Multiple Positive Solutions for Semilinear Elliptic Equations with Sign-Changing Weight Functions in ℝ𝑁 - Tsing-San Hsu (Abstract and Applied Analysis)
Existence and multiplicity of positive solutions for the following semilinear elliptic equation: −Δ𝑢+𝑢=𝑎(𝑥)|𝑢|𝑝−2𝑢+𝜆𝑏(𝑥)|𝑢|𝑞−2𝑢 in ℝ𝑁, 𝑢∈𝐻1(ℝ𝑁), are established, where 𝜆>0,1<𝑞<2<𝑝<2∗(2∗=2𝑁/(𝑁−2) if 𝑁≥3,2∗=∞ if 𝑁=1,2), 𝑎, 𝑏 satisfy suitable conditions, and 𝑏 maybe changes sign in ℝ𝑁. The study is based on the extraction of the Palais-Smale sequences in the Nehari manifold.