Positive solutions to nonlinear elliptic problems involving Sobolev exponent

In this paper we consider nonlinear elliptic PDEs of the type $$-{\mathrm{\Delta }}_{p}u+a(x)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=\mathrm{\mid }u{\mathrm{\mid }}^{p^{\ast }-2}uin\mathrm{\Omega },$$ where $1<p<N$ and $p^{\ast }=Np\mathrm{/}(N-p)$ is the critical Sobolev exponent, and allowing the asymptotic behavior of the weight function $a$ to be sensitive to the direction. We provide a unified variational approach to obtain existence of distinct solutions in either the unbounded case $\mathrm{\Omega }={\mathbb{R}}^N$ or when $\mathrm{\Omega }$ is a smooth bounded domain. A key point is a precise description of the compactness properties of certain sequences of approximating solutions (Palais-Smale sequences), for which we use novel observations on nonexistence in certain regimes. Most of our main results are new in the case of the classical Laplace operator, $p=2$.