In this paper we give sufficient conditions to obtain continuity results of solutions for the so called ϕ-Laplacian Δϕ with respect to domain perturbations. We point out that this kind of results can be extended to a more general class of operators including, for instance, nonlocal nonstandard growth type operators.

We show that local minimizers of functionals of the form $\int_{\Omega} \left[f(Du(x)) + g(x\,,u(x))\right]\,{\rm d}x$ ,  $u \in u_0 + W_0^{1,p}(\Omega)$ ,are locally Lipschitz continuous provided f is a convex function with $p-q$ growth satisfying a condition of qualified convexity at infinity and g is Lipschitz continuous in u. As a consequence of this, we obtain an existence result for a related nonconvex functional.

We prove Lipschitz continuity for localminimizers of integral functionals of the Calculus of Variationsin the vectorial case, where the energy density depends explicitlyon the space variables and has general growth with respect to thegradient. One of the models is $$ F\left(u\right)=\int_{\Omega}a(x)[h\left(|Du|\right)]^{p(x)}{\rm d}x $$ with h a convex function with general growth (also exponential behaviouris allowed).