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An elliptic boundary-value problem with a discontinuous nonlinearity. II. (English) Zbl 0647.35029
Summary: [For Part I, see the first author, ibid. 91, 161-174 (1981; Zbl 0511.35032).]
Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ 2.\) The study, begun in part I, of the boundary-value problem, for (\(\lambda\) /k,\(\psi)\), \[ - \Delta \psi \in \lambda H(\psi -k)\quad in\quad \Omega \subset {\mathbb{R}}^ 2,\quad \psi =0\quad on\quad \partial \Omega, \] is continued. Here \(\Delta\) denotes the Laplacian, H is the Heaviside step function and one of \(\lambda\) or k is a given positive constant. The solutions considered always have \(\psi >0\) in \(\Omega\) and \(\lambda /k>0\), and have cores \(A=\{(x,y)\in \Omega |\psi (x,y)>k\}.\)
In the special case \(\Omega =B(0,R)\), a disc, the explicit exact solutions of the branch \(\tau_ e\) have connected cores A and the diameter of A tends to zero when the area of A tends to zero. This result is established here for other convex domains \(\Omega\) and solutions with connected cores A.
An adaptation of the maximum principles and of the domain folding arguments of B. Gidas, W. M. Ni and L. Nirenberg [Commun. Math. Phys. 68, 209-243 (1979; Zbl 0425.35020)] is an important step in establishing the above result.

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
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References:
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