Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection |
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Authors: | Xue Luo Feng Zhou |
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Affiliation: | a Department of Mathematics, East China Normal University, 200241 Shanghai, PR China b LMAM, UMR 7122, Université Paul Verlaine de Metz, 57045 Metz Cedex 1, France |
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Abstract: | In this note, we investigate the regularity of the extremal solution u? for the semilinear elliptic equation −△u+c(x)⋅∇u=λf(u) on a bounded smooth domain of Rn with Dirichlet boundary condition. Here f is a positive nondecreasing convex function, exploding at a finite value a∈(0,∞). We show that the extremal solution is regular in the low-dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two. |
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Keywords: | 35B65 35B45 35J60 |
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