Existence, multiplicity and concentration of bound states for a quasilinear elliptic field equation

In this paper we study existence, multiplicity and concentration of solutions for the following nonlinear field equation where the potential V is positive and W is an appropriate singular function. Here is regarded as a small parameter. Under suitable conditions on V and W we find solutions exhibiting a concentration behaviour at an absolute minimum of V as Such solutions are obtained as local minima for the associated functional; the proofs of our results rely on a careful analysis of the behaviour of minimizing sequences and use arguments inspired by the concentration-compactness principle. Received July 21, 1999; Accepted April 9, 2000 / Published online September 14, 2000     

BOUNDARY CONCENTRATION IN RADIAL SOLUTIONS TO A SYSTEM OF SEMILINEAR ELLIPTIC EQUATIONS

We study concentration phenomena for the system in the unit ball B₁ of ³ with Dirichlet boundaryconditions. Here , , > 0 and p > 1. We prove the existenceof positive radial solutions (, ) such that concentrates ata distance (/2)|log | away from the boundary B₁ as the parameter tends to 0. The approach is based on a combination of Lyapunov–Schmidtreduction procedure together with a variational method.     

On the existence of some new positive interior spike solutions to a semilinear Neumann problem

In this paper we are concerned with the following Neumann problem

     

Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem

We study the following system of Maxwell-Schrödinger equations $ \Delta u - u - \delta u \psi+ f(u)=0, \quad \Delta \psi + u^2 = 0 \mbox{in} {\mathbb R}^N , u, \;\psi > 0, \quad u, \;\psi \to 0 \ \mbox{as} \ |x| \to + \infty, $ where δ > 0, u, ψ : $\psi: {\mathbb R}^N \to {\mathbb R}We study the following system of Maxwell-Schr?dinger equations where δ > 0, u, ψ : , f : , N ≥ 3. We prove that the set of solutions has a rich structure: more precisely for any integer K there exists δ_K > 0 such that, for 0 < δ < δ_K, the system has a solution (u_δ, ψ_δ) with the property that u_δ has K spikes centered at the points . Furthermore, setting , then, as δ → 0, approaches an optimal configuration for the following maximization problem: Subject class: Primary 35B40, 35B45; Secondary 35J55, 92C15, 92C40     

Layered solutions for a semilinear elliptic system in a ball

We consider the following system of Schrödinger-Poisson equations in the unit ball B₁ of R³:

     

Solitary charged waves interacting with the electrostatic field

In this paper we study the semiclassical limit for the following system of Schrödinger-Maxwell equations in the unit ball B₁ of R³: