ON A GROUND STATE AND TWO SYMMETRIC GROUND STATE SOLUTIONS IN A DOMAIN

Abstract. Let Ω be a domain in RN, N ≥ 1, and 2 ∗ = ∞ if N = 1, 2, 2 ∗ = 2N N−2 if N> 2, 2 < p < 2∗. Consider the semilinear elliptic equation −∆u+ u = |u|p−2u in Ω; (*) u ∈ H10 (Ω). The existence, the nonexistence, and the multiplicity of positive solutions of equation (*) are affected by the geometry and the topology of the domain Ω. In the article, we first present various analyses and use them to characterize which domain Ω is a ground state domain or a non-ground state domain. Secondly, for a y-symmetric domain Ω, we study their index α(Ω) and y-symmetric index αs(Ω). We determine whether α(Ω) = αs(Ω) or α(Ω) < αs(Ω). In case that α(Ω) < αs(Ω) and that both α(Ω) and αs(Ω) admits ground state solutions, then we obtain that in Ω, the Equation (*) has three positive solutions, of which one is y-symmetric and other two are not y-symmetric. 1. Introduction. Let Ω be a domain in RN, N ≥ 1, and 2 ∗ = ∞ if N = 1, 2, 2 ∗ = 2NN−2 if N> 2, 2 < p < 2 ∗. Consider the semilinear elliptic equatio