Abstract. In this paper we deal with a bifurcation result for the following parametric one-dimensional mean curvature problem: where and is a Carathéodory function vanishing at zero. More precisely, a critical point theorem (local minimum result) for differentiable functionals is exploited in order to prove that the above problem admits at least one nontrivial and nonnegative weak solution under an asymptotical behaviour of the nonlinear datum at zero. A concrete example of an application is then presented.

Abstract. Let be a measure space. We show that if and there is at least one nonnegative function (not necessarily ) such that then and This property is trivial when but not when even if μ is σ-finite, which is not assumed. The discrete case when is spelled out in the last section.

Abstract. We consider a parabolic problem with degeneracy in the interior of the spatial domain, and we focus on Carleman estimates for the associated adjoint problem. The novelty of interior degeneracy does not let us adapt previous Carleman estimates to our situation. As an application, observability inequalities are established.

Abstract. Aim of this paper is to study the following elliptic equation driven by a general non-local integrodifferential operator such that in , in , where , is an open bounded set of ℝ n , , with Lipschitz boundary, λ is a positive real parameter, is a fractional critical Sobolev exponent, while is the non-local integrodifferential operator As a concrete example, we consider the case when , which gives rise to the fractional Laplace operator . In this framework, in the existence result proved along the paper, we show that our problem admits a non-trivial solution for any , provided and λ is different from the eigenvalues of . This result may be read as the non-local fractional counterpart of the one obtained by Capozzi, Fortunato and Palmieri and by Gazzola and Ruf for the classical Laplace equation with critical nonlinearities. In this sense the present work may be seen as the extension of some classical results for the Laplacian to the case of non-local fractional operators.

Abstract. The present paper deals with an anisotropic Kirchhoff problem under homogeneous Dirichlet boundary conditions, set in a bounded smooth domain of ℝ N ( ). The problem studied is a stationary version of the original Kirchhoff equation, involving the anisotropic -Laplacian operator, in the framework of the variable exponent Lebesgue and Sobolev spaces. The question of the existence of weak solutions is treated. Applying the Mountain Pass Theorem of Ambrosetti and Rabinowitz, the existence of a nontrivial weak solution is obtained in the anisotropic variable exponent Sobolev space , provided that the positive parameter λ that multiplies the nonlinearity f is small enough.

Abstract. We prove a version of the Trudinger–Moser inequality in the hyperbolic space ℍ N , which gives a sharper version of the Trudinger–Moser inequality on the Euclidean unit ball, as well as a hyperbolic space version of the Onofri inequality, and prove the existence of extremal functions to some related problems.

Abstract. Given a Carnot–Carathéodory space with associated frame of vector fields , we derive the subelliptic ∞-Laplace system for mappings , which reads in the limit of the subelliptic p-Laplacian as . Here is the horizontal gradient and is the projection on its nullspace. Next, we identify the variational principle characterizing the subelliptic ∞-Laplacian system, which is the “Euler–Lagrange PDE” of the supremal functional for an appropriately defined notion of horizontally ∞-minimal mappings. We also establish a maximum principle for for solutions to the subelliptic ∞-Laplacian system. These results extend previous work of the author [J. Differential Equations 253 (2012), no. 7, 2123–2139; Proc. Amer. Math. Soc., to appear] on vector-valued calculus of variations in L ∞ from the Euclidean to the subelliptic setting.

Abstract. We consider a general class of eigenvalue problems where the leading elliptic term corresponds to a convex homogeneous energy function that is not necessarily differentiable. We derive a strong maximum principle and show uniqueness of the first eigenfunction. Moreover we prove the existence of a sequence of eigensolutions by using a critical point theory in metric spaces. Our results extend the eigenvalue problem of the p-Laplace operator to a much more general setting.

Abstract. In this work we consider the p -Laplacian type parabolic equation with Dirichlet boundary condition and establish the existence of weak solutions using Leray–Schauder's fixed point theorem and semi-discretization process.