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We prove an abstract version of concentration compactness principle in Hilbert space and show its applications to a range of elliptic problems on unbounded domains.

We investigate the following quasilinear and singular problem, $$\text{t}o2.7cm\left\{\begin{array}{cc}-{\Delta }_{p}u=\frac{\lambda }{u^{\delta }}+u^q\hfill & \text{in}\Omega ;\hfill \\ {u|}_{\partial \Omega }=0,u\>0\hfill & \text{in}\Omega ,\hfill \end{array}\right.\text{t}o2.7cm\text{(P)}$$ where $\Omega$ is an open bounded domain with smooth boundary, $1\<p\<\infty$, $p-1\<q\le p^{*}-1$, $\lambda \>0$, and $0\<\delta \<1$. As usual, $p^{*}=\frac{Np}{N-p}$ if $1\<p\<N$, $p^{*}\in (p,\infty )$ is arbitrarily large if $p=N$, and $p^{*}=\infty$ if $p\>N$. We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in $W_0^{1,p}\left(\Omega \right)$. While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a...