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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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