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Multiple Positive Solutions for Semilinear Elliptic Equations in ℝ 𝑁 Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions

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Abstract and Applied Analysis
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Abstract

We study the existence and multiplicity of positive solutions for the following semilinear elliptic equation − Δ 𝑢 + 𝑢 = 𝜆 𝑎 ( 𝑥 ) | 𝑢 | 𝑞 − 2 𝑢 + 𝑏 ( 𝑥 ) | 𝑢 | 𝑝 − 2 𝑢 in ℝ 𝑁 , 𝑢 ∈ 𝐻 1 ( ℝ 𝑁 ) , where 𝜆 > 0 , 1 < 𝑞 < 2 < 𝑝 < 2 ∗ ( 2 ∗ = 2 𝑁 / ( 𝑁 − 2 ) if 𝑁 ≥ 3 , 2 ∗ = ∞ if 𝑁 = 1 , 2 ), 𝑎 ( 𝑥 ) , 𝑏 ( 𝑥 ) satisfy suitable conditions, and 𝑎 ( 𝑥 ) may change sign in ℝ 𝑁 .

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