Papers by Everaldo Medeiros
Topological Methods in Nonlinear Analysis, 2009
In this paper, we study multiplicity of weak solutions for the following class of quasilinear ell... more In this paper, we study multiplicity of weak solutions for the following class of quasilinear elliptic problems of the form where Ω is a bounded domain in R n with smooth boundary ∂Ω, 1 < q < 2 < p ≤ n, λ is a real parameter, ∆pu = div(|∇u| p-2 ∇u) is the p-Laplacian and the nonlinearity g(u) has subcritical growth. The proofs of our results rely on some linking theorems and critical groups estimates.
Proceedings of the Edinburgh Mathematical Society, 2016
We consider the semilinear problemwhere λ is a positive parameter and f has exponential critical ... more We consider the semilinear problemwhere λ is a positive parameter and f has exponential critical growth. We first establish the existence of a non-zero weak solution. Then, by assuming that f is odd, we prove that the number of solutions increases when the parameter λ becomes large. In the proofs we apply variational methods in a suitable weighted Sobolev space consisting of functions with rapid decay at infinity.
Partial Differential Equations and Applications, 2022
Asymptotic Analysis, 2022
In this paper we deal with the following class of nonlinear Schrödinger equations − Δ u + V ( | x... more In this paper we deal with the following class of nonlinear Schrödinger equations − Δ u + V ( | x | ) u = λ Q ( | x | ) f ( u ) , x ∈ R 2 , where λ > 0 is a real parameter, the potential V and the weight Q are radial, which can be singular at the origin, unbounded or decaying at infinity and the nonlinearity f ( s ) behaves like e α s 2 at infinity. By performing a variational approach based on a weighted Trudinger–Moser type inequality proved here, we obtain some existence and multiplicity results.
Journal of Mathematical Analysis and Applications, 2018
We study the existence of solutions for the nonlinear Schrödinger equation −Δu + V (x)u = f (x, u... more We study the existence of solutions for the nonlinear Schrödinger equation −Δu + V (x)u = f (x, u) in R 2 , where the potential V is 1-periodic, 0 lies in a spectral gap from the spectrum of the Schrödinger operator S = −Δ + V and the nonlinearity f (x, t) has exponential growth in the sense of Trudinger-Moser. The main feature here is that f (x, t) is allowed to be both periodic and nonperiodic in the x variable. Our proofs rely on a linking theorem and the Lions concentration compactness principle.
Journal of Differential Equations, 2021
Abstract In this paper we consider existence, nonexistence and multiplicity of solutions for a cl... more Abstract In this paper we consider existence, nonexistence and multiplicity of solutions for a class of indefinite quasilinear elliptic problems in the upper half-space involving weights in anisotropic Lebesgue spaces. One of our basic tools consists in a Hardy type inequality proved in the present paper that allows us to establish Sobolev embeddings into Lebesgue spaces with weights in anisotropic Lebesgue spaces.
Matemática Contemporânea, 2007
We study the existence of positive and negative weak solutions for the equation −∆ p u + V (x)|u|... more We study the existence of positive and negative weak solutions for the equation −∆ p u + V (x)|u| p−2 u = λf (u) in R N , where −∆ p u = div(|∇u| p−2 ∇u) is the p-Laplacian operator, 1 < p < N , λ is a positive real parameter and the potential V : R N → R is bounded from below for a positive constant and "large" at infinity. It is assumed that the nonlinearity f : R → R is continuous and just superlinear in a neighborhood of the origin.
Zeitschrift für angewandte Mathematik und Physik, 2015
This paper is concerned with the study of solutions for a class of Hamiltonian elliptic system in... more This paper is concerned with the study of solutions for a class of Hamiltonian elliptic system involving nonlinear Schrödinger equations with critical growth. We use an approach based on a dual variational formulation to prove existence and multiplicity of solutions for sufficiently small values of the parameter.
Journal of Differential Equations, 2014
ABSTRACT In line with the Concentration–Compactness Principle due to P.-L. Lions [19], we study t... more ABSTRACT In line with the Concentration–Compactness Principle due to P.-L. Lions [19], we study the lack of compactness of Sobolev embedding of W1,n(Rn)W1,n(Rn), n⩾2n⩾2, into the Orlicz space LΦαLΦα determined by the Young function Φα(s)Φα(s) behaving like eα|s|n/(n−1)−1eα|s|n/(n−1)−1 as |s|→+∞|s|→+∞. In the light of this result we also study existence of ground state solutions for a class of quasilinear elliptic problems involving critical growth of the Trudinger–Moser type in the whole space RnRn.
Journal of Mathematical Analysis and Applications, 2014
We study the existence and multiplicity of solutions for the following class of nonlinear Schrödi... more We study the existence and multiplicity of solutions for the following class of nonlinear Schrödinger equations −∆u + V (| x |)u = Q (| x |)f (u) in R 2 , where V and Q are unbounded or decaying radial potentials and the nonlinearity f (s) has exponential critical growth. The approaches used here are based on a version of the Trudinger-Moser inequality and a minimax theorem.
