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Semilinear equations with exponential nonlinearity and measure data. (English) Zbl 1148.35318
Summary: We study the existence and non-existence of solutions of the problem
$-\Delta u+e^ u-1=\mu \text{ in } \Omega,\qquad u=0 \text{ on } \partial\Omega,\tag{1}$
where $$\Omega$$ is a bounded domain in $$\mathbb R^ N$$, $$N\geq3$$, and $$\mu$$ is a Radon measure. We prove that if $$\mu\leq 4\pi\mathcal H^{N-2}$$, then (1) has a unique solution. We also show that the constant $$4\pi$$ in this condition cannot be improved.

##### MSC:
 35J60 Nonlinear elliptic equations 35R05 PDEs with low regular coefficients and/or low regular data 35J25 Boundary value problems for second-order elliptic equations
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##### References:
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