Abstract:

In this paper, we study the following Hénon-type elliptic problem $$\begin{cases} -\Delta u = |x|^\alpha f(u) + \lambda |x|^\beta u^q, & \text{in } {B_1(0)}, \\ u > 0, & \text{in } {B_1(0)}, \\ u = 0, & \text{on } \partial {B_1(0)}, \end{cases} $$ where $\alpha > 0$, $\beta \geq 0$, $\lambda > 0$, $0<q<1$, ${B_1(0)}$ is the unit ball in $\mathbb{R}^N$ with $N \geq 3$, and $f$ satisfies $$0 \leq tf(t) \leq C_0 t^{2_\alpha^*}, t \in \mathbb{R}$$ with $2_\alpha^* = \frac{2(N+\alpha)}{N-2}$. Without any compactness conditions, we employ the Galerkin method to investigate the existence of positive solutions.The main result is that there exists a $\lambda_\ast > 0$such that for $\lambda \in (0, \lambda_\ast)$, the problem has a positive radial solution$u \in \ H_0^1( B_1(0)) \cap C^{1, \theta}(\overline{{B_1(0)}}), \theta \in (0, 1)$.

Key words: Galerkin method, H énon equation, no compactness conditions