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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:
 [1] Keady, Maximum principles and an application to an elliptic boundary-value problem with a discontinuous nonlinearity. With corrigenda. (1984) [2] Figueiredo, J. Math. Pures Appl. 61 pp 41– (1982) [3] DOI: 10.1017/S0334270000004677 · Zbl 0586.76029 · doi:10.1017/S0334270000004677 [4] Keady, Proc. Roy. Soc. Edinburgh Sect. A 91 pp 161– (1981) · Zbl 0511.35032 · doi:10.1017/S0308210500012713 [5] Kawohl, Lecture Notes in Mathematics 1150 (1985) [6] Hayman, Subharmonic functions (1976) [7] Gilbarg, Elliptic partial differential equations of second order (1984) [8] DOI: 10.1080/03605308308820293 · Zbl 0523.76014 · doi:10.1080/03605308308820293 [9] Sperb, Maximum principles and their applications (1981) · Zbl 0454.35001 [10] DOI: 10.1007/BF00250468 · Zbl 0222.31007 · doi:10.1007/BF00250468 [11] DOI: 10.1080/03605307708820040 · Zbl 0371.35017 · doi:10.1080/03605307708820040 [12] DOI: 10.1007/BF01221125 · Zbl 0425.35020 · doi:10.1007/BF01221125 [13] Fraenkel, Proc. Roy. Soc. Edinburgh Sect. A 88 pp 267– (1981) · Zbl 0466.31007 · doi:10.1017/S0308210500020114 [14] DOI: 10.1112/plms/s3-39.3.385 · Zbl 0406.46026 · doi:10.1112/plms/s3-39.3.385 [15] DOI: 10.1016/0022-247X(81)90090-1 · Zbl 0482.31001 · doi:10.1016/0022-247X(81)90090-1 [16] Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large. (1985) [17] DOI: 10.1007/BF01982715 · Zbl 0454.35087 · doi:10.1007/BF01982715 [18] Bandle, Exposition. Math. 4 pp 75– (1986) [19] DOI: 10.1007/BF00251252 · Zbl 0609.76018 · doi:10.1007/BF00251252 [20] Potter, Maximum principles in differential equations (1967) [21] DOI: 10.1512/iumj.1985.34.34036 · Zbl 0549.35025 · doi:10.1512/iumj.1985.34.34036 [22] Keady, Proceedings of the Mini-conference in Nonlinear Analysis (1984) · Zbl 0568.35040
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