Uniform Continuity and Brézis-Lieb Type Splitting for Superposition Operators in Sobolev Space

Denote by \(\mathcal{F}\) a superposition (or Nemyckii-)
operator induced by a continuous function
\(f\colon\mathbb{R}\to\mathbb{R}\) that satisfies a polynomial growth
condition with exponent \(\mu>0\).  If \(\nu\ge 1\) is such that \(\mu\nu>2\)
and that the Sobolev embedding of \(H^1(\mathbb{R}^N)\) in
\(L^{\mu\nu}(\mathbb{R}^N)\) is locally compact we prove that
\(\mathcal{F}\colon H^1(\mathbb{R}^N)\to L^\nu(\mathbb{R}^N)\) is
uniformly continuous on any bounded subset of \(H^1(\mathbb{R}^N)\).  This
result implies a variant of the Brézis-Lieb Lemma that applies to more
general nonlinear superposition operators within this range of growth
exponents.  In particular, no convexity or Hölder continuity
assumptions are imposed on \(f\), in contrast to previously known results.