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By using linking and ∇ -theorems in this paper we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet–Neumann boundary data, (− Δ) s u = λ u + f (x , u) in  Ω , u = 0 on  Σ , ∂ u ∂ ν = 0 on  Σ , where (− Δ) s , s ∈ (1 / 2 , 1) , is the spectral fractional Laplacian operator, Ω ⊂ ℝ N , N > 2 s , is a smooth bounded domain, λ > 0 is a real parameter, ν is the outward normal to ∂ Ω , Σ , Σ are smooth (N − 1) -dimensional submanifolds of ∂ Ω such that Σ ∪ Σ = ∂ Ω , Σ ∩ Σ = ∅ and Σ ∩ Σ ¯ = Γ is a smooth (N − 2) -dimensional submanifold of ∂ Ω. [ABSTRACT FROM AUTHOR]

In this note, we present the state-of-the-art theory of bi-Sobolev mappings. We recall that f is a Sobolev homeomorphism if f belongs to W loc 1 , 1 ∩ Hom (Ω ; Ω ′) , and f is a bi-Sobolev map if and only if f and f - 1 are Sobolev homeomorphisms. This concept, introduced in Hencl et al. (J. Math. Anal. Appl. 355, 22–32 2009), plays a central role in Geometric Function Theory. For instance, we just mention here that maps of bi-Sobolev type are strictly related to the notion of mappings of finite distortion; see, among others, the papers (Hencl and Koskela 2014) Pratelli (Nonlinear Anal. 154, 258–268 2017). [ABSTRACT FROM AUTHOR]

We consider the following class of fractional Schrödinger equations:

The aim of this paper is to establish the existence of multiple solutions for a perturbed Kirchhoff-type problem depending on two real parameters. More precisely, we show that an appropriate oscillating behaviour of the nonlinear part, even under small perturbations, ensures the existence of at least three nontrivial weak solutions. Our approach combines variational methods with properties of nonlocal fractional operators.

Some existence results for a parametric Dirichlet problem defined on the Sierpiński fractal are proved. More precisely, a critical point result for differentiable functionals is exploited in order to prove the existence of a well-determined open interval of positive eigenvalues for which the problem admits at least one non-trivial weak solution.

We investigate complete hypersurfaces with some positive higher order mean curvature in a semi-Riemannian warped product space. Under standard curvature conditions on the ambient space and appropriate constraints on the higher order mean curvatures, we establish rigidity and nonexistence results via Liouville type results and suitable maximum principles related to the divergence of smooth vector fields on a complete noncompact Riemannian manifold. Applications to standard warped product models, like the Schwarzschild, Reissner-Nordström and pseudo-hyperbolic spaces, as well as steady state type spacetimes, are given and a particular study of entire graphs is also presented. [ABSTRACT FROM AUTHOR]

In this paper we deal with a bifurcation result for the following parametric one-dimensional mean curvature problem:

In this paper by exploiting critical point theory, the existence of two distinct nontrivial solutions for a nonlinear algebraic system with a parameter is established. Our goal is achieved by requiring an appropriate behavior of the nonlinear term f at zero and at infinity. Some applications to difference equations are also presented.

Our purpose in this paper is to study solitons of the spacelike mean curvature flow in a generalized Robertson-Walker (GRW) spacetime -IxfMn. Under suitable constraints on the warping function f and on the curvatures of the Riemannian fiber Mn, we apply suitable maximum principles in order to obtain nonexistence and uniqueness results concerning these solitons. Applications to standard models of GRW spacetimes, namely, the Einstein-de Sitter spacetime, steady state type spacetimes, de Sitter and anti-de Sitter spaces, are given. Furthermore, we establish new Calabi-Bernstein type results related to entire spacelike mean curvature flow graphs constructed over the Riemannian fiber of the ambient spacetime. [ABSTRACT FROM AUTHOR]