417 research outputs found
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
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
Nodal solutions for the Choquard equation
We consider the general Choquard equations where is a
Riesz potential. We construct minimal action odd solutions for and minimal action nodal solutions for
. 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
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
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