On elliptic problems with a nonlinearity depending on the gradient

We investigate the solvability of the Neumann problem $(1.1)$ involving the nonlinearity depending on the gradient. We prove the existence of a solution when the right hand side $f$ of the equation belongs to $L^m(\Omega )$ with $1 \leq m \lt 2$

On nodal radial solutions of an elliptic problem involving critical Sobolev exponent

summary:In this paper we construct radial solutions of equation (1) (and (13)) having prescribed number of nodes

Multiple solutions for a nonlinear Neumann problem involving

We study the nonlinear Neumann problem (1) involving a critical Sobolev exponent and a nonlinearity of lower order. Our main results assert that for every k∈ℕ problem (1) admits at least k pairs of nontrivial solutions provided a parameter μ; belongs to some interval (0, μ*)

Representing and Coping with Early Twentieth-Century Chongqing: “Guide Songs” as Maps, Memory Cells, and Means of Creating Cultural Imagery

Chongqing’s “guide songs” form an interesting subgenre among the broad category of haozi 號子 (workers’ songs). These early twentieth-century songs were a form of rhythm-based oral narrative describing Chongqing’s urban spaces, river docks, and harbors. Each toponym mentioned in the lyrics was followed by a depiction of the characteristic associations, whether visible or symbolic, of the place. This article aims to analyze the verbal images of Chongqing presented in these songs in order to understand how the city was remembered, reproduced, and represented. The article deconstructs representations of the city produced by the lower classes, mainly by Sichuan boatmen, and links culturally meaningful images of urban spaces with the historical experiences of work, religion, and historical-mythical memory. It also points to the functions that oral narratives had in the urban environment of early twentieth-century Chongqing. Rhythmic and easy to remember, the songs provided ready-to-use guides and repositories of knowledge useful to anyone living or working there. A cross between utilitarian resource books and cultural representations, they shaped modes of thinking and visualizations of urban spaces and Chongqing. Finally, this article responds to the need to employ popular culture in our thinking about Chinese cities and the multiplicity of meanings they were given in pre-Communist times

Sur un systeme non lineaire d'inegalites aux derivees partielles du type parabolique

"Rozważa się układ nierówności różniczkowych typu parabolicznego (w sensie J. Szarskiego) postaci (...)" (fragm. streszczenia

Asymptotic estimates for solutions of the second boundary value problem for parabolic equations

Let .Q be an unbounded domain in Rn . We denote the boundary of Q by dQ. We consider the second boundary value problem n n (1 ) | f - = ) U i j ^ ' ^ V x. + X L + c ^ t » x , u in (0, oo ) x Q , ( 2 ) diiigi = 0 f o r (t,x) 6 (0, oo)x ai? , (3) u(0,x) = f i x ) for x e Q , where ¿^(fr x j denotes the inward conormal derivative to ( 0 , < » ) x 3 i P a t the point ( t , x ) (Fragment tekstu)

Existence results for quasilinear Dirichlet problem

This paper deals with the Dirichlet problem (1) -Σ D(a(x, u) D1u) + c(x)u = b(x, u, Du) in Q. (2) n(x)=ф(x) on ∂Q. in a bounded domain Q⊂R,n with the boundary ∂Q of class C2 and a function ф which, in general, ..

Magnetic Monopoles, Electric Neutrality and the Static Maxwell-Dirac Equations

We study the full Maxwell-Dirac equations: Dirac field with minimally coupled electromagnetic field and Maxwell field with Dirac current as source. Our particular interest is the static case in which the Dirac current is purely time-like -- the "electron" is at rest in some Lorentz frame. In this case we prove two theorems under rather general assumptions. Firstly, that if the system is also stationary (time independent in some gauge) then the system as a whole must have vanishing total charge, i.e. it must be electrically neutral. In fact, the theorem only requires that the system be {\em asymptotically} stationary and static. Secondly, we show, in the axially symmetric case, that if there are external Coulomb fields then these must necessarily be magnetically charged -- all Coulomb external sources are electrically charged magnetic monopoles