Multi-Bump Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity: The Topological Effect of Potential Wells

In this article, we are interested in multi-bump solutions of the singularly perturbed problem - ε 2 ⁢ Δ ⁢ v + V ⁢ ( x ) ⁢ v = f ⁢ ( v ) in ⁢ ℝ N . -\varepsilon^{2}\Delta v+V(x)v=f(v)\quad\text{in }\mathbb{R}^{N}. Extending previous results, we prove the existence of multi-bump solutions for an optimal class of nonlinearities f satisfying the Berestycki–Lions conditions and, notably, also for more general classes of potential wells than those previously studied. We devise two novel topological arguments to deal with general classes of potential wells. Our results prove the existence of multi-bump solutions in which the centers of bumps converge toward potential wells as ε → 0 {\varepsilon\rightarrow 0} . Examples of potential wells include the following: the union of two compact smooth submanifolds of ℝ N {\mathbb{R}^{N}} where these two submanifolds meet at the origin and an embedded topological submanifold of ℝ N {\mathbb{R}^{N}} .

On the Algebra of Operators Corresponding to the Union of Smooth Submanifolds

For a pair of smooth transversally intersecting submanifolds in some enveloping smooth manifold, we study the algebra generated by pseudodifferential operators and (co)boundary operators corresponding to submanifolds. We establish that such an algebra has 18 types of generating elements. For operators from this algebra, we define the concept of symbol and obtain the composition formula.

Lipschitz–Killing curvatures and polar images

We relate the Lipschitz–Killing measures of a definable set X ⊂ ℝn in an o-minimal structure to the volumes of generic polar images. For smooth submanifolds of ℝn, such results were established by Langevin and Shifrin. Then we give infinitesimal versions of these results. As a corollary, we obtain a relation between the polar invariants of Comte and Merle and the densities of generic polar images.