Absence and presence of Lavrentiev’s phenomenon for double phase functionals upon every choice of exponents

We study classes of weights ensuring the absence and presence of the Lavrentiev’s phenomenon for double phase functionals upon every choice of exponents. We introduce a new sharp scale for weights for which there is no Lavrentiev’s phenomenon up to a counterexample we provide. This scale embraces the sharp range for \(\alpha \) $\alpha$ -Hölder continuous weights. Moreover, it allows excluding the gap for every choice of exponents \(q,p>1\) $q,p>1$ .