We consider the elliptic problem ∆u+up = 0, u > 0 in an exterior domain, Ω = RN \D under zero Dirichlet and vanishing conditions, where D is smooth and bounded, and p is supercritical, namely p > N+2 N−2 . We prove that the associated Dirichlet problem has infinitely many positive solutions with slow decay O(|x| 2 p−1 ) at infinity. In addition, a fast decay solution exists if p is close enough...

In this paper, oscillation and asymptotic properties of solutions of the Dirichlet boundary value problem for hyperbolic and parabolic equations are considered. We demonstrate that introducing an arbitrary constant delay essentially changes the above properties. For instance, the delay equation does not inherit the classical properties of the Dirichlet boundary value problem for the heat equati...

We study how the solution of the two-dimensional Dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem. We show that the Hadamard formula for the variation of the Dirichlet Green function under deformations of the domain reveals an integrable structure. The independent variables corresponding to the infinite set of commuting flows are ...

We construct examples of strictly convex functions f on (?1; 1) satisfying f 0 (?1) < n 2 < f 0 (1) such that the Dirichlet problem u 00 + f(u) = h(x) in 0; ], u(0) = u() = 0, has an innnite number of solutions, for any choice of h(x). Kaper and Kwong earlier have presented examples with ve solutions to settle a conjecture raised by Lazer and McKenna. Here, we also give a suucient condition for...

The convex envelope of a given function was recently characterized as the solution of a fully nonlinear partial differential equation (PDE). In this article we study a modified problem: the Dirichlet problem for the underlying PDE. The main result is an optimal regularity result. Differentiability (C1,α regularity) of the boundary data implies the corresponding result for the solution in the in...

The paper reviews some of the early ideas behind the development of the theory of discrete harmonic functions. The connection to random walks is pointed out. Then a current and more general construction using weight functions is described. Discrete analogues of the Laplace operator are defined for Zn and a discrete planar hexagonal structure H. Discrete analogues of the Dirichlet problem and Po...

Introduction. A fundamental paper in the theory of solving partial differential equations by iteration has been written by Frankel.1 In this paper Frankel discusses the Richardson and Liebmann procedures and their corresponding accelerated procedures. The latter are termed the extrapolated Liebmann and the second-order Richardson procedures. From his paper three disadvantages of the second-orde...

where Ω is a bounded domain in RN (N ≥ 1) with smooth boundary ∂Ω. The conditions imposed on f (x, t) are as follows: (f1) f ∈ C(Ω×R,R); f (x,0) = 0, for all x ∈Ω. (f2) lim|t|→0( f (x, t)/t) = μ, lim|t|→∞( f (x, t)/t) = uniformly in x ∈Ω. Since we assume (f2), problem (1.1) is called asymptotically linear at both zero and infinity. This kind of problems have captured great interest since the pi...

We study the Lp Dirichlet problem for the Stokes system on Lipschitz domains. For any fixed p > 2, we show that a reverse Hölder condition with exponent p is sufficient for the solvability of the Dirichlet problem with boundary data in LpN (∂Ω, R d). Then we obtain a much simpler condition which implies the reverse Hölder condition. Finally, we establish the solvability of the Lp Dirichlet prob...

We consider the biharmonic Dirichlet problem on a polygonal domain. Regularity estimates in terms of Sobolev norms of fractional order are proved. The analysis is based on new interpolation results which generalizes Kellogg’s method for solving subspace interpolation problems. The Fourier transform and the construction of extension operators to Sobolev spaces on R are used in the proof of the i...