The Emergence of Law Consultants

In this paper we study a slightly subcritical Choquard problem on a bounded domain A. We prove that the number of positive solutions depends on the topology of the domain. In particular when the exponent of the nonlinearity approaches the critical one, we show the existence of cat (A) + 1 solutions. Here cat (A) denotes the Lusternik–Schnirelmann category

A multiplicity result for double singularly perturbed elliptic systems

We show that the number of low energy solutions of a double singularly perturbed Schroedinger Maxwell system type on a smooth 3 dimensional manifold (M,g) depends on the topological properties of the manifold. The result is obtained via Lusternik Schnirelmann category theory

Nodal solutions for the Choquard equation

We consider the general Choquard equations $$-\Delta u + u = (I_\alpha \ast |u|^p) |u|^{p - 2} u$$ where $I_\alpha$ is a Riesz potential. We construct minimal action odd solutions for $p \in (\frac{N + \alpha}{N}, \frac{N + \alpha}{N - 2})$ and minimal action nodal solutions for $p \in (2,\frac{N + \alpha}{N - 2})$. We introduce a new minimax principle for least action nodal solutions and we develop new concentration-compactness lemmas for sign-changing Palais--Smale sequences. The nonlinear Schr\"odinger equation, which is the nonlocal counterpart of the Choquard equation, does not have such solutions.Comment: 23 pages, revised version with additional details and symmetry properties of odd solution

Low energy solutions for singularly perturbed coupled nonlinear systems on a Riemannian manifold with boundary

Let (M,g) be asmooth, compact Riemannian manifold with smooth boundary, with n= dim M= 2,3. We suppose the boundary of M to be a smooth submanifold of M with dimension n-1. We consider a singularly perturbed nonlinear system, namely Klein-Gordon-Maxwell-Proca system, or Klein-Gordon-Maxwell system of Scrhoedinger-Maxwell system on M. We prove that the number of low energy solutions, when the perturbation parameter is small, depends on the topological properties of the boundary of M, by means of the Lusternik Schnirelmann category. Also, these solutions have a unique maximum point that lies on the boundary

Positive solutions for singularly perturbed nonlinear elliptic problem on manifolds via Morse theory

Given (M, g0) we consider the problem -{\epsilon}^2Delta_{g0+h}u + u = (u+)^{p-1} with ({\epsilon}, h) \in (0, {\epsilon}0) \times B{\rho}. Here B{\rho} is a ball centered at 0 with radius {\rho} in the Banach space of all Ck symmetric covariant 2-tensors on M. Using the Poincar\'e polynomial of M, we give an estimate on the number of nonconstant solutions with low energy for ({\epsilon}, h) belonging to a residual subset of (0, {\epsilon}0) \times B{\rho}, for ({\epsilon}0, {\rho}) small enough