τ −(p+ 1)(n−α)−q(n−β) = 0, we show that the positive solution of above system are radical symmetric and decreasing about some point by using the method of Moving planes in Rn Mathematics Subject Classification (2020): 35R11, 35A10, 35B06.

Supercaloric functions for the parabolic p-Laplace equation in the fast diffusion case

We study a generalized class of supersolutions, so-called p-supercaloric functions, to the parabolic p-Laplace equation. This class of functions is defined as lower semicontinuous functions that are finite in a dense set and satisfy the parabolic comparison principle. Their properties are relatively well understood for $$p\ge 2$$ p ≥ 2 , but little is known in the fast diffusion case $$1<p<2$$ 1 < p < 2 . Every bounded p-supercaloric function belongs to the natural Sobolev space and is a weak supersolution to the parabolic p-Laplace equation for the entire range $$1<p<\infty $$ 1 < p < ∞ . Our main result shows that unbounded p-supercaloric functions are divided into two mutually exclusive classes with sharp local integrability estimates for the function and its weak gradient in the supercritical case $$\frac{2n}{n+1}<p<2$$ 2 n n + 1 < p < 2 . The Barenblatt solution and the infinite point source solution show that both alternatives occur. Barenblatt solutions do not exist in the subcritical case $$1<p\le \frac{2n}{n+1}$$ 1 < p ≤ 2 n n + 1 and the theory is not yet well understood.

Minimal and Locally Edge Minimal Fluid Models for Resource-Sharing Networks

We investigate minimal and locally edge minimal fluid models for real-time resource-sharing networks, which are natural counterparts of pathwise minimal and locally edge minimal performance processes for the corresponding real-time stochastic systems. The models under study arise as optimizers of appropriate idleness-based criteria within a suitable family of fluid models for a given resource-sharing network. The class of minimal fluid models is fairly general, corresponding to efficient service protocols in which transmission on each route takes place in the earliest deadline first (EDF) order. For such a model, the distribution of the current lead times of the fluid mass on each route coincides with the fluid arrival measure for this route, truncated below on the current frontier level. Locally edge minimal fluid models may be regarded, in some sense, as fluid counterparts of EDF resource-sharing networks. Under mild assumptions, a locally edge minimal fluid model is uniquely determined by its data. We also show stability of such models in the strictly subcritical case. More generally, each such a subcritical model converges to the invariant manifold in finite time.

Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity

<p style='text-indent:20px;'>In this paper, we study the existence of positive solutions for the following quasilinear Schrödinger equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} -\triangle u+V(x)u+\frac{\kappa}{2}[\triangle|u|^{2}]u = \lambda K(x)h(u)+\mu|u|^{2^*-2}u, \quad x\in\mathbb{R}^{N}, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \kappa&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \lambda&gt;0, \mu&gt;0, h\in C(\mathbb{R}, \mathbb{R}) $\end{document}</tex-math></inline-formula> is superlinear at infinity, the potentials <inline-formula><tex-math id="M3">\begin{document}$ V(x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ K(x) $\end{document}</tex-math></inline-formula> are vanishing at infinity. In order to discuss the existence of solutions we apply minimax techniques together with careful <inline-formula><tex-math id="M5">\begin{document}$ L^{\infty} $\end{document}</tex-math></inline-formula>-estimates. For the subcritical case (<inline-formula><tex-math id="M6">\begin{document}$ \mu = 0 $\end{document}</tex-math></inline-formula>) we can deal with large <inline-formula><tex-math id="M7">\begin{document}$ \kappa&gt;0 $\end{document}</tex-math></inline-formula>. For the critical case we treat that <inline-formula><tex-math id="M8">\begin{document}$ \kappa&gt;0 $\end{document}</tex-math></inline-formula> is small.</p>

