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A semilinear elliptic boundary-value problem describing small patches of vorticity in an otherwise irrotational flow. (English) Zbl 0568.35040
Nonlinear analysis, Miniconf. Canberra/Aust. 1984, Proc. Cent. Math. Anal. Aust. Natl. Univ. 8, 123-132 (1984).
[For the entire collection see Zbl 0552.00008.]
Let $$\Omega$$ be a bounded convex domain in $$R^ 2$$ and the boundary $$\partial \Omega$$ of class $$C^ 2$$ has the curvature bounded away from zero. The problem $(P)\quad -\Delta \psi \in \lambda H(\psi -k)\quad in\quad \Omega,\quad \psi =0\quad on\quad \partial \Omega$ is considered, H being the set-valued Heaviside stepfunction $$(H(t)=0$$ if $$t<0$$, $$H(0)=[0,1]$$ and $$H(t)=1$$ for $$t>0)$$. let $$(\lambda_ n,k_ n,\psi_ n)$$ be a sequence of the positive solutions of the problem (P) such that area of $$\{$$ $$x\in \Omega:$$ $$\psi_ n(x)>k_ n\}$$ tends to zero. Then $$(\sup_{x\in \Omega}\psi_ n(x)-k_ n)/k_ n\to 0$$.
Reviewer: V.Tsalyuk

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35R05 PDEs with low regular coefficients and/or low regular data 35B40 Asymptotic behavior of solutions to PDEs 35J25 Boundary value problems for second-order elliptic equations