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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.

35J60 Nonlinear elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI Numdam Numdam EuDML
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