Nonhomogeneous Elliptic Equations Research Articles

On a localization-in-frequency approach for a class of elliptic problems with singular boundary data

We consider a class of nonhomogeneous elliptic equations in the half-space with critical singular boundary potentials and nonlinear fractional derivative terms. The forcing terms are considered on the boundary and can be taken as singular measure. Employing a functional setting and approach based on localization-in-frequency and Littlewood–Paley decomposition, we obtain results on solvability, regularity, and symmetry of solutions.

Multiplicity of positive solutions for a class of nonhomogeneous elliptic equations in the hyperbolic space

The paper is concerned with positive solutions to problems of the type \[ -\Delta_{\mathbb{B}^{N}} u - \lambda u = a(x) |u|^{p-1}\;u + f \text{ in }\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, \] where $\mathbb {B}^N$ denotes the hyperbolic space, $1< p<2^*-1:=\frac {N+2}{N-2}$ , $\;\lambda < \frac {(N-1)^2}{4}$ , and $f \in H^{-1}(\mathbb {B}^{N})$ ( $f \not \equiv 0$ ) is a non-negative functional. The potential $a\in L^\infty (\mathbb {B}^N)$ is assumed to be strictly positive, such that $\lim _{d(x, 0) \rightarrow \infty } a(x) \rightarrow 1,$ where $d(x,\, 0)$ denotes the geodesic distance. First, the existence of three positive solutions is proved under the assumption that $a(x) \leq 1$ . Then the case $a(x) \geq 1$ is considered, and the existence of two positive solutions is proved. In both cases, it is assumed that $\mu ( \{ x : a(x) \neq 1\}) > 0.$ Subsequently, we establish the existence of two positive solutions for $a(x) \equiv 1$ and prove asymptotic estimates for positive solutions using barrier-type arguments. The proofs for existence combine variational arguments, key energy estimates involving hyperbolic bubbles.

Cost of observability inequalities for elliptic equations in 2-d with potentials and applications to control theory

The goal of this article is to obtain observability estimates for non-homogeneous elliptic equations in the presence of a potential, posed on a smooth bounded domain Ω in and observed from a non-empty open subset More precisely, for our main result shows that, when has a finite number of holes, the observability constant of the elliptic operator with domain is of the form where C is a positive constant depending only on Ω and ω. Our methodology of proof is crucially based on the one recently developed by Logunov, Malinnikova, Nadirashvili, and Nazarov [1], in the context of the Landis conjecture on exponential decay of solutions to homogeneous elliptic equations in the plane The main difference and additional difficulty compared to [1] is that the zero set of the solutions to elliptic equations with source term can be very intricate and should be dealt with carefully. As a consequence of these new observability estimates, we obtain new results concerning control of semi-linear elliptic equations in the spirit of Fernández-Cara, Zuazua’s open problem concerning small-time global null-controllability of slightly super-linear heat equations.

Existence of Normalized Positive Solutions for a Class of Nonhomogeneous Elliptic Equations

Existence of Normalized Positive Solutions for a Class of Nonhomogeneous Elliptic Equations

An approach to elliptic equations with nonlinear gradient terms via a modulation framework

We consider a class of nonhomogeneous elliptic equations with fractional Laplacian and nonlinear gradient terms, namely [Formula: see text] in [Formula: see text], where [Formula: see text], [Formula: see text] is the nonlinearity, [Formula: see text] the potential and [Formula: see text] is a forcing term. Some examples of nonlinearities dealt with are [Formula: see text], [Formula: see text] and [Formula: see text], covering large values of [Formula: see text], and particularly variational supercritical powers for [Formula: see text] and super-[Formula: see text] ones for [Formula: see text] (superquadratic if [Formula: see text]). Moreover, we are able to consider some exponential growths, [Formula: see text] belonging to certain classes of power series, or [Formula: see text] satisfying some conditions in the Lipschitz spirit. We obtain results on existence, uniqueness, symmetry, and other qualitative properties in a new framework, namely modulation-type spaces based on Lorentz spaces. For that, we need to develop properties and estimates in those spaces such as complex interpolation, Hölder-type inequality, estimates for product, convolution and Riesz potential operators, among others. In order to handle the nonlinearity, other ingredients are estimates for composition operators in our setting.

Strong Maximum Principle and Boundary Estimates for Nonhomogeneous Elliptic Equations

We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear nonhomogeneous degenerate elliptic equations on the formF(x,u,Du,D2u)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ F(x,u,Du,D^{2}u) = 0 $$\\end{document} under suitable assumptions allowing for non-Lipschitz growth in the gradient term. In case of smooth boundaries, we also prove a Hopf lemma, a boundary Harnack inequality, and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply, e.g., to weak solutions of an eigenvalue problem for the variable exponent p-Laplacian.

