Papers by João Marcos Do Ó
Mathematische Nachrichten, Sep 12, 2016
For the following singularly perturbed problem urn:x-wiley:0025584X:media:mana201500074:mana20150... more For the following singularly perturbed problem urn:x-wiley:0025584X:media:mana201500074:mana201500074-math-0001we construct a solution which concentrates at several given isolated positive local minimum components of V as . Here, the nonlinearity f is of critical growth. Moreover, the monotonicity of and the so‐called Ambrosetti–Rabinowitz condition are not required.
arXiv (Cornell University), Mar 30, 2015
We consider a fractional Schrödinger-Poisson system with a general nonlinearity in subcritical an... more We consider a fractional Schrödinger-Poisson system with a general nonlinearity in subcritical and critical case. The Ambrosetti-Rabinowitz condition is not required. By using a perturbation approach, we prove the existence of positive solutions. Moreover, we study the asymptotics of solutions for a vanishing parameter.
Zeitschrift für Angewandte Mathematik und Physik, Sep 5, 2015
AbstractIn this paper, we deal with the following singularly perturbed elliptic problem $$\begin{... more AbstractIn this paper, we deal with the following singularly perturbed elliptic problem $$\begin{array}{ll}-\varepsilon^2\Delta u+V(x)u=f(u),\quad u \in H^1(\mathbb{R}^2),\end{array}$$-ε2Δu+V(x)u=f(u),u∈H1(R2),where f(s) has critical growth of Trudinger–Moser type. In this paper, we construct a localized bound-state solution concentrating at an isolated component of the positive local minimum points of V as $${\varepsilon \rightarrow 0}$$ε→0 under certain conditions on f(s). Our results complete the analysis made in Byeon et al. (Commun Partial Differ Equ 33: 1113–1136, 2008) for the two-dimensional case, in the sense that, in that paper only the subcritical growth was considered.
Proceedings of the Edinburgh Mathematical Society, Jul 17, 2018
We are concerned with the following Kirchhoff type equation −ε 2 M ε 2−N R N |∇u| 2 dx ∆u + V (x)... more We are concerned with the following Kirchhoff type equation −ε 2 M ε 2−N R N |∇u| 2 dx ∆u + V (x)u = f (u), x ∈ R N , N ≥ 2, where M ∈ C(R + , R +), V ∈ C(R N , R +) and f (s) is of critical growth. In this paper, we construct a localized bound state solution concentrating at a local minimum of V as ε → 0 under certain conditions on f (s), M and V. In particular, the monotonicity of f (s)/s and the Ambrosetti-Rabinowitz condition are not required.
arXiv (Cornell University), Jun 25, 2019
In this paper we study the following class of nonlocal problems involving Caffarelli-Kohn-Nirenbe... more In this paper we study the following class of nonlocal problems involving Caffarelli-Kohn-Nirenberg type critical growth L(u) − λh(x)|x| −2(1+a) u = µ f (x)|u| q−2 u + |x| −pb |u| p−2 u in R N , where h(x) ≥ 0, f (x) is a continuous function which may change sign, λ, µ are positive real parameters and 1 < q < 2, 4 < p = 2N/[N + 2(b − a) − 2], 0 ≤ a < b < a + 1 < N/2, N ≥ 3. Here L(u) = −M R N |x| −2a |∇u| 2 dx div(|x| −2a ∇u) and the function M : R + ∪ {0} → R + is exactly as in the Kirchhoff model, given by M(t) = α + βt, α, β > 0. Using the idea of the constrained minimization on Nehari manifold we show the existence of at least two positive solutions for suitable choices of λ and µ.
arXiv (Cornell University), Nov 11, 2018
In this paper, we study the following nonlocal nonautonomous Hamiltonian system on whole R
Topological Methods in Nonlinear Analysis, 2019
In this paper we prove the existence of positive ground state solution for a class of linearly co... more In this paper we prove the existence of positive ground state solution for a class of linearly coupled systems involving Kirchhoff-Schr\"odinger equations. We study the subcritical and critical case. Our approach is variational and based on minimization technique over the Nehari manifold. We also obtain a nonexistence result using a Pohozaev identity type.
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.
Springer eBooks, 2014
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.
Proceedings of the American Mathematical Society, Mar 30, 2016
In this paper we study some weighted Trudinger-Moser type problems, namely where B ⊂ R 2 represen... more In this paper we study some weighted Trudinger-Moser type problems, namely where B ⊂ R 2 represents the open unit ball centered at zero in R 2 and H stands either for H 1 0,rad (B) or H 1 rad (B). We present the precise balance between h(r) and F (t) that guarantees s F,h to be finite. We prove that s F,h is attained up to the h(r)-radially critical case. In particular, we solve two open problems in the critical growth case.
Journal of Differential Equations, Feb 1, 2010
In this paper we establish the existence of standing wave solutions for quasilinear Schrödinger e... more In this paper we establish the existence of standing wave solutions for quasilinear Schrödinger equations involving critical growth. By using a change of variables, the quasilinear equations are reduced to semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v. In the proof that v is nontrivial, the main tool is the concentration-compactness principle due to P.L. Lions together with some classical arguments used by H. Brezis and L. Nirenberg (1983) in [9].
arXiv (Cornell University), Apr 27, 2015
We investigate the existence of ground state solutions for a class of nonlinear scalar field equa... more We investigate the existence of ground state solutions for a class of nonlinear scalar field equations defined on whole real line, involving a fractional Laplacian and nonlinearities with Trudinger-Moser critical growth. We handle the lack of compactness of the associated energy functional due to the unboundedness of the domain and the presence of a limiting case embedding.
