Given an open and bounded set Ω ⊆ ℝ n {\Omega\subseteq\mathbb{R}^{n}} and a family 𝐗 = ( X 1 , … , X m ) {\mathbf{X}=(X_{1},\ldots,X_{m})} of Lipschitz vector fields on Ω, with m ≤ n {m\leq n} , we characterize three classes of local functionals defined on first-order X-Sobolev spaces, which admit an integral representation in terms of X, i.e. F ( u , A ) = ∫ A f ( x , u ( x ) , X u ( x ) ) 𝑑 x , F(u,A)=\int_{A}f(x,u(x),Xu(x))\,dx, with f being a Carathéodory integrand.
This paper deals with some classes of Kirchhoff type problems on a double phase setting and with nonlinear boundary conditions. Under general assumptions, we provide multiplicity results for such problems in the case when the perturbations exhibit a suitable behavior in the origin and at infinity, or when they do not necessarily satisfy the Ambrosetti–Rabinowitz condition. To this aim, we combine variational methods, truncation arguments and topological tools.
Given a 𝐶 2 family of vector fields 𝑋 1 , … , 𝑋 𝑚 which induces a continuous Carnot-Carathéodory distance, we show that any absolute minimizer of a supremal functional defined by a 𝐶 2 quasiconvex Hamiltonian 𝑓(𝑥, 𝑠, 𝑝), allowing 𝑠-variable dependence, is a viscosity solution to the Aronsson equation − 𝑚 ∑ 𝑖=1 𝑋 𝑖 (𝑓(𝑥, 𝑢(𝑥), 𝑋𝑢(𝑥))) 𝜕𝑓 𝜕𝑝 𝑖 (𝑥, 𝑢(𝑥), 𝑋𝑢(𝑥)) = 0, M S C 2 0 2 0 35D40, 35R03 (primary)
This paper deals with some classes of Kirchhoff type problems on a double phase setting and with nonlinear boundary conditions. Under general assumptions, we provide multiplicity results for such problems in the case when the perturbations exhibit a suitable behavior in the origin and at infinity, or when they do not necessarily satisfy the Ambrosetti-Rabinowitz condition. To this aim, we combine variational methods, truncation arguments and topological tools.
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