Vol. 33, No. 1, 1970

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Bounds for the solution of a certain class of nonlinear partial differential equations

Vinod B. Goyal

Vol. 33 (1970), No. 1, 117–138
Abstract

This paper is a study of boundedness and other properties of the solutions of nonlinear partial differential equations of the form

Δu = P (x1,x2,⋅⋅⋅ ,xn)f(u )
(1.1)

where P(x1,x2,,xn) is positive, and u(x1,x2,xn) is to be defined in some region of Euclidean n-space, and Δu = i=1n2u∕∂xi2 is the Laplacian of u. In particular, we consider the case f(u) = eu.

Our principal result is concerned with the nonexistence of entire solutions. An entire solution u = u(x1,x2,,xn) will be defined as a solution which though continuous for 0 r < is twice continuously differentiable for 0 < 7 < . Other results are concerned with the general form of and explicit bounds for solutions.

Mathematical Subject Classification
Primary: 35.36
Milestones
Received: 20 January 1966
Revised: 27 January 1967
Published: 1 April 1970
Authors
Vinod B. Goyal