Local Lipschitz continuity for p,q−PDEs with explicit u−dependence

Local Lipschitz continuity for energy integrals with slow growth

“…Moreover this class of functionals has been already considered in literature, see for instance [4,7] concerning regularity of local minimizers of a class of integrals of the Calculus of Variations, see also [8] where the functionals considered do not necessarily satisfy the Euler-Lagrange equation. On the other hand, in [20] the motivation to introduce an explicit u-dependence on the coefficients in the differential equation comes from several recent studies on nonlinear elliptic and parabolic equations with general growth conditions. In this work we have been inspired by the papers [18,11] dealing with functionals depending only on Du with general growth assumptions, respectively fast and slow.…”

Non‐autonomous double phase eigenvalue problems with indefinite weight and lack of compactness

“…For such problems, there is no global (that is, up to the boundary) regularity theory. There are only interior regularity results, which are primarily due to Baroni et al [3] and Marcellini [10,20,21]. In fact, most of works dealt with double phase problems having unbalanced growth in bounded domains of ℝ 𝑁 , we refer the readers to [12][13][14][15][22][23][24] and references therein.…”

“…There are only interior regularity results, which are primarily due to Baroni et al. [3] and Marcellini [10, 20, 21]. In fact, most of works dealt with double phase problems having unbalanced growth in bounded domains of $$\mathbb {R}^N$$, we refer the readers to [12–15, 22–24] and references therein.…”

Regularity properties for quasiminimizers of a (p, q)-Dirichlet integral

“…Harnack inequalities for double phase problems were obtained by Di Benedetto-Trudinger [11], Baroni-Colombo-Mingione [1] and the references therein. See also Kinnunen-Lehrbäck-Vähäkangas [26], Marcellini [33,34] and Mingione-Rǎdulescu [39] for other results and literature reviews.…”