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Adisak Seesanea

Publications and source records attributed to Adisak Seesanea.

15 recordsLinked to original sources
arXiv2026

Dirichlet problem for Lane-Emden type equations with several sublinear terms

We prove the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} L u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in } \Omega, \liminf \limits_{x \rightarrow y} u(x) = f(y), & y \in \partial^\infty\Omega, \end{cases} \] where $0 < q_{i} < 1$. Here $Lu = - \text{div}(A \nabla u)$ is a uniformly elliptic operator with bounded coefficients, $\sigma_{i}$ is a nonnegative locally finite Borel measure on an $A$-regular domain $\Omega \subset \mathbb{R}^n$ which possesses a positive Green function associated with $L$, and $f$ is a nonnegative continuous function on the boundary $\partial^\infty\Omega$. An analogous result for positive continuous solutions to the problem is also illustrated. Our method can be adapted to address related sublinear problems with zero boundary conditions involving the fractional Laplace operator $(-\Delta)^{\alpha}$ for $0< \alpha < n/2$, in place of $L$, in $\mathbb{R}^n$ as well.

math.AP↗
arXiv2025

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

We establish a Sobolev-type inequality in Lorentz spaces for $\mathcal{L}$-superharmonic functions \[ \|u\|_{L^{\frac{nq}{n-\alpha q},t}(\mathbb{R}^n)} \leq c \left\| \frac{u(x) - u(y)}{|x-y|^{\frac{n}{q}+\alpha}} \right\|_{L^{q,t}(\mathbb{R}^n \times \mathbb{R}^n)} \] 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 $(- \Delta_{p})^{\alpha} $ with $0 < \alpha < 1$ and $1<p<2$. Related Gagliardo-Nirenberg interpolation for $\mathcal{L}$-superharmonic functions is also derived.

math.AP↗
arXiv2024

Calderon-Zygmund theory for strongly coupled linear system of nonlocal equations with Holder-regular coefficient

We extend the Calderón-Zygmund theory for nonlocal equations to strongly coupled system of linear nonlocal equations $\mathcal{L}^{s}_{A} u = f$, where the operator $\mathcal{L}^{s}_{A}$ is formally given by \[ \mathcal{L}^s_{A}u = \int_{\mathbb{R}^n}\frac{A(x, y)}{\vert x-y\vert ^{n+2s}} \frac{(x-y)\otimes (x-y)}{\vert x-y\vert ^2}(u(x)-u(y))dy. \] For $0 < s < 1$ and $A:\mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R}$ taken to be symmetric and serving as a variable coefficient for the operator, the system under consideration is the fractional version of the classical Navier-Lamé linearized elasticity system. The study of the coupled system of nonlocal equations is motivated by its appearance in nonlocal mechanics, primarily in peridynamics. Our regularity result states that if $A(\cdot, y)$ is uniformly Holder continuous and $\inf_{x\in \mathbb{R}^n}A(x, x) > 0$, then for $f\in L^{p}_{loc},$ for $p\geq 2$, the solution vector $u\in H^{2s-δ,p}_{loc}$ for some $δ\in (0, s)$.

math.AP↗
arXiv2023

Nonlocal Sublinear Elliptic Problems Involving Measures

We study Dirichlet problems for fractional Laplace equations of the form $(-\Delta)^{\frac{\alpha}{2}} u = f(x,u)$ in $\mathbb{R}^{n}$ for $0<\alpha<n$ where the nonlinearity $f(x,u) = \sum_{i=1}^{M} \sigma_{i} u^{q_i} + \omega$ involves sublinear terms with $0<q_{i}<1$ and the coefficients $\sigma_{i}, \omega$ 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 $\sigma_{i}$ and $\omega$. The uniqueness properties of such solutions are discussed. Our techniques are also applicable to similar sublinear problems on uniform bounded domains when $0<\alpha< 2$, or on arbitrary domains with positive Green's functions in the classical case $\alpha =2$.

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↗
arXiv2023

Homogenization of diffusion processes with singular drifts and potentials via unfolding method

This work is concerned with homogenization problems for elliptic equations of the type \[ \begin{cases} \mathfrak{L}_{\delta} u_{\delta} + \lambda u_{\delta} = f_{\delta} \qquad \text{in} \;\; D, \\ \qquad \quad \;\, u = 0 \qquad \, \text{on} \;\; \partial D, \end{cases} \] where $\delta > 0$, $\lambda \in \mathbb{R}$, $D$ is a bounded open set in $\mathbb{R}^{d}$, and $f_{\delta} \in H^{-1}(D)$. The operator $ \mathfrak{L}_{\delta} u = -{\rm div} \left( A^\delta \nabla u + C^\delta u \right) + B^\delta \nabla u +k^\delta u $ involved uniformly bounded diffusion coefficients $A^\delta$, where drifts $B^\delta$, $C^\delta$, and potential $k^\delta$ are possibly unbounded. An application to homogenization of the corresponding diffusion processes is also discussed.

