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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:
[1] H. Brezis, M. Marcus, A.C. Ponce, Nonlinear elliptic equations with measures revisited, Ann. of Math. Stud., Princeton University Press, in press · Zbl 1151.35034
[2] Brezis, H.; Merle, F., Uniform estimates and blow-up behavior for solutions of \(- \operatorname{\Delta} u = V(x) \operatorname{e}^u\) in two dimensions, Comm. partial differential equations, 16, 1223-1253, (1991) · Zbl 0746.35006
[3] Evans, L.C.; Gariepy, R.F., Measure theory and fine properties of functions, (1992), CRC Press · Zbl 0626.49007
[4] Federer, H., Geometric measure theory, (1969), Springer-Verlag · Zbl 0176.00801
[5] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer-Verlag Berlin · Zbl 0691.35001
[6] Mattila, P., Geometry of sets and measures in Euclidean spaces, (1995), Cambridge University Press
[7] A.C. Ponce, How to construct good measures, Proceedings of the Fifth European Conference on Elliptic and Parabolic Problems. A special tribute to the work of Haïm Brezis · Zbl 1160.35414
[8] Stampacchia, G., Équations elliptiques du second ordre à coefficientes discontinus, (1966), Les Presses de l’Université de Montréal Montréal · Zbl 0151.15501
[9] Vázquez, J.L., On a semilinear equation in \(\mathbb{R}^2\) involving bounded measures, Proc. roy. soc. Edinburgh sect. A, 95, 181-202, (1983) · Zbl 0536.35025
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