Paulo Cesar Carrião

Existence of Homoclinic Orbits for Asymptotically Periodic Systems Involving Duffing-Like Equation

Abstract We are concerned with perturbations of the Hamiltonian system of the type q-L(t) q +W q (t,q)=0, t ϵ R , where q = (q1, … , qN) e R N, W ϵ C1 ( R × R N, R ), and L(t) ϵ C ( R , R N2) is a positive definite symmetric matrix. Variational arguments are used to prove the existence of homoclinic solutions for system (HS).

Multiplicity of solutions for a class of quasilinear problem in exterior domains with Neumann conditions

We study the existence and multiplicity of solutions for a class of quasilinear elliptic problem in exterior domain with Neumann boundary conditions.

On the existence of positive solutions of a perturbed Hamiltonian system in Rn

Using the Legendre–Fenchel transformation and the Mountain Pass Theorem due to Ambrosetti and Rabinowitz, we establish an existence result for perturbations of periodic and asymptotically periodic semilinear Hamiltonian systems of the type (PW)−Δu+u=W2(x)|v|p−1vinRN,−Δv+v=W1(x)|u|q−1uinRN,u(x),v(x)→0as|x|→∞,u>0,v>0inRN,N⩾2. Here, the numbers p,q>1 are below the critical hyperbola if N⩾3, that is, they satisfy 1/(p+1)+1/(q+1)>(N−2)/N, while no additional restrictions on p and q are required if N=2. The functions Wi, i=1,2, are bounded positive continuous functions.

Soliton Solutions to a Class of Quasilinear Elliptic Equations on ℝ

Abstract This paper is concerned with the existence of positive solutions for a class of quasilinear elliptic equations on ℝ. The results are proved by combining the concentration-compactness principle due to Lions with a minimization approach.

Multiplicity of solutions for critical singular problems

Abstract In this work we deal with the class of critical singular quasilinear elliptic problems in R N of the form (P) − div ( | x | − a p | ∇ u | p − 2 ∇ u ) = α | x | − b q | u | q − 2 u + β | x | − d r k | u | r − 2 u x ∈ R N , where 1 p N , a N / p , a ≤ b a + 1 , α and β are positive parameters, q = q ( a , b ) ≡ N p / [ N − p ( a + 1 − b ) ] and d ∈ R . Moreover, 1 r p ∗ = N p / ( N − p ) and 0 ≤ k ∈ L r ( d − b ) q / ( q − r ) ( R N ) . Multiplicity results are established by combining a version of the concentration–compactness lemma due to Lions with the Krasnoselski genus and the symmetric mountain-pass theorem due to Rabinowitz.

Multiplicity of Solutions For a Convex-concave Problem With a Nonlinear Boundary Condition

Abstract We study the existence of multiple positive solutions for a convex-concave problem, denoted by (Pλμ), with a nonlinear boundary condition involving two critical exponents and two positive parameters λ and μ. We obtain a continuous strictly decreasing function f such that K1 ≡ {(f(μ), μ) : μ ∈ [0, ∞)} divides [0, ∞) × [0, ∞) \ {(0, 0)} in two connected sets K0 and K2 such that problem (Pλμ) has at least two solutions for (λ, μ) ∈ K2, at least one solution for (λ, μ) ∈ K1 and no solution for (λ, μ) ∈ K0.

Remarks on a Class of Neumann Problems Involving Critical Exponents

This paper deals with a class of elliptic problems with double critical exponents involving convex and concave-convex nonlinearities. Existence results are obtained by exploring some properties of the best Sobolev trace constant together with an approach developed by Brezis and Nirenberg.

POSITIVE RADIAL SOLUTIONS FOR A CLASS OF ELLIPTIC SYSTEMS CONCENTRATING ON SPHERES WITH POTENTIAL DECAY

We deal with the existence of positive radial solutions concentrating on spheres for the following class of elliptic system where is a small positive parameter; , and are radially symmetric potentials; Q is a -homogeneous function and p is subcritical, that is, 1 , where is the critical Sobolev exponent for .

A class of elliptic systemsinvolving N-functions

Abstract In this work, we state a result of compactness due to Lions in Orlicz spaces. Wegive an application proving an existence result for a gradient type elliptic systems in ℝ N involving N -functions.