math.AP↗
arXiv2020

The Dirichlet problem for sublinear elliptic equations with source

We present a necessary and sufficient condition on nonnegative Radon measures $μ$ and $ν$ for the existence of a positive continuous solution of the Dirichlet problem for the sublinear elliptic equation $-Δu=μu^q+ν$ with prescribed nonnegative continuous boundary data in a general domain. Moreover, two-sided pointwise estimates of Brezis-Kamin type for positive bounded solutions and the uniqueness of a positive continuous $L^q$-solution are investigated.

math.AP↗
arXiv2020

Existence of minimal solutions to quasilinear elliptic equations with several sub-natural growth terms

We study the existence of positive solutions to quasilinear elliptic equations of the type \[ -Δ_{p} u = σu^{q} + μ\quad \text{in} \ \mathbb{R}^{n}, \] in the sub-natural growth case $0 < q < p - 1$, where $Δ_{p}u = \nabla \cdot ( |\nabla u|^{p - 2} \nabla u )$ is the $p$-Laplacian with $1 < p < n$, and $σ$ and $μ$ are nonnegative Radon measures on $\mathbb{R}^{n}$. We construct minimal generalized solutions under certain generalized energy conditions on $σ$ and $μ$. To prove this, we give new estimates for interaction between measures. We also construct solutions to equations with several sub-natural growth terms using the same methods.

math.AP↗
arXiv2018

Solutions to sublinear elliptic equations with finite generalized energy

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation \[ \mathcal{L}u = σu^{q} + μ\quad \text{in} \;\; Ω, \] in the sublinear case $0 0$. In this case $u \in L^{γ+q}(Ω, σ)\cap L^γ(Ω, μ)$, where $γ=1$ corresponds to finite energy solutions. Here $\mathcal{L} u:= -\,\text{div}(\mathcal{A}\nabla u)$ is a linear uniformly elliptic operator with bounded measurable coefficients, and $σ$, $μ$ are nonnegative functions (or Radon measures), on an arbitrary domain $Ω\subseteq \mathbb{R}^n$ which possesses a positive Green function associated with $\mathcal{L}$. When $0<γ\leq 1$, this result yields sufficient conditions for the existence of a positive solution to the above problem which belongs to the Dirichlet space $\dot{W}_{0}^{1,p}(Ω)$ for $1<p\leq 2$.

math.AP↗
arXiv2018

Solutions in Lebesgue spaces to nonlinear elliptic equations with sub-natural growth terms

We study the existence problem for positive solutions $u \in L^{r}(\mathbb{R}^{n})$, $0<r<\infty$, to the quasilinear elliptic equation \[ -Δ_{p} u = σu^{q} \quad \text{in} \;\; \mathbb{R}^n \] in the sub-natural growth case $0<q< p-1$, where $Δ_{p}u = \text{div}( |\nabla u|^{p-2} \nabla u )$ is the $p$-Laplacian with $1<p<\infty$, and $σ$ is a nonnegative measurable function (or measure) on $\mathbb{R}^n$. Our techniques rely on a study of general integral equations involving nonlinear potentials and related weighted norm inequalities. They are applicable to more general quasilinear elliptic operators such as the $\mathcal{A}$-Laplacian $\text{div} \mathcal{A}(x,\nabla u)$, and the fractional Laplacian $(-Δ)^α$ on $\mathbb{R}^n$, as well as linear uniformly elliptic operators with bounded measurable coefficients $\text{div}(\mathcal{A} \nabla u)$ on an arbitrary domain $Ω\subseteq \mathbb{R}^n$ with a positive Green function.

math.AP↗
arXiv2018

Extensions of the Heisenberg group by two-parameter groups of dilations

We introduce extensions of the multidimensional Heisenberg group $\mathbb{H}^n$ by two-parameter groups of dilations, and then classify the extended groups up to isomorphism, by employing Lie algebra techniques. We show that the groups are isomorphic to subgroups of the symplectic group $\textit{Sp(}n+1,\mathbb{R})$ as well as subgroups of the affine group $\textit{Aff}(n+1,\mathbb{R})$. Thus, they possess both, a metaplectic and a wavelet representation. Moreover, the metaplectic representation splits into a sum of two subrepresentations which both are equivalent to the same subrepresentation of the wavelet representation.

math.RT↗
arXiv2017

Finite energy solutions to inhomogeneous nonlinear elliptic equations with sub-natural growth terms

We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation \[ -Δ_{p} u = σu^{q} + μ\quad \text{on} \;\; \mathbb{R}^n \] in the sub-natural growth case $0<q<p-1$, where $Δ_{p}$ ($1<p<\infty$) is the $p$-Laplacian, and $σ$, $μ$ are positive Borel measures on $\mathbb{R}^n$. Uniqueness of such a solution is established as well. Similar inhomogeneous problems in the sublinear case $0<q<1$ are treated for the fractional Laplace operator $(-Δ)^α$ in place of $-Δ_{p}$, on $\mathbb{R}^n$ for $0<α<\frac{n}{2}$, and on an arbitrary domain $Ω\subset \mathbb{R}^n$ with positive Green's function in the classical case $α= 1$.

math.AP↗