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Aye Chan May

Publications and source records attributed to Aye Chan May.

5 recordsLinked to original sources
arXiv2025

Fractional sublinear Sobolev inequality for $\mathcal{L}-$superharmonic functions

We establish a Sobolev-type inequality in Lorentz spaces for $\mathcal{L}$-superharmonic functions \[ \|u\|_{L^{\frac{nq}{n-αq},t}(\mathbb{R}^n)} \leq c \left\| \frac{u(x) - u(y)}{|x-y|^{\frac{n}{q}+α}} \right\|_{L^{q,t}(\mathbb{R}^n \times \mathbb{R}^n)} \] in the sublinear case $p-1 < q < 1$ and $p-1\leq t\leq \infty$. The nonlocal nonlinear elliptic operator $\mathcal{L}$ is modeled from the fractional $p$-Laplacian $(- Δ_{p})^α $ with $0 < α< 1$ and $1<p<2$. Related Gagliardo-Nirenberg interpolation for $\mathcal{L}$-superharmonic functions is also derived.

math.AP↗
arXiv2025

Nonlocal Sublinear Elliptic Problems Involving Measures

We study Dirichlet problems for fractional Laplace equations of the form $(-Δ)^{\fracα{2}} u = f(x,u)$ in $\mathbb{R}^{n}$ for $0<α<n$ where the nonlinearity $f(x,u) = \sum_{i=1}^{M} σ_{i} u^{q_i} + ω$ involves sublinear terms with $0<q_{i}<1$ and the coefficients $σ_{i}, ω$ are nonnegative locally finite Borel measures on $\mathbb{R}^n$. We develop a potential theoretic approach for the existence of positive minimal solutions in Lorentz spaces to the problems under certain assumptions on $σ_{i}$ and $ω$. The uniqueness properties of such solutions are discussed. Our techniques are also applicable to similar sublinear problems on uniform bounded domains when $0<α< 2$, or on arbitrary domains with positive Green's functions in the classical case $α=2$.

math.AP↗
arXiv2023

Minimal L^p-Solutions to Singular Sublinear Elliptic Problems

We solve the existence problem for the minimal positive solutions $u\in L^{p}(Ω, dx)$ to the Dirichlet problems for sublinear elliptic equations of the form \[ \begin{cases} Lu=σu^q+μ\qquad \quad \text{in} \quad Ω, \\ \liminf\limits_{x \rightarrow y}u(x) = 0 \qquad y \in \partial_{\infty}Ω, \end{cases} \] where $0<q<1$ and $Lu:=-\text{div} (\mathcal{A}(x)\nabla u)$ is a linear uniformly elliptic operator with bounded measurable coefficients. The coefficient $σ$ and data $μ$ are nonnegative Radon measures on an arbitrary domain $Ω\subset \mathbb{R}^n$ with a positive Green function associated with $L$. Our techniques are based on the use of sharp Green potential pointwise estimates, weighted norm inqualities, and norm estimates in terms of generalized energy.

math.AP↗