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We extend, in many variables, results established by Cegrell [2] on the continuation of the plurisubharmonic functions defined outside closed subsets, of open sets of C n , having zero Ronkin [8] gamma capacity. This will be achieved by the continuation of some plurisubharmonic functions, having on "a priori" growth, through some thin (meaning to be precised) closed subsets of C n (n 1).

We prove that the holomorphic differential equation ϕ (ϕ + c) = γ(ϕ) 2 (ϕ : C → C be a holomorphic function and (γ, c) ∈ C 2) plays a classical role on many problems of real and complex convexity. The condition exactly γ ∈ {1, s−1 s /s ∈ N\{0}} (independently of the constant c) is of great importance in this paper. On the other hand, let n ≥ 1, (A 1 , A 2) ∈ C 2 , and g 1 , g 2 : C n → C be two analytic functions. Put u(z, w) =| A 1 w − g 1 (z) | 2 + | A 2 w − g 2 (z) | 2 , v(z, w) =| A 1 w − g 1 (z) | 2 + | A 2 w − g 2 (z) | 2 , for (z, w) ∈ C n × C. We prove that u is strictly plurisubharmonic and convex on C n ×C if and only if n = 1, (A 1 , A 2) ∈ C 2 \{0} and the functions g 1 and g 2 have a classical representation form described in the present paper. Now v is convex and strictly psh on C n × C if and only if (A 1 , A 2) ∈ C 2 \{0}, n ∈ {1, 2} and g 1 , g 2 have several representations investigated in this paper.