Existence of singular solutions for a Dirichlet problem containing a Dirac mass
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Date
2011
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PERGAMON-ELSEVIER SCIENCE LTD
Abstract
We give general existence results of solutions (u, v) to the Dirichlet problem
{-Delta u = f (x, u, v) + c delta(0), -Delta u = g(x, u, v) + d delta(0) u = v = 0 on partial derivative B, in D'(B), (P)
where B is the unit ball centered at zero in R(N), N >= 3, delta(0) is the Dirac mass at 0 and c, d are nonnegative constants. No assumptions on the sign of the functions f and g are required. We also characterize the set of (c, d) such that problem (P) admits a solution in some particular cases of the nonlinearities f and g. (C) 2011 Elsevier Ltd. All rights reserved.
{-Delta u = f (x, u, v) + c delta(0), -Delta u = g(x, u, v) + d delta(0) u = v = 0 on partial derivative B, in D'(B), (P)
where B is the unit ball centered at zero in R(N), N >= 3, delta(0) is the Dirac mass at 0 and c, d are nonnegative constants. No assumptions on the sign of the functions f and g are required. We also characterize the set of (c, d) such that problem (P) admits a solution in some particular cases of the nonlinearities f and g. (C) 2011 Elsevier Ltd. All rights reserved.
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Keywords
Existence, Dirac mass, Singular solutions, EQUATIONS, NONEXISTENCE, SYSTEMS