arXiv subjects
Publications and source records attributed to J. M. Do Ó.
This study establishes Liouville-type theorems for indefinite quasilinear elliptic equations in the upper half-space. Additionally, we demonstrate the existence of solutions for this class of problems using the fibering method. Our approach relies on a novel weighted Sobolev embedding developed for the upper half-space.
In this paper we study the problem $-\mathrm{div}(ρ(x_N)\nabla u)=a|u|^{p-2}u$ in $\mathbb{R}^N_+$, $-\partial u/\partial x_N=b|u|^{q-2}u$ in $\mathbb{R}^{N-1}$ where $a,b \in \mathbb{R}$, $p,q\in (1,\infty)$ and $ρ$ is a positive weight. We establish regularity results for weak solutions and, using a variational approach combined with a new Pohozaev-type identity, we show that the introduction of the weighted operator $-\mathrm{div}(ρ(x_N)\nabla u)$ can reverse the known solvability behavior of the classical Laplacian case. Specifically, we identify regimes where the problem admits solutions despite nonexistence for the corresponding case with $-Δ$, and vice versa, thus inverting the classical existence and nonexistence results.
This article investigates the existence and properties of ground state solutions to the following nonlocal Hamiltonian elliptic system: \begin{align*} \begin{cases} (-Δ)^\frac12 u +V_0 u =g(v),~x\in \mathbb{R} (-Δ)^\frac12 v +V_0 v =f(u),~x\in \mathbb{R}, \end{cases} \end{align*} where $(-Δ)^\frac12$ is the square root Laplacian operator, $V_0 >0$ and $f,~g$ have critical exponential growth in $\mathbb{R}$. Using minimization technique over some generalized Nehari manifold, we show that the set $\mathcal{S}$ of ground state solutions is non empty. Moreover for $(u,v) \in \mathcal{S}$, $u,~v$ are uniformly bounded in $L^\infty(\mathbb{R})$ and uniformly decaying at infinity. We also show that the set $\mathcal{S}$ is compact in $H^\frac12(\mathbb{R}) \times H^\frac12(\mathbb{R})$ up to translations. Furthermore under locally lipschitz continuity of $f$ and $g$ we obtain a suitable Pohožaev type identity for any $(u,v) \in \mathcal{S}$. We deduce the existence of semi-classical ground state solutions to the singularly perturbed system \begin{align*} \begin{cases} ε(-Δ)^\frac12 φ+V(x) φ=g(ψ),~x\in \mathbb{R} ε(-Δ)^\frac12 ψ+V(x) ψ=f(φ),~x\in \mathbb{R}, \end{cases} \end{align*} where $ε>0$ and $V \in C(\mathbb{R})$ satisfy the assumption $(V)$ given below (see Section 1). Finally as $ε\rightarrow 0$, we prove the existence of minimal energy solutions which concentrate around the closest minima of the potential $V$.