Weak Comparison Principle Research Articles

A singular system involving mixed local and non-local operators

This article sets forth results on the existence, non-existence, uniqueness, and regularities properties, as well as boundary behavior of solutions for singular systems involving mixed local and non-local elliptic operators (see System (S) below). More precisely, we first establish a new weak comparison principle for a singular equation. Afterward, we discuss the non-existence of positive classical solutions, as well as construct suitable ordered pairs of sub-solutions and super-solutions. This allows us to obtain the existence of a pair of positive weak solutions for System (S) by employing Schauder’s fixed-point theorem in the associated conical shell. Finally, we adapt a method of Krasnoselsky to establish the uniqueness of such a positive pair of solutions.

Symmetry via the moving plane method for a class of quasilinear elliptic problems involving the Hardy potential

We consider positive solutions to a class of quasilinear elliptic problems involving the Hardy potential under zero Dirichlet boundary condition. Via moving plane method, proving a weak comparison principle, we prove symmetry and monotonicity properties for the solutions defined on strictly convex symmetric domains.

Positive weak solutions of uniformly elliptic inequalities and applications to population models

By using the uniformly ellipticity of linear partial differential operators (which contain p-Laplacian operators with 1 < p < ∞ ), weak comparison principle and Urysohn's lemma, in the paper, we introduce and study a class of new variational inequalities involving uniformly elliptic operators and demicontinuous S-contractive operators to extend some corresponding tasks and scope on population genetics. Further, as applications, we employ the main results presented in this paper to second-order uniformly elliptic inequalities and a population model of one species arising in disrupted environments.

Existence and global behavior of weak solutions to a doubly nonlinear evolution

In this article, we study a class of doubly nonlinear parabolic problems involving the fractional p-Laplace operator. For this problem, we discuss existence, uniqueness and regularity of the weak solutions by using the time-discretization method and monotone arguments. For global weak solutions, we also prove stabilization results by using the accretivity of a suitable associated operator. This property is strongly linked to the Picone identity that provides further a weak comparison principle, barrier estimates and uniqueness of the stationary positive weak solution. For more information see https://ejde.math.txstate.edu/Volumes/2021/09/abstr.html

Existence and Hölder regularity of infinitely many solutions to a p-Kirchhoff-type problem involving a singular nonlinearity without the Ambrosetti–Rabinowitz (AR) condition

We carry out an investigation of the existence of infinitely many solutions to a fractional p-Kirchhoff-type problem with a singularity and a superlinear nonlinearity with a homogeneous Dirichlet boundary condition. Further, the solution(s) will be proved to be bounded and a weak comparison principle has also been proved. A ‘\(C^1\) versus \(W_0^{s,p}\)’ analysis has also been discussed.

Weak Comparison Principle for Weighted Fractional p -Laplacian Equation

The aim of this paper is to establish a weak comparison principle for a class fractional p -Laplacian equation with weight. The nonlinear term f x , s &gt; 0 is a Carathéodory function which is possibly unbounded both at the origin and at infinity and such that f x , s s 1 − p decreases with respect to s for a.e. x ∈ Ω .

Generalized anisotropic Neumann problems of Ambrosetti–Prodi type with nonstandard growth conditions

We investigate the realization of a general class of nonlinear boundary value problems of Ambrosetti–Prodi type involving anisotropic elliptic operators with bounded measurable coefficients, nonstandard growth conditions, and generalized Neumann boundary conditions. By combining several tools, such as priori estimates, regularity theory, a sub-supersolution method, weak comparison principles, and the Leray-Schauder degree theory, we obtain the existence of an unique parameter controlling conditions for the non-existence, existence, minimality, and multiplicity of solutions.

Singular Doubly Nonlocal Elliptic Problems with Choquard Type Critical Growth Nonlinearities

The theory of elliptic equations involving singular nonlinearities is well-studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we study the very singular and doubly nonlocal singular problem $$(P_\lambda )$$ (See below). Firstly, we establish a very weak comparison principle and the optimal Sobolev regularity. Next using the critical point theory of nonsmooth analysis and the geometry of the energy functional, we establish the global multiplicity of positive weak solutions.

A weak comparison principle in tubular neighbourhoods of embedded manifolds

We study weak solutions to degenerate quasilinear elliptic equations, involving first order terms, in unbounded tubular domains. In particular we show that, under suitable hypotheses, the weak comparison principle holds if the domain is narrow enough.

A Picone identity for variable exponent operators and applications

Abstract In this work, we establish a new Picone identity for anisotropic quasilinear operators, such as the p(x)-Laplacian defined as div(|∇ u| p(x)−2 ∇ u). Our extension provides a new version of the Diaz-Saa inequality and new uniqueness results to some quasilinear elliptic equations with variable exponents. This new Picone identity can be also used to prove some accretivity property to a class of fast diffusion equations involving variable exponents. Using this, we prove for this class of parabolic equations a new weak comparison principle.

Regularity and weak comparison principles for double phase quasilinear elliptic equations

We consider the Euler equation of functionals involving a term of the form \begin{document}$ \int_{\Omega}(| \nabla u|^p+a(x)| \nabla u|^q) \,{ {\rm{d}}} x, $\end{document} with \begin{document}$ 1 and \begin{document}$ a(x)\geq 0 $\end{document} . We prove weak comparison principle and summability results for the second derivatives of solutions.

