Class Of Nonlinear Elliptic Equations Research Articles

Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations

We study the uniqueness and expansion properties of the positive solutions u of (E) Δu + hu − kup = 0 in a non-smooth domain Ω, subject to the condition u(x) → ∞ when dist (x, ∂Ω) → 0, where h and k are continuous functions in Ω¯, k > 0 and p > 1. When ∂Ω has the local graph property, we prove that the solution is unique. When ∂Ω has a singularity of conical or wedge-like type, we give the asymptotic behavior of u. When ∂Ω has a re-entrant cuspidal singularity, we prove that the rate of blow-up may not be of the same order as in the previous more regular cases.

On positive solutions of some nonlinear differential equations — A probabilistic approach

By using connections between superdiffusions and partial differential equations (established recently by Dynkin, 1991), we study the structure of the set of all positive (bounded or unbounded) solutions for a class of nonlinear elliptic equations. We obtain a complete classification of all bounded solutions. Under more restrictive assumptions, we prove the uniqueness property of unbounded solutions, which was observed earlier by Cheng and Ni (1992).

On a class of equations of monge-ampere type

We prove that the Dirichlet problem is solvable in a generalized sense for a class of nonlinear elliptic equations and related equations of Monge-Ampère type. Bibliography: 14 titles.

BEHAVIOR OF SOLUTIONS OF A NONLINEAR VARIATIONAL PROBLEM IN A NEIGHBORHOOD OF SINGULAR POINTS OF THE BOUNDARY AND AT INFINITY

A study is made of the functions realizing a minimum for the functional where has power order of growth with respect to . The rate of decrease is established for the th-order derivatives of an extremal in the integral metric in a neighborhood of a singularity on the boundary of power cusp type and at infinity in domains having outside some ball the structure of a cylinder or layer, and also domains constricting or expanding at infinity in a power manner. Estimates are obtained under the assumption that homogeneous Dirichlet conditions or Neumann conditions are given on the indicated part of the boundary. The estimates depend on the geometry of the domain. The results obtained are new also for a broad class of nonlinear elliptic equations. Bibliography: 19 titles.

On a class of nonlinear ellipticequations in Hilbert spaces

We consider elliptic nonlinear equations in a separable Hilbert space and their solutions in spaces of Sobolev type.

On a class of nonlinear elliptic equations

On a class of nonlinear elliptic equations

Phragmen-Lindelöf decay theorems for classes of nonlinear Dirichlet problems in a circular cylinder

Classes of nonlinear elliptic equations in a long circular cylinder of radius one are considered. The equations are of the form ▽ 2 u = S( u, u′) u″ + T( u) u′ 2, where u = u( x 1, x 2, x 3), and u′, u″ represent general partial derivatives of the indicated order. Homogeneous Dirichlet data are prescribed on the long sides of the cylinder, and throughout the cylinder u is a priori assumed to be sufficiently small while u′ (and, for some classes, also u″) is assumed to be bounded in absolute value by one. With the above assumptions, it is proved that every solution u decays exponentially with distance from the nearer end with a decay constant k which depends on the smoothness properties of S and T but is independent of the length of the cylinder.

Aprioribounds for a class of nonlinear elliptic equations and applications to physical problems

SynopsisUpper and lower bounds for the solutions of a nonlinear Dirichlet problem are given and isoperimetric inequalities for the maximal pressure of an ideal charged gas are constructed. The method used here is based on a geometrical result for two-dimensional abstract surfaces.