Positive solutions for semilinear elliptic equations with singular forcing terms

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Abstract

We consider the existence of solutions to the semilinear elliptic problem(*)κ{(- Δ u + u = up + κ ∑i = 1m ci δai in  D′ (RN),; u > 0 a.e. in RN and u (x) → 0 as | x | → ∞,). with prescribed given finite points {ai}i = 1m in RN and positive numbers {ci}i = 1m, where N ≥ 3, 1 < p < N / (N - 2), κ ≥ 0 is a parameter, and δa is the Dirac delta function supported at a ∈ RN. We reduce the problem (*)κ to the problem in H1 (RN) ∩ C0 (RN) in terms of auxiliary functions, and then show the existence of a positive constant κ* > 0 such that (*)κ has at least two solutions if κ ∈ (0, κ*), a unique solution if κ = κ*, and no solution if κ > κ*. © 2007 Elsevier Inc. All rights reserved.

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APA

Naito, Y., & Sato, T. (2007). Positive solutions for semilinear elliptic equations with singular forcing terms. Journal of Differential Equations, 235(2), 439–483. https://doi.org/10.1016/j.jde.2007.01.006

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