A nonlinear problem witha weight and a nonvanishing boundary datum

We consider the problem: $$\inf_{{u}\in {H}^{1}_{g}(\Omega),\|u\|_{q}=1} \int_{\Omega}{p(x)}|\nabla{u(x)}|^{2}dx-\lambda\int_{\Omega}| u(x)|^{2}dx$$ where $\Omega$ is a bounded domain in $\R^{n}$, ${n}\geq{4}$, $p : \bar{\Omega}\longrightarrow \R$ is a given positive weight such that $p\in H^{1}(\Omega)\cap C(\bar{\Omega})$, $0< c_1 \leq p(x) \leq c_2$, $\lambda$ is a real constant and $q=\frac{2n}{n-2}$ and $g$ a given positive boundary data. The goal of this present paper is to show that minimizers do exist. We distinguish two cases, the first is solved by a convex argument while the second is not so straightforward and will be treated using the behavior of the weight near its minimum and the fact that the boundary datum is not zero.