On the Kuratowski measure of noncompactness for duality mappings (Q1947836)

In a real Banach space \(X\) having a Fréchet differentiable norm, the duality mapping on \(X\) is considered, namely \(J_\varphi:X\rightarrow X^*\), with \(\varphi:\mathbb{R}_+\rightarrow \mathbb{R}_+\) being a gauge function. The main result of the paper is that, for the Kuratowski measure on noncompactness of \(J_\varphi\), one has \[ \alpha(J_\varphi)\geq \sup\left\{\frac{\varphi(r)}{r}: r>0\right\} \text{ if } \dim X=\infty\;, \] whereas \[ \alpha(J_\varphi)=0 \;\text{ if and only if } \dim X<\infty\;. \] \smallskip As an application, the minus \(p\)-Laplacian \(-\Delta_p:W^{1,p}_0(\Omega)\rightarrow W^{-1,p'}(\Omega)\), \(1<p<\infty\), \(1/p+1/p'=1\), is viewed as the duality mapping on \(W^{1,p}_0(\Omega)\) corresponding to the gauge function \(\varphi(t)=t^{p-1}\). It is deduced that the Kuratowski measure of noncompactness for the \(p\)-Laplacian satisfies \[ \alpha(-\Delta_p)=1 \text{ for } p=2 \] and \[ \alpha(-\Delta_p)=\infty \text{ for } p\in (1,2)\cup (2,\infty)\;. \]