Volume 13, Issue 4

We study the branch set of a mapping between subsets of ℝn{\mathbb{R}^{n}}, i.e., the set where a given mapping is not defining a local homeomorphism. We construct several sharp examples showing that the branch set or its image can have positive measure.

We characterize lower bounds for the Bakry–Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy and line integrals on the L2{L^{2}}-Wasserstein space, and define a curvature-dimension condition for general metric measure spaces together with a square integrable 1-form in the sense of [N. Gigli, Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc. 251 2018, 1196, 1–161]. This extends the Lott–Sturm–Villani approach for lower Ricci curvature bounds of metric measure spaces. In generalized smooth context, consequences are new Bishop–Gromov estimates, pre-compactness under measured Gromov–Hausdorff convergence, and a Bonnet–Myers theorem that generalizes previous results by Kuwada [K. Kuwada, A probabilistic approach to the maximal diameter theorem, Math. Nachr. 286 2013, 4, 374–378]. We show that N -warped products together with lifted vector fields satisfy the curvature-dimension condition. For smooth Riemannian manifolds, we derive an evolution variational inequality and contraction estimates for the dual semigroup of nonsymmetric diffusion operators. Another theorem of Kuwada [K. Kuwada, Duality on gradient estimates and Wasserstein controls, J. Funct. Anal. 258 2010, 11, 3758–3774], [K. Kuwada, Space-time Wasserstein controls and Bakry–Ledoux type gradient estimates, Calc. Var. Partial Differential Equations 54 2015, 1, 127–161] yields Bakry–Emery gradient estimates.

This paper deals with the existence of multiple solutions for the quasilinear equation -div⁡𝐀⁢(x,∇⁡u)+|u|α⁢(x)-2⁢u=f⁢(x,u) in ℝN,{-\operatorname{div}\mathbf{A}(x,\nabla u)+|u|^{\alpha(x)-2}u=f(x,u)\quad\text% {in ${\mathbb{R}^{N}}$,}} which involves a general variable exponent elliptic operator 𝐀{\mathbf{A}} in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has various types of behavior like |ξ|q⁢(x)-2⁢ξ{|\xi|^{q(x)-2}\xi} for small |ξ|{|\xi|} and like |ξ|p⁢(x)-2⁢ξ{|\xi|^{p(x)-2}\xi} for large |ξ|{|\xi|}, where 1<α⁢(⋅)≤p⁢(⋅)<q⁢(⋅)<N{1<\alpha(\,\cdot\,)\leq p(\,\cdot\,)<q(\,\cdot\,)<N}. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works [A. Azzollini, P. d’Avenia and A. Pomponio, Quasilinear elliptic equations in ℝN\mathbb{R}^{N} via variational methods and Orlicz–Sobolev embeddings, Calc. Var. Partial Differential Equations 49 2014, 1–2, 197–213] and [N. Chorfi and V. D. Rădulescu, Standing wave solutions of a quasilinear degenerate Schrödinger equation with unbounded potential, Electron. J. Qual. Theory Differ. Equ. 2016 2016, Paper No. 37] from cases where the exponents p and q are constant, to the case where p⁢(⋅){p(\,\cdot\,)} and q⁢(⋅){q(\,\cdot\,)} are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the weighting method.

The purpose of this paper is to study the existence of weak solutions for some classes of Schrödinger equations defined on the Euclidean space ℝd{\mathbb{R}^{d}} (d≥3{d\geq 3}). These equations have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using the Palais principle of symmetric criticality and a group-theoretical approach used on a suitable closed subgroup of the orthogonal group O⁢(d){O(d)}. In addition, if the nonlinear term is odd, and d>3{d>3}, the existence of (-1)d+[d-32]{(-1)^{d}+[\frac{d-3}{2}]} pairs of sign-changing solutions has been proved. To make the nonlinear setting work, a certain summability of the L∞{L^{\infty}}-positive and radially symmetric potential term W governing the Schrödinger equations is requested. A concrete example of an application is pointed out. Finally, we emphasize that the method adopted here should be applied for a wider class of energies largely studied in the current literature also in non-Euclidean setting as, for instance, concave-convex nonlinearities on Cartan–Hadamard manifolds with poles.

We prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.