Band 12, Heft 3

We prove a general magnetic Bourgain–Brezis–Mironescu formula which extends the one obtained in [37] in the Hilbert case setting. In particular, after developing a rather complete theory of magnetic bounded variation functions, we prove the validity of the formula in this class.

The paper is concerned with existence of nonnegative solutions of a Schrödinger–Choquard–Kirchhoff-type fractional p -equation. As a consequence, the results can be applied to the special case (a+b⁢∥u∥sp⁢(θ-1))⁢[(-Δ)ps⁢u+V⁢(x)⁢|u|p-2⁢u]=λ⁢f⁢(x,u)+(∫ℝN|u|pμ,s*|x-y|μ⁢𝑑y)⁢|u|pμ,s*-2⁢u in ⁢ℝN,(a+b\|u\|_{s}^{p(\theta-1)})[(-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u]=\lambda f(x,u)% +\Bigg{(}\int_{\mathbb{R}^{N}}\frac{|u|^{p_{\mu,s}^{*}}}{|x-y|^{\mu}}\,dy% \Biggr{)}|u|^{p_{\mu,s}^{*}-2}u\quad\text{in }\mathbb{R}^{N}, where ∥u∥s=(∬ℝ2⁢N|u⁢(x)-u⁢(y)|p|x-y|N+p⁢s⁢𝑑x⁢𝑑y+∫ℝNV⁢(x)⁢|u|p⁢𝑑x)1p,\|u\|_{s}=\Bigg{(}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}% \,dx\,dy+\int_{\mathbb{R}^{N}}V(x)|u|^{p}\,dx\Biggr{)}^{\frac{1}{p}}, a,b∈ℝ0+{a,b\in\mathbb{R}^{+}_{0}}, with a+b>0{a+b>0}, λ>0{\lambda>0} is a parameter, s∈(0,1){s\in(0,1)}, N>p⁢s{N>ps}, θ∈[1,N/(N-p⁢s)){\theta\in[1,N/(N-ps))}, (-Δ)ps{(-\Delta)^{s}_{p}} is the fractional p -Laplacian, V:ℝN→ℝ+{V:\mathbb{R}^{N}\rightarrow\mathbb{R}^{+}} is a potential function, 0<μ<N{0<\mu<N}, pμ,s*=(p⁢N-p⁢μ/2)/(N-p⁢s){p_{\mu,s}^{*}=(pN-p\mu/2)/(N-ps)} is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality, and f:ℝN×ℝ→ℝ{f:\mathbb{R}^{N}\times\mathbb{R}\rightarrow\mathbb{R}} is a Carathéodory function. First, via the Mountain Pass theorem, existence of nonnegative solutions is obtained when f satisfies superlinear growth conditions and λ is large enough. Then, via the Ekeland variational principle, existence of nonnegative solutions is investigated when f is sublinear at infinity and λ is small enough. More intriguingly, the paper covers a novel feature of Kirchhoff problems, which is the fact that the parameter a can be zero. Hence the results of the paper are new even for the standard stationary Kirchhoff problems.

We consider perturbed eigenvalue problems of the 1-Laplace operator and verify the existence of a sequence of solutions. It is shown that the eigenvalues of the perturbed problem converge to the corresponding eigenvalue of the unperturbed problem as the perturbation becomes small. The results rely on nonsmooth critical point theory based on the weak slope.

We define the notion of higher-order colocally weakly differentiable maps from a manifold M to a manifold N . When M and N are endowed with Riemannian metrics, p≥1{p\geq 1} and k≥2{k\geq 2}, this allows us to define the intrinsic higher-order homogeneous Sobolev space W˙k,p⁢(M,N){\dot{W}^{k,p}(M,N)}. We show that this new intrinsic definition is not equivalent in general with the definition by an isometric embedding of N in a Euclidean space; if the manifolds M and N are compact, the intrinsic space is a larger space than the one obtained by embedding. We show that a necessary condition for the density of smooth maps in the intrinsic space W˙k,p⁢(M,N){\dot{W}^{k,p}(M,N)} is that π⌊k⁢p⌋⁢(N)≃{0}{\pi_{\lfloor kp\rfloor}(N)\simeq\{0\}}. We investigate the chain rule for higher-order differentiability in this setting.