D1,p(RN) versus Cb(RN,1+|x|N−pp−1α) local minimizers

Let $X=D^{1,p}\left(R^N\right)$ be the Beppo-Levi space (homogeneous Sobolev space) with $2\le p<N$, and for $\frac{p-1}{p}<\alpha \le 1$ let $V_{\alpha }=X\cap C_b\left(R^N,1+{\left|x\right|}^{\frac{N-p}{p-1}\alpha }\right)$ be the subspace of bounded continuous functions with weight $1+{\left|x\right|}^{\frac{N-p}{p-1}\alpha }$. In this paper we prove a Brezis-Nirenberg type result for the energy functional $\Phi :X\to R$ related to the quasilinear elliptic equation in $R^N$ of the form $u\in X:-{\Delta }_{p}u=a\left(x\right)g\left(u\right)\text{in}{R}^N,$which states that a local minimizer of $\Phi$ in the $V_{\alpha }$-topology must be a local minimizer in the ”bigger” $X$-topology.

Global $L^{\infty }$-estimates for solutions of general quasilinear elliptic equations of divergence type in $R^N$ on the one hand, and decay estimates for solutions of $p$-Laplace equations via nonlinear Wolff potentials as well as comparison theorems for $p$-Laplacian type operators on the other hand play an important role in the proofs.