arXiv subjects

Antonio J. Fernández

Publications and source records attributed to Antonio J. Fernández.

21 records · Page 2Linked to original sources

arXiv2018

Nonlinear fractional Laplacian problems with nonlocal "gradient terms"

Let $Ω\subset \mathbb{R}^N$, $N \geq 2$, be a smooth bounded domain. For $s \in (1/2,1)$, we consider a problem of the form \[ \left\{\begin{aligned} (-Δ)^s u & = μ(x)\, \mathbb{D}_s^{2}(u) + λf(x)\,, & \quad \mbox{in} Ω,\\ u & = 0\,, & \quad \mbox{in} \mathbb{R}^N \setminus Ω, \end{aligned} \right. \] where $λ> 0$ is a real parameter, $f$ belongs to a suitable Lebesgue space, $μ\in L^{\infty}(Ω)$ and $\mathbb{D}_s^2$ is a nonlocal "gradient square" term given by \[ \mathbb{D}_s^2 (u) = \frac{a_{N,s}}{2}\mbox{p.v.} \int_{\mathbb{R}^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}} dy \,. \] Depending on the real parameter $λ> 0$, we derive existence and non-existence results. The proof of our existence result relies on sharp Calderón-Zygmund type regularity results for the fractional Poisson equation with low integrability data. We also obtain existence results for related problems involving different nonlocal diffusion terms.

math.AP↗

arXiv2018

A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient

Let $Ω\subset \mathbb R^N$, $N \geq 2$, be a smooth bounded domain. We consider a boundary value problem of the form $$-Δu = c_λ(x) u + μ(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(Ω)\cap L^{\infty}(Ω)$$ where $c_λ$ depends on a parameter $λ\in \mathbb R$, the coefficients $c_λ$ and $h$ belong to $L^q(Ω)$ with $q>N/2$ and $μ\in L^{\infty}(Ω)$. Under suitable assumptions, but without imposing a sign condition on any of these coefficients, we obtain an a priori upper bound on the solutions. Our proof relies on a new boundary weak Harnack inequality. This inequality, which is of independent interest, is established in the general framework of the $p$-Laplacian. With this a priori bound at hand, we show the existence and multiplicity of solutions.

math.AP↗

arXiv2018

Existence and Multiplicity for elliptic p-Laplacian problems with critical growth in the gradient

We consider the boundary value problem $-Δ_p u = λc(x) |u|^{p-2}u + μ(x) |\grad u|^p + h(x)$, $u \in W^{1,p}_0(Ω) \cap L^{\infty}(Ω)$, where $Ω\subset \mathbb R^N$, $N \geq 2$, is a bounded domain with smooth boundary. We assume $c$, $h \in L^q(Ω)$ for some $q > \max\{N/p,1\}$ with $c \gneqq 0$ and $μ\in L^{\infty}(Ω)$. We prove existence and uniqueness results in the coercive case $ λ\leq 0$ and existence and multiplicity results in the non-coercive case $ λ>0$. Also, considering stronger assumptions on the coefficients, we clarify the structure of the set of solutions in the non-coercive case.

math.AP↗