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Existence of Positive Bounded Solutions of Semilinear Elliptic Problems

International Journal of Differential Equations
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This paper is concerned with the existence of bounded positive solution for the semilinear elliptic problem Δ 𝑢 = 𝜆 𝑝 ( 𝑥 ) 𝑓 ( 𝑢 ) in Ω subject to some Dirichlet conditions, where Ω is a regular domain in ℝ 𝑛   ( 𝑛 ≥ 3 ) with compact boundary. The nonlinearity 𝑓 is nonnegative continuous and the potential 𝑝 belongs to some Kato class 𝐾 ( Ω ) . So we prove the existence of a positive continuous solution depending on 𝜆 by the use of a potential theory approach.

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