Blowing-up solutions for a supercritical elliptic equation

<p style='text-indent:20px;'>This paper concerns the existence of solutions of the following supercritical PDE: <inline-formula><tex-math id="M1">\begin{document}$ (P_\varepsilon) $\end{document}</tex-math></inline-formula>: <inline-formula><tex-math id="M2">\begin{document}$ -\Delta u = K|u|^{\frac{4}{n-2}+\varepsilon}u\; \mbox{ in }\Omega, \; u = 0 \mbox{ on }\partial\Omega, $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ n\geq 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ K $\end{document}</tex-math></inline-formula> is a <inline-formula><tex-math id="M7">\begin{document}$ C^3 $\end{document}</tex-math></inline-formula> positive function and <inline-formula><tex-math id="M8">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> is a small positive real. Our method is inspired from the work of Bahri-Li-Rey. It consists to reduce the existence of a critical point to a finite dimensional system. Using a fixed-point theorem, we are able to construct positive solutions of <inline-formula><tex-math id="M9">\begin{document}$ (P_\varepsilon) $\end{document}</tex-math></inline-formula> having the form of two bubbles with non comparable speeds and which have only one blow-up point in <inline-formula><tex-math id="M10">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. That means that this blow-up point is non simple. This represents a new phenomenon compared with the subcritical case.</p>

ON ASYMPTOTIC RELATIONS FOR THE CRITICAL GALTON – WATSON PROCESS

Determining the asymptotics of the continuation probability for a Galton–Watson branching process is one of the most important problems in the theory of branching processes. This problem was solved by A.N. Kolmogorov (1938) in the case when the process starts with a single particle, and the classical result is obtained. A similar result for continuous branching processes was proved by B.A. Sevastyanov (1951). The next term in the expansion for continuous branching processes was obtained by V.M. Zolotarev (1957). The next term in the expansion for continuous branching processes in the critical case was obtained by V.P. Chistyakov (1957); the asymptotic expansion in the subcritical case under the condition of finiteness of the k-factorial moment was obtained by R. Mukhamedkhanova (1966). Asymptotic expansions for discrete branching processes in the subcritical and supercritical cases, provided that any m-factorial moment is finite, were obtained by S.V. Nagaev and R. Mukhamedkhanova (1966). In the critical case, the weak convergence of the conditional distribution of the quantity P(Z(n) > 0)Z(n) under the condition Z(n) > 0 to the exponential distribution was proved by A.M. Yaglom (1947) for processes starting with a single particle in the case of finiteness of the third moment of the number of generations. Subsequently, Spitzer, Kesten, and Ney (1966) proved this result under the condition that the second moment is finite. A similar result for branching processes with continuous parameters was established by V.M. Zolotarev (1957). In this paper, we study the asymptotics of the probability of continuation of the critical Galton-Watson process, starting with η particles. In addition, we prove an analogue of Yaglom’s theorem for critical Galton – Watson processes starting with a random number of particles.

Skeletal stochastic differential equations for superprocesses

It is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP).In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever.Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.

Wave equations associated with Liouville-type problems: global existence in time and blow-up criteria

We are concerned with wave equations associated with some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated with the latter problems and second, to substantially refine the analysis initiated in Chanillo and Yung (Adv Math 235:187–207, 2013) concerning the mean field equation. In particular, by exploiting the variational analysis recently derived for Liouville-type problems we prove global existence in time for the subcritical case and we give general blow-up criteria for the supercritical and critical case. The strategy is mainly based on fixed point arguments and improved versions of the Moser–Trudinger inequality.

Constant Q-curvature metrics on conic 4-manifolds

We consider the constant Q-curvature metric problem in a given conformal class on a conic 4-manifold and study related differential equations. We define subcritical, critical, and supercritical conic 4-manifolds. Following [M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 1991, 2, 793–821] and [S.-Y. A. Chang and P. C. Yang, Extremal metrics of zeta function determinants on 4-manifolds, Ann. of Math. (2) 142 1995, 1, 171–212], we prove the existence of constant Q-curvature metrics in the subcritical case. For conic 4-spheres with two singular points, we prove the uniqueness in critical cases and nonexistence in supercritical cases. We also give the asymptotic expansion of the corresponding PDE near isolated singularities.