Journal of Differential Equations, Jun 1, 2015
Abstract We prove a sharp form of the Trudinger–Moser inequality for the Sobolev space H 1 , n ( ... more Abstract We prove a sharp form of the Trudinger–Moser inequality for the Sobolev space H 1 , n ( R n ) . The sharpness comes from the introduction of an extra factor ‖ u ‖ n n in the classical Trudinger–Moser inequality. Let l ( α ) : = sup { u ∈ H 1 , n ( R n ) : ‖ u ‖ 1 , n = 1 } ∫ R n Φ ∘ ν α ( u ) d x , where Φ ( t ) : = e t − ∑ i = 0 n − 1 t i i ! and ν α ( u ) : = β n ( 1 + α ‖ u ‖ n n ) 1 / ( n − 1 ) | u | n / ( n − 1 ) . The main results read: (1) for 0 ≤ α 1 we have l ( α ) ∞ , (2) for α > 1 , l ( α ) = ∞ and (3) moreover, we prove that for 0 ≤ α 1 , an extremal function for l ( α ) exists.
Differential and Integral Equations, 2020
Advances in Differential Equations
Proceedings of the Edinburgh Mathematical Society, 2022
We consider the following class of quasilinear Schrödinger equations proposed in plasma physics a... more We consider the following class of quasilinear Schrödinger equations proposed in plasma physics and nonlinear optics $-\Delta u+V(x)u+\frac {\kappa }{2}[\Delta (u^{2})]u=h(u)$ in the whole two-dimensional Euclidean space. We establish the existence and qualitative properties of standing wave solutions for a broader class of nonlinear terms $h(s)$ with the critical exponential growth. We apply the dual approach to obtain solutions in the usual Sobolev space $H^{1}(\mathbb {R}^{2})$ when the parameter $\kappa >0$ is sufficiently small. Minimax techniques, Trudinger–Moser inequality and the Nash–Moser iteration method play an essential role in establishing our results.
Mathematical Methods in the Applied Sciences, 2021
The purpose of this paper is to investigate the existence and multiplicity of weak solutions for ... more The purpose of this paper is to investigate the existence and multiplicity of weak solutions for a class of nonlocal problems involving the fractional magnetic operator and nonlinearities which may have critical or subcritical growth in the sense of Trudinger–Moser inequality. By using variational methods based on minimax argument and assuming suitable conditions on the nonlinearity, we address essential geometric cases of the associated energy functional, such as the mountain pass geometry and the linking geometry. Some of the main results established in this work are new even when the magnetic potential is equal to zero, which corresponds to the usual fractional Laplacian operator.
Journal of Differential Equations, 2015
We prove a sharp form of the Trudinger-Moser inequality for the Sobolev space H 1,n (R n ). The s... more We prove a sharp form of the Trudinger-Moser inequality for the Sobolev space H 1,n (R n ). The sharpness comes from the introduction of an extra factor u n n in the classical Trudinger-Moser inequality. Let where Φ(t) := e tn-1 i=0 t i i! and ν α (u) := β n (1 + α u n n ) 1/(n-1) |u| n/(n-1) . The main results read: (1) for 0 ≤ α < 1 we have (α) < ∞, (2) for α > 1, (α) = ∞ and (3) moreover, we prove that for 0 ≤ α < 1, an extremal function for (α) exists.
Proceedings of the American Mathematical Society, 2016
In this paper we study some weighted Trudinger-Moser type problems, namely s F , h = sup u ∈ H , ... more In this paper we study some weighted Trudinger-Moser type problems, namely s F , h = sup u ∈ H , ‖ u ‖ H = 1 ∫ B F ( u ) h ( | x | ) d x , \begin{equation*} \displaystyle {s_{F,h} = \sup _{u \in H, \, \| u\|_H =1 } \int _{B} F(u) h(|x|) dx}, \end{equation*} where B ⊂ R 2 B \subset {\mathbb R}^2 represents the open unit ball centered at zero in R 2 {\mathbb R}^2 and H H stands either for H 0 , rad 1 ( B ) H^1_{0, \textrm {rad}}(B) or H rad 1 ( B ) H^1_{\textrm {rad}}(B) . We present the precise balance between h ( r ) h(r) and F ( t ) F(t) that guarantees s F , h s_{F,h} to be finite. We prove that s F , h s_{F,h} is attained up to the h ( r ) h(r) -radially critical case. In particular, we solve two open problems in the critical growth case.
Advanced Nonlinear Studies, 2015
In this paper we study the existence of weak positive solutions for the following class of quasil... more In this paper we study the existence of weak positive solutions for the following class of quasilinear Schrödinger equations −Δu + V(x)u − [Δ(u2)]u = h(u) in ℝN, where h satisfies some “mountain-pass” type assumptions and V is a nonnegative continuous function. We are interested specially in the case where the potential V is neither bounded away from zero, nor bounded from above. We give a special attention to the case when V may eventually vanish at infinity. Our arguments are based on penalization techniques, variational methods and Moser iteration scheme.
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Papers by João Marcos Do Ó