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On a semilinear equation in \({\mathbb{R}}^ 2\) involving bounded measures. (English) Zbl 0536.35025
In this paper we study the existence of solutions for the equation \((1)\quad -\Delta u+\beta(u)=f\) in \({\mathbb{R}}^ 2:\) f is a bounded measure, \(\Delta\) is the Laplace operator and \(\beta\) is a continuous, nondecreasing real function with \(0=\beta(0)\) [or more generally a maximal monotone graph and 0\(\in \beta(0)]\). An example of such an equation is the Poisson-Boltzmann equation where \(\beta(u)=\exp(cu)-1\), \(c>0\). Typically the second member is either an \(L^ 1\) function or a sum of Dirac masses. In the first case existence of a class of unique solutions has been proved by P. Bénilan, H. Brézis and M. G. Crandall [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 2, 523-555 (1975; Zbl 0314.35077)] (in all dimensions \(n\geq 1).\)
The existence of solutions of (1) when f is a measure depends on the presence of Dirac masses in f and the rate of growth of \(\beta\) as \(| u| \to \infty\) as we show in this paper, the critical growth being exponential. Here is a simple version of the main result: ”i) if \(\beta(u)={\mathbb{O}}(\exp(a| u|))\) for every \(a>0\) as \(| u| \to \infty\) then there is a solution for every f, ii) if on the contrary \(\beta\) (u)exp(-\(a| u|)\to \infty\) as \(| u| \to \infty\) for every \(a>0\) then there is no solution of (1) if f has Dirac masses, iii) finally if \(\beta\) (u) behaves as \(| u| \to \infty\) like ex\(p(a| u|)\) for an \(a>0\) then there is a solution if and only if the Dirac masses \(c_ i\delta(x-x_ i)\) contained in f do not exceed the critical value, i.e. \(| c_ i| \leq 4\pi /a''\). And we are able to define a coefficient a (and a critical value 4\(\pi\) /a) for any \(\beta\) thus making the theorem general. The phenomenon of critical values is particular to dimension \(n=2\). A discussion of the case \(n=1\) is included. The case \(n\geq 3\) is studied by Bénilan and Brézis in connection with the Thomas Fermi equation [cf. H. Brézis, Free boundary problems, Proc. Semin. Pavia 179, Vol. II., 85-91 (1980; Zbl 0464.49032)].

MSC:
35J60 Nonlinear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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References:
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[2] DOI: 10.1016/0001-8708(77)90108-6 · Zbl 0938.81568
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[5] DOI: 10.2969/jmsj/02540565 · Zbl 0278.35041
[6] DOI: 10.1007/BF01197552 · Zbl 0451.35101
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[10] DOI: 10.1016/0362-546X(81)90028-6 · Zbl 0457.35031
[11] DOI: 10.1007/BF02761946 · Zbl 0516.35009
[12] DOI: 10.1007/BF01197882
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