Approximation of a semilinear elliptic problem in an unbounded domain

Kolli, Messaoud; Schatzman, Michelle

ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 1, pp. 117-132

Let $f$ be an odd function of a class ${\mathrm{C}}^2$ such that $f\left(1\right)=0,f^{\text{'}}\left(0\right)<0,f^{\text{'}}\left(1\right)>0$ and $x\mapsto f\left(x\right)/x$ increases on $[0,1]$. We approximate the positive solution of $-\Delta u+f\left(u\right)=0,$ on ${ℝ}_{+}^2$ with homogeneous Dirichlet boundary conditions by the solution of $-\Delta u_L+f\left(u_L\right)=0,$ on ${]0,L[}^2$ with adequate non-homogeneous Dirichlet conditions. We show that the error $u_L-u$ tends to zero exponentially fast, in the uniform norm.

Classification : 35J60, 35P15
Keywords: semilinear elliptic equations, full-space problems, approximation by finite domains

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