Ground state solutions for a semilinear elliptic problem with critical-subcritical growth

Abstract

Abstract We prove the existence of at least one ground state solution for the semilinear elliptic problem { - Δ ⁢ u = u p ⁢ ( x ) - 1 , u > 0 , in ⁢ G ⊆ ℝ N , N ≥ 3 , u ∈ D 0 1 , 2 ⁢ ( G ) , \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=u^{p(x)-1},\quad u% >0,\quad\text{in}\ G\subseteq\mathbb{R}^{N},\ N\geq 3,\\ \displaystyle u&\displaystyle\in D_{0}^{1,2}(G),\end{aligned}\right. where G is either ℝ N {\mathbb{R}^{N}} or a bounded domain, and p : G → ℝ {p\colon G\to\mathbb{R}} is a continuous function assuming critical and subcritical values.