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We consider the elliptic problem -Δu - λu = a(x)g(u), with a(x) sign-changing and g(u) behaving like u^p, p > 1. Under suitable conditions on g(u) and a(x), we extend the multiplicity, existence and nonexistence results known to hold for this equation on a bounded domain (with standard homogeneous boundary conditions) to the case that the bounded domain is replaced by the entire space ℝ^N. More precisely, we show that there exists Λ > 0 such that this equation on ℝ^N has no positive solution for λ > Λ, at least two positive solutions for λ ∈ (o,Λ), and at least one positive solution for λ ∈ (-∞,0]U{A}. Our approach is based on some descriptions of mountain pass solutions of semilinear elliptic problems on bounded domains obtained by a special version of the mountain pass theorem. These results are of independent interests. |
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