A detailed study of the notions of convexity for a hypersurface in a Finsler manifold is carried out. In particular, the infinitesimal and local notions of convexity are shown to be equivalent. Our approach differs from Bishop's one in his classical result (Bishop, Indiana Univ Math J 24:169-172, 1974) for the Riemannian case. Ours not only can be extended to the Finsler setting but it also reduces the typical requirements of differentiability for the metric and it yields consequences on the multiplicity of connecting geodesics in the convex domain defined by the hypersurface.Comment: 22 pages, AMSLaTex. Typos corrected, references update

The aim of this paper is t o study the geodesic connectedness of a complete static Lorentzian manifold (M.〈·, ·〉

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Abstract. The aim of this paper is investigating the existence and the multiplicity of weak solutions of the quasilinear elliptic problemwith smooth boundary ∂Ω and the nonlinearity g behaves as u p−1 at infinity. The main tools of the proof are some abstract critical point theorems in Bartolo et al. (Nonlinear Anal. 7: 981-1012, 1983, but extended to Banach spaces, and two sequences of quasi-eigenvalues for the p-Laplacian operator as in Candela and Palmieri (Calc.

The aim of this paper is investigating the existence and multiplicity of weak solutions to non-local equations involving the magnetic fractional Laplacian, when the nonlinearity is subcritical and asymptotically linear at infinity. We prove existence and multiplicity results by using variational tools, extending to the magnetic local and non-local setting some known results for the classical and the fractional Laplace operators.