Existence of multiple solutions for a nonhomogeneous p-Laplacian elliptic equation with critical Sobolev-Hardy exponent

This paper is concerned with the existence of multiple nontrivial solutions for nonhomogeneous p-Laplacain elliptic problems involving the critical Hardy-Sobolev exponent. The method used here is based on the Nehari manifold.

Multiplicity results for nonhomogeneous elliptic equations with singular nonlinearities

<p style='text-indent:20px;'>This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M2">\begin{document}$ q $\end{document}</tex-math></inline-formula> type and singular nonlinearities</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{alignedat}{2} {} - \mathcal{L}_{p,q} u & {} = \lambda \frac{f(u)}{u^\gamma}, \ u>0 && \quad\mbox{ in } \, \Omega, \\ u & {} = 0 && \quad\mbox{ on } \partial\Omega, \end{alignedat} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M5">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> boundary, <inline-formula><tex-math id="M6">\begin{document}$ N \geq 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \lambda >0 $\end{document}</tex-math></inline-formula> is a real parameter,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \mathcal{L}_{p,q} u : = {\rm{div}}(|\nabla u|^{p-2} \nabla u + |\nabla u|^{q-2} \nabla u), $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'><inline-formula><tex-math id="M8">\begin{document}$ 1<p<q< \infty $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ \gamma \in (0,1) $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M10">\begin{document}$ f $\end{document}</tex-math></inline-formula> is a continuous nondecreasing map satisfying suitable conditions. By constructing two distinctive pairs of strict sub and super solution, and using fixed point theorems by Amann [<xref ref-type="bibr" rid="b1">1</xref>], we prove existence of three positive solutions in the positive cone of <inline-formula><tex-math id="M11">\begin{document}$ C_\delta(\overline{\Omega}) $\end{document}</tex-math></inline-formula> and in a certain range of <inline-formula><tex-math id="M12">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>.</p>

On nonhomogeneous elliptic equations with the Hardy—Leray potentials

In this paper, we present some suitable distributional identities of solutions for nonhomogeneous elliptic equations involving the Hardy—Leray potentials and study qualitative properties of the solutions to the corresponding nonhomogeneous problems in this distributional sense. We address some applications on the nonexistence of some nonhomogeneous problems with the Hardy—Leray potentials and the nonexistence principle eigenvalue with some indefinite potentials.

The sub-supersolution method for a nonhomogeneous elliptic equation involving Lebesgue generalized spaces

In this paper, a nonhomogeneous elliptic equation of the form −A(x,|u|Lr(x))div(a(|∇u|p(x))|∇u|p(x)−2∇u)=f(x,u)|∇u|Lq(x)α(x)+g(x,u)|∇u|Ls(x)γ(x)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& - \\mathcal{A}\\bigl(x, \\vert u \\vert _{L^{r(x)}}\\bigr) \\operatorname{div}\\bigl(a\\bigl( \\vert \\nabla u \\vert ^{p(x)}\\bigr) \\vert \\nabla u \\vert ^{p(x)-2} \\nabla u\\bigr) \\\\& \\quad =f(x, u) \\vert \\nabla u \\vert ^{\\alpha (x)}_{L^{q(x)}}+g(x, u) \\vert \\nabla u \\vert ^{ \\gamma (x)}_{L^{s(x)}} \\end{aligned}$$ \\end{document} on a bounded domain Ω in {mathbb{R}}^{N} (N >1) with C^{2} boundary, with a Dirichlet boundary condition is considered. Using the sub-supersolution method, the existence of at least one positive weak solution is proved. As an application, the existence of at least one solution of a generalized version of the logistic equation and a sublinear equation are shown.

Sign-Changing Solutions for Resonant and Superlinear Nonhomogeneous Elliptic Equations

Sign-Changing Solutions for Resonant and Superlinear Nonhomogeneous Elliptic Equations

Existence and regularity results for nonlinear and nonhomogeneous elliptic equation

Existence and regularity results for nonlinear and nonhomogeneous elliptic equation

Infinite number of solutions for some elliptic eigenvalue problems of Kirchhoff-type with non-homogeneous material

In this paper, using variational method, we study the existence of an infinite number of solutions (some are positive, some are negative, and others are sign-changing) for a non-homogeneous elliptic Kirchhoff equation with a nonlinear reaction term.

