Decay estimates for quasilinear elliptic equations and a Brezis–Nirenberg result in \(D^{1,p}({\mathbb R}^N)\)

We prove decay estimates for solutions of quasilinear elliptic equations in the whole \({\mathbb R}^N\) ${\mathbb{R}}^N$ of the form \(\begin{aligned} u\in X: -\text{ div }\,A(x,\nabla u)=a(x) f(x,u), \end{aligned}\) $$\begin{array}{c}\hfill u\in X:-\text{div}A(x,\mathrm{\nabla }u)=a(x)f(x,u),\end{array}$$ where \(X=D^{1,p}({\mathbb R}^N)\) $X=D^{1,p}({\mathbb{R}}^N)$ is the Beppo-Levi space (also called homogeneous Sobolev space). Based on these decay estimates we are able to prove a Brezis–Nirenberg type result for the energy functional \(\Phi : X\rightarrow {\mathbb R}\) $\mathrm{\Phi }:X\to \mathbb{R}$ related to the p-Laplacian equation in \({\mathbb R}^N\) ${\mathbb{R}}^N$ in the form \(\begin{aligned} u\in X: -\Delta _p u=a(x) g(u), \end{aligned}\) $$\begin{array}{c}\hfill u\in X:-{\mathrm{\Delta }}_{p}u=a(x)g(u),\end{array}$$ saying that for the subspace V of bounded continuous functions with weight \(1+|x|^{\frac{N-p}{p}},\) $1+{|x|}^{\frac{N-p}{p}},$ a local minimizer of \(\Phi \) $\mathrm{\Phi }$ in the finer V topology is also a local minimizer in the X-topology. Global \(L^\infty \) $L^{\infty }$ -estimates on the one hand and pointwise estimates for solutions of quasilinear elliptic equations in terms of nonlinear Wolff potentials on the other hand play a crucial role in the proofs.