This paper studies the existence, nonexistence and uniqueness of positive solutions for a class o... more This paper studies the existence, nonexistence and uniqueness of positive solutions for a class of quasilinear equations. We also analyze the behavior of this solutions with respect to two parameters $\kappa$ and $\lambda$ that appears in the equation. The proof of our main results relies on bifurcation techniques, the sub and supersolution method and a construction of an appropriate large solutions.
We establish some multiplicity results for p-sublinear and p-superlinear p-biharmonic problems us... more We establish some multiplicity results for p-sublinear and p-superlinear p-biharmonic problems using Morse theory. AMS (MOS) Subject Classification. Primary 35J60, Secondary 47J10, 47J30, 58E05
Analysis and Topology in Nonlinear Differential Equations, 2014
We investigate the asymptotic behavior of best constants in expanding domains \( \Omega_{\varepsi... more We investigate the asymptotic behavior of best constants in expanding domains \( \Omega_{\varepsilon}= \varepsilon^{-1}\Omega(\varepsilon > 0),\) for the Sobolev trace embedding \( H^{1}(\Omega_{\varepsilon})\hookrightarrow L^{p}(\partial\Omega_{\varepsilon}),\quad 1\leq p \leq 2_{*}:=2(N-1)/(N-2).\) We provide a detailed description of the shape for extremal \( u_{\varepsilon} \) of the best constant and prove that the maximum of \( u_{\varepsilon} \) is achieved on the boundary \( \partial_{\Omega} \), and concentrates around a maximum point of the mean curvature of the boundary. The nonexistence of extremal is obtained for large \( {\varepsilon} \).
Nonlinear Analysis: Theory, Methods & Applications, 2010
ABSTRACT We establish some multiplicity results for a class of boundary value problems involving ... more ABSTRACT We establish some multiplicity results for a class of boundary value problems involving the Hardy–Sobolev operator using Morse theory.
Journal of Mathematical Analysis and Applications, 2008
This paper deals with a semilinear Schrödinger equation whose nonlinear term involves a positive ... more This paper deals with a semilinear Schrödinger equation whose nonlinear term involves a positive parameter λ and a real function f (u) which satisfies a superlinear growth condition just in a neighborhood of zero. By proving an a priori estimate (for a suitable class of solutions) we are able to avoid further restrictions on the behavior of f (u) at infinity in order to prove, for λ sufficiently large, the existence of one-sign and sign-changing solutions. Minimax methods are employed to establish this result.
Journal of Differential Equations, 2009
In this paper, Ekeland variational principle, mountain-pass theorem and a suitable Trudinger-Mose... more In this paper, Ekeland variational principle, mountain-pass theorem and a suitable Trudinger-Moser inequality are employed to establish sufficient conditions for the existence of solutions of quasilinear nonhomogeneous elliptic partial differential equations of the form − N u + V (x)|u| N−2 u = f (x, u) + εh(x) in R N , N 2, where V : R N → R is a continuous potential, f : R N × R → R behaves like exp(α|u| N/(N−1)) when |u| → ∞ and h ∈ (W 1,N (R N)) * = W −1,N , h ≡ 0. As an application of this result we have existence of two positive solutions for the following elliptic problem involving critical growth − u + V (x)u = λu e u 2 − 1 + εh(x) in R 2 , where λ > 0 is large, ε > 0 is a small parameter and h ∈ H −1 (R 2), h 0.
Communications in Contemporary Mathematics, 2000
This paper deals with the following class of quasilinear elliptic problems in radial form [Formul... more This paper deals with the following class of quasilinear elliptic problems in radial form [Formula: see text] where α, β, δ, ℓ, γ, q are given real numbers, λ > 0 is a parameter and 0 < R < ∞. Some results on the existence of positive solutions are obtained by combining the Mountain Pass Theorem with an argument used by Brézis and Nirenberg to overcome the lack of compactness due to the presence of critical Sobolev exponents.
ABSTRACT We establish some multiplicity results for a class of boundary value problems involving ... more ABSTRACT We establish some multiplicity results for a class of boundary value problems involving the Hardy–Sobolev operator using Morse theory.
Annales de l'Institut Henri Poincare (C) Non Linear …, 2003
The main results of this paper establish, via the variational method, the multiplicity of solutio... more The main results of this paper establish, via the variational method, the multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents under the presence of symmetry. The concentration-compactness principle allows to prove that the Palais-Smale condition is satisfied below a certain level.
Journal of Differential Equations, 2010
In this paper we study existence and properties of solutions of the problem ∆w = 0 on the half-sp... more In this paper we study existence and properties of solutions of the problem ∆w = 0 on the half-space R N + with nonlinear boundary condition ∂w/∂η + w = |w| p−2 w where 2 < p < 2(N − 1)/(N − 2) and N ≥ 3. We obtain a ground state solution w = w(x1, ..., xN−1, t) which is radial and has exponential decay in the first N − 1 variables. Moreover, w has sharp polynomial decay in the variable t.
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Papers by Everaldo Medeiros