*Published Paper*

**Inserted:** 23 jan 2024

**Last Updated:** 23 jan 2024

**Journal:** Rendiconti Lincei - Matematica e Applicazioni

**Volume:** 34

**Number:** 2

**Pages:** 451-463

**Year:** 2023

**Doi:** 10.4171/RLM/1014

**Links:**
doi

**Abstract:**

The aim of this paper is to propose some results which we hope could contribute to understand better the Lavrentiev's phenomenon for energy integrals as $ F\left( u,\Omega \right) =\int_{\Omega }\left\{ f\left( x,Du\right) +\left \langle b\left( x\right) ,u\right \rangle \right\} \ dx, $ under some $p,q-$growth conditions as $c_{1}\left\vert z\right\vert ^{p}-c_{2}\leq f\left( x,z\right) \leq c_{3}\left\vert z\right\vert ^{q}+c_{4\,}$; in fact we expect that the Lavrentiev's phenomenon does not occur if the quotient $q/p$ is not too large in dependence of $n$, for instance as in the cases - either scalar or vectorial ones - that we consider in this manuscript.

**Keywords:**
calculus of variations, Elliptic equations, Local minimizer, Elliptic systems, $p,q$-growth, Lavrentievâ€™s phenomenon