Volume 4, Issue 3

We consider semilinear elliptic problems of the form Δu + g(u) = f(x) with Neumann boundary conditions or Δu+λ1u+g(u) = f(x) with Dirichlet boundary conditions, and we derive conditions on g and f under which an upper bound on the number of solutions can be obtained.

In this work we prove that for some class of initial data the solution of the Cauchy problem u t = (u m ) xx + a(u m ) x + u(1 - u m-1 ), x ∈ ℝ; t > 0 u(0; x) = u 0 (x), u 0 (x) ≥ 0 approaches the travelling solution, spreading either to the right or to the left, or two travelling waves moving in opposite directions.

In this paper we study a generalized porous medium equation where the diffusion rate, say m(x) -spatially heterogeneous- is assumed to be linear, m = 1, on a piece of the support domain, Ω 1 and slow nonlinear, m(x) > 1, in its complement, Ω m := Ω \ Ω̄ 1 . More precisely, we characterize the existence of positive solutions and construct the corresponding global bifurcation diagram as one of the parameters of the model changes, showing that a continuous transition occurs between the diagrams of the completely linear case (Ω = Ω 1 ) and of the completely nonlinear case (­Ω m = Ω­). As a result, the effect of a localized slow diffusion rate with varying support is completely characterized. Our analysis is imperative in order to design porous media multi-components systems with changing diffusion rates.

There are many notions of solutions of nonlinear elliptic partial differential equations. This paper is concerned with solutions which are obtained as suprema (or infima) of so-called subfunctions (superfunctions) or viscosity subsolutions (viscosity supersolutions). The paper also explores the relationship of these (generalized) solutions of differential inequalities and provides a relevant example for which existence questions have been studied using these concepts.

In this paper we obtain some non-existence results for the Klein-Gordon equation coupled with the electrostatic field. The method relies on the deduction of some suitable Pohožaev identity which provides necessary conditions to get existence of nontrivial solutions. The case of Maxwell-Schrödinger type coupled equations is also considered.

An elementary approach, based on a systematic use of lower and upper solutions, is employed to detect the qualitative properties of solutions of first order scalar periodic ordinary differential equations. This study is carried out in the Carathéodory setting, avoiding any uniqueness assumption, in the future or in the past, for the Cauchy problem. Various classical and recent results are recovered and generalized.