Compactly supported solutions for a semilinear elliptic problem in ℝⁿ with sign-changing function and non-Lipschitz nonlinearity

Abstract

For a sign-changing function a(x) we consider the solutions of the following semilinear elliptic problem in ℝⁿ with n ≥ 3:

where γ > 0 and 0 < q < 1 < p < (n + 2)/(n − 2). Under an appropriate growth assumption on a⁻ at infinity, we show that all solutions are compactly supported. When Ω⁺ = {x ∈ ℝⁿ | a(x) > 0} has several connected components, we prove that there exists an interval on γ in which the solutions exist. In particular, if a(x) = a(|x|), by applying the mountain-pass theorem there are at least two solutions with radial symmetry that are positive in Ω⁺.