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SEMI-CLASSICAL LIMIT FOR THE ONE DIMENSIONAL NONLINEAR SCHRÖDINGER EQUATION
Preview AbstractIn this article we study standing wave solutions for the nonlinear Schrödinger equation, which correspond to solutions of the equation
We are interested in solutions having a prescribed L2 norm, exhibiting high oscillatory behavior and concentrating in an interval. We prove existence of such solutions and we study their asymptotic behavior as the parameter ε goes to zero. In particular we obtain an envelope function describing the amplitude of the solutions and we identify their asymptotic density.
THE COMPETITION BETWEEN INCOMING AND OUTGOING FLUXES IN AN ELLIPTIC PROBLEM
Preview AbstractIn this work, we consider existence and uniqueness of positive solutions to the elliptic equation -Δu = λu in Ω, with the nonlinear boundary conditions
on Γ1, 
on Γ2, where Ω is a smooth bounded domain, ∂Ω = Γ1 ∪ Γ2, 
, ν is the outward unit normal, p, q > 0 and λ is a real parameter. We obtain a complete picture of the bifurcation diagram of the problem, depending on the values of p, q and the parameter λ. Our proofs are based on different techniques: variational arguments, bifurcation techniques or comparison arguments, depending on the range of parameters considered.EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO ELLIPTIC PROBLEMS WITH SUBLINEAR MIXED BOUNDARY CONDITIONS
Preview AbstractIn this work, we consider a class of semilinear elliptic problems with nonlinear boundary conditions of mixed type. Under some monotonicity properties of the nonlinearities involved, we show that positive solutions are unique, and that their existence is characterized by the sign of some associated eigenvalues. One of the most important contributions of this work relies on the fact that we deal with boundary conditions of the form ∂u/∂ν = g(x,u) on Γ and u = 0 on Γ', where ν is the outward unit normal to Γ while Γ,Γ' are open, Γ ∩ Γ' = ∅,
, but 
need not be disjoint.Some global results for a class of homogeneous nonlocal eigenvalue problems
Preview AbstractThis paper studies the global bifurcation phenomenon for the following homogeneous nonlocal eigenvalue problem
{−(∫Ω|∇u|2dx)Δu=λu3+h(x,u,λ)in Ω,u=0on Ω.Under some natural hypotheses on h and Ω, we show that (μ1,0) is a bifurcation point of the nontrivial solution set of the above problem. As application of the above result, we determine the interval of λ, in which there exist positive solutions for the following Kirchhoff type problem{−(∫Ω|∇u|2dx)Δu=λf(x,u)in Ω,u=0on ∂Ω,where f is asymptotically 3-linear at zero and infinity. Our results provide a positive answer to an open problem. Moreover, we also study the spectral structure for a homogeneous nonlocal eigenvalue problem.