ON THE UNIQUENESS OF SOLUTIONS OF A SEMILINEAR EQUATION IN AN ANNULUS
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Date
2021
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Abstract
We establish the uniqueness of positive radial solutions of
(Delta u + f(u) = 0, x is an element of A
u(x) = 0 x is an element of partial derivative A (P)
where A := A(a,b) = {x is an element of R-n : a < vertical bar x vertical bar < bg, 0 < a < b <= infinity. We assume that the nonlinearity f is an element of C[0,infinity) boolean AND C-1(0,infinity) is such that f(0) = 0 and satisfies some convexity and growth conditions, and either f(s) > 0 for all s > 0, or has one zero at B > 0, is non positive and not identically 0 in (0,B) and it is positive in (B,infinity).
(Delta u + f(u) = 0, x is an element of A
u(x) = 0 x is an element of partial derivative A (P)
where A := A(a,b) = {x is an element of R-n : a < vertical bar x vertical bar < bg, 0 < a < b <= infinity. We assume that the nonlinearity f is an element of C[0,infinity) boolean AND C-1(0,infinity) is such that f(0) = 0 and satisfies some convexity and growth conditions, and either f(s) > 0 for all s > 0, or has one zero at B > 0, is non positive and not identically 0 in (0,B) and it is positive in (B,infinity).
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Keywords
Uniqueness, annulus, semilinear equation, subcritical, energy function