Positive maximal and minimal solutions for non-homogeneous elliptic equations depending on the gradient

We are concerned with positive maximal and minimal solutions for non-homogeneous elliptic equations of the form−div(a(|∇u|p)|∇u|p−2∇u)=f(x,u,∇u) in Ω, supplied with Dirichlet boundary conditions. First we localize maximal and minimal solutions between not necessarily bounded sub-super solutions. Then using a uniform gradient estimate, which seems of independent interest, we show the existence of positive maximal and minimal solutions in some situations. More precisely, we obtain positive maximal and minimal solution to some classes of non-homogeneous equations depending on the gradient which may be perturbed by unbounded, singular or logistic sources.

On Solvability in the Sense of Sequences for some Non-Fredholm Operators with Drift and Anomalous Diffusion

We study the solvability of certain linear nonhomogeneous elliptic equations and establish that, under some technical assumptions, the L2-convergence of the right-hand sides yields the existence and convergence of solutions in an appropriate Sobolev space. The problems involve differential operators with or without Fredholm property, in particular, the one-dimensional negative Laplacian in a fractional power, on the whole real line or on a finite interval with periodic boundary conditions. We prove that the presence of the transport term in these equations provides regularization of the solutions.

Lorentz improving estimates for the [formula omitted]-Laplace equations with mixed data

The aim of this paper is to develop the regularity theory for a weak solution to a class of quasilinear nonhomogeneous elliptic equations, whose prototype is the following mixed Dirichlet p-Laplace equation of type div(|∇u|p−2∇u)=f+div(|F|p−2F)inΩ,u=gon∂Ω, in Lorentz space, with given data F∈Lp(Ω;Rn), f∈Lpp−1(Ω), g∈W1,p(Ω) for 1<p<∞ and Ω⊂Rn (n≥2) satisfying a Reifenberg flat domain condition or a p-capacity uniform thickness condition, which are considered in several recent papers. To better specify our result, the proofs of regularity estimates involve fractional maximal operators and valid for a more general class of quasilinear nonhomogeneous elliptic equations with mixed data. This paper not only deals with the Lorentz estimates for a class of more general problems with mixed data but also improves the good-λ approach technique proposed in our preceding works (Tran, 2019; Tran and Nguyen, 2019; Tran and Nguyen, 2020; Tran and Nguyen (in press)), to achieve the global Lorentz regularity estimates for gradient of weak solutions in terms of fractional maximal operators.

The Dirichlet problem for nonlocal elliptic equations

We study a class of nonlocal elliptic equations on a bounded domain . For (), by the Lax–Milgram theorem and the De Giorgi iteration, we prove the existence and uniqueness of solution. Furthermore, we investigate the existence of densities for measure-valued solutions to nonhomogeneous measure-valued nonlocal elliptic equations.

Existence of solutions for a nonhomogeneous elliptic Kircchoff type equation

For a bounded domain Ω in RN, a bounded Carathéodory function g in Ω×R, and a nonnegative and nonzero locally integrable measurable function h in Ω, we give sufficient conditions on the C1-function M:[0,∞)→R for the existence of a solution of the nonlocal elliptic problem{−(1+γM′(∫Ω|∇u|2))Δu=λu+g(x,u)−h(x)|u|p−1u+f(x),x∈Ωu(x)=0,x∈∂Ω for every p>1, γ>0, λ∈R and f∈L2(Ω).

Nonhomogeneous polyharmonic elliptic problems involving GJMS operator on Riemannian manifold

Let [Formula: see text] be a closed Riemannian manifold, under some assumptions on [Formula: see text] and [Formula: see text] by applying a method used in [Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincaré 9 (1992) 281–304], we show the existence and multiplicity of solutions of the semi-linear elliptic equation: [Formula: see text]. When [Formula: see text] is an Einsteinian manifold of positive scalar curvature, under additional conditions, we obtain the existence of positive solutions.

Multiplicity of solutions for a nonlocal nonhomogeneous elliptic equation with critical exponential growth

In this paper we are interested in the following nonlocal nonhomogeneous elliptic equation in R2,−Δu+V(x)u=(1|x|μ⁎F(u)|x|β)f(u)|x|β+εh(x)inR2, where V is a positive continuous potential, 0<μ<2, β≥0, 2β+μ≤2, ε is a small parameter and F(s) is the primitive function of f(s). Suppose that the nonlinearity f(s) is of critical exponential growth in the sense of Trudinger-Moser inequality, we prove the existence and multiplicity of solutions by variational methods.