Narrow Results

The purpose of this paper is to prove some existence and non-existence theorems for the nonlinear elliptic problems of the form -Δ_pu = λk(x)u^q ± h(x)u^σ if x ∈ Ω, subject to the Dirichlet condition u = 0 on ∂Ω. In the proof of our results we use the sub-solution, super-solution methods and variational arguments.

We discuss existence, non-existence and multiplicity of positive solutions of the Dirichlet problem for the one-dimensional prescribed curvature equation

In this paper, by using the Alexandrov–Serrin method of moving plane combined with integral inequality, we prove some non-existence results for positive weak solution of semilinear elliptic system in the half-space .

Let Ω be a smooth bounded domain in ℝ^N with N ≥ 1. In this paper we study the Hardy–Poincaré inequality with weight function singular at the boundary of Ω. In particular we provide sufficient and necessary conditions on the existence of minimizers.

Let $({ℳ},g)$ be a smooth compact Riemannian manifold of dimension $N\ge 3$ and let $\mathrm{\Sigma }$ to be a closed submanifold of dimension $1\le k\le N-2$. In this paper, we study existence and non-existence of minimizers of Hardy inequality with weight function singular on $\mathrm{\Sigma }$ within the framework of Brezis–Marcus–Shafrir [Extremal functions for Hardy’s inequality with weight, J. Funct. Anal.171 (2000) 177–191]. In particular, we provide necessary and sufficient conditions for existence of minimizers.

We explore how some ideas that have been used before for optimal design problems with two materials through a variational approach, can be extended to cover three or more materials. In particular, we focus on two paradigmatic situations where we consider a linear-in-the-gradient cost functional, and a typical quadratic situation. In both cases, we are able to formulate, quite explicitly, a full relaxation of the problem through which optimal microstructures for the original non-convex problem can be understood.

This paper is concerned with the traveling wave solutions for a discrete SIR epidemic model with a saturated incidence rate. We show that the existence and non-existence of the traveling wave solutions are determined by the basic reproduction number $R_0$ of the corresponding ordinary differential system and the minimal wave speed $c^{\ast }$. More specifically, we first prove the existence of the traveling wave solutions for $R_0>1$ and $c>c^{\ast }$ via considering a truncated initial value problem and using the Schauder’s fixed point theorem. The existence of the traveling wave solutions with speed $c=c^{\ast }$ is then proved by using a limiting argument. The main difficulty is to show that the limit of a decreasing sequence of the traveling wave solutions with super-critical speeds is non-trivial. Finally, the non-existence of the traveling wave solutions for $R_0>1,$$0<c<c^{\ast }$ and $R_0\le 1,$$c>0$ is proved.

Since the concept of dark matter and dark energy was proposed, it is still unclear what its essence is. This paper will discuss the nature of dark matter and dark energy based on the dualism and its derivative theories or viewpoints. In the process of photon degradation to eight-subphotons in the edge region of sub-galaxy force, the traction force is superimposed on the gravity of adjacent sub-galaxies. Then the so-called dark matter is the dormant generalized photon in space. In addition to the previously known factors, there are three other types of redshift of light waves, namely, low density of space photons, photon degradation and strong light interference. Therefore, it is also concluded that the farther away from the observer, the greater the redshift, without assuming that dark energy drives the expansion of the universe — energy cannot be continuously generated out of thin air.

The existence is one of the ultimate concepts in the studies of western philosophy. Almost all well-known philosophers in history had their own understandings of this concept. As a new philosophical form in new era, the philosophy of information states that philosophy of information and information science should develop together. The existence in philosophy needs to be redefined. The segmentation of existential field should be the first step to tackle with the issue of existence and non-existence. In the view of the philosophy of information, there are four levels of the existence: A. objective direct existence, B. objective indirect existence, C. subjective for-itself existence and D. subjective regenerated existence. The existential levels of a certain thing cover four levels in part or in its totality. Consequently, the nonexistence, as the opposite concept of the existence, has also changed. The existence and non-existence can be transformed into each other, with five properties in the process, those are: continuity, developmental feature, contingency, retrospective feature and predictability.

Inspired by the construction of blow-up solutions of the heat flow of harmonic maps from D² into S² via maximum principle (Chang et al., J. Diff. Geom., 36, 1992, pp. 507-515.) we provide examples of nonexistence of smooth axially symmetric harmonic maps from B³ into S² with smooth boundary maps of degree zero.

In this paper we consider the problem: ∂_tu − Δu = f (u), u(0) = u₀ ∈ exp L^p (ℝ^N), where p > 1 and f : ℝ → ℝ having an exponential growth at infinity with f (0) = 0. We prove local well-posedness in $\exp L_0^p\left({ℝ}^N\right)$ for $f\left(u\right)\sim e^{{\left|u\right|}^q},0<q\le p,\left|u\right|\to \infty$. However, if for some λ > 0, $\underset{s\to \infty }{\lim }\inf \left(f\left(s\right)e^{-\lambda s^p}\right)>0$ then non-existence occurs in exp L^p (ℝ^N ). Under smallness condition on the initial data and for exponential nonlinearity f such that |f(u)| ∼ |u|^m as u → 0, $\frac{N\left(m-1\right)}{2}\ge p$, we show that the solution is global. In particular, p – 1 > 0 sufficiently small is allowed. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on m.