A simple proof of a strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation

A strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation is considered. The difficulties of the problem come from the fact that this nonlinear equation is non-uniformly elliptic, does not depend on the value of unknown functions, depends on spatial variables and solutions are semicontinuous. Our simple proof of the strong comparison principle consists only of three ingredients, the definition of viscosity solutions, the inf and sup convolutions of functions, and the theory of classical solutions of quasilinear elliptic equations. Once we have the strong comparison principle, we can prove a weak comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation in a bounded domain.

Comparison principle for stochastic heat equation on $\mathbb{R}^{d}$

We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on $\mathbb{R}^{d}$ \[\biggl(\frac{\partial}{\partial t}-\frac{1}{2}\Delta \biggr)u(t,x)=\rho\bigl(u(t,x)\bigr)\dot{M}(t,x),\] for measure-valued initial data, where $\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and $\rho$ is Lipschitz continuous. These results are obtained under the condition that $\int_{\mathbb{R}^{d}}(1+|\xi|^{2})^{\alpha-1}\hat{f}(\text{d}\xi)<\infty$ for some $\alpha\in(0,1]$, where $\hat{f}$ is the spectral measure of the noise. The weak comparison principle and nonnegativity of solutions to the same equation are obtained under Dalang’s condition, that is, $\alpha=0$. As some intermediate results, we obtain handy upper bounds for $L^{p}(\Omega)$-moments of $u(t,x)$ for all $p\ge2$, and also prove that $u$ is a.s. Hölder continuous with order $\alpha-\varepsilon$ in space and $\alpha/2-\varepsilon$ in time for any small $\varepsilon>0$.

English

We shall prove a weak comparison principle for quasilinear elliptic operators −div(a(x,∇u)) that includes the negative p-Laplace operator, where a: × ℝN → ℝN satisfies certain conditions frequently seen in the research of quasilinear elliptic operators. In our result, it is characteristic that functions which are compared belong to different spaces.

Analysis of the autonomous problem about coupled active non-Newtonian multi-seepage in sparse medium

The flow field of non-Newtonian fluid in sparse medium was analyzed by computational fluid dynamics (CFD) method. The results show that the axial velocity and radial velocity of the non-Newtonian fluid are larger than those of the Newtonian fluid due to the coupling of the viscosity of the non-Newtonian fluid and the shear rate, and the tangential velocity is less than that of the Newtonian fluid. These differences lead to the difference in the sparse medium Non-Newtonian fluids are of a special nature. The influence of the weight function on the global existence and blasting of the problem is discussed by analyzing the non-Newtonian percolation equation with nonlocal and weighted non-local Dirichlet boundary conditions. According to the non-Newtonian percolation equation, we define the weak solution of the problem and expound the local existence of the weak solution. Then we construct the test function and prove the weak comparison principle by using the Grown well inequality. The overall existence and blasting are analyzed by constructing the upper and lower solutions.

Nonuniqueness and multi-bump solutions in parabolic problems with the p-Laplacian

The validity of the weak and strong comparison principles for degenerate parabolic partial differential equations with the p-Laplace operator Δp is investigated for p>2. This problem is reduced to the comparison of the trivial solution (≡0, by hypothesis) with a nontrivial nonnegative solution u(x,t). The problem is closely related also to the question of uniqueness of a nonnegative solution via the weak comparison principle. In this article, realistic counterexamples to the uniqueness of a nonnegative solution, the weak comparison principle, and the strong maximum principle are constructed with a nonsmooth reaction function that satisfies neither a Lipschitz nor an Osgood standard “uniqueness” condition. Nonnegative multi-bump solutions with spatially disconnected compact supports and zero initial data are constructed between sub- and supersolutions that have supports of the same type.

Maximum Principles for Dynamic Equations on Time Scales and Their Applications

We consider the second dynamic operators of elliptic type on time scales. We establish basic generalized maximum principles and apply them to obtain weak comparison principle for second dynamic elliptic operators and to obtain the uniqueness of Dirichlet boundary value problems for dynamic elliptic equations.

A weak comparison principle for solutions of very degenerate elliptic equations

We prove a comparison principle for weak solutions of elliptic quasilinear equations in divergence form whose ellipticity constants degenerate at every point where \(\nabla u\in K\), where \(K\subset \mathbb{R }^N\) is a Borel set containing the origin.

Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces

We prove a weak comparison principle in narrow unbounded domains for solutions to \(-\Delta _p u=f(u)\) in the case \(2<p< 3\) and \(f(\cdot )\) is a power-type nonlinearity, or in the case \(p>2\) and \(f(\cdot )\) is super-linear. We exploit it to prove the monotonicity of positive solutions to \(-\Delta _p u=f(u)\) in half spaces (with zero Dirichlet assumption) and therefore to prove some Liouville-type theorems.

A singular parabolic equation: Existence, stabilization

We investigate the following quasilinear parabolic and singular equation,(Pt){ut−Δpu=1uδ+f(x,u)in (0,T)×Ω,u=0on (0,T)×∂Ω,u>0in (0,T)×Ω,u(0,x)=u0(x)in Ω, where Ω is an open bounded domain with smooth boundary in RN, 1<p<∞, 0<δ and T>0. We assume that (x,s)∈Ω×R+→f(x,s) is a bounded below Caratheodory function, locally Lipschitz with respect to s uniformly in x∈Ω and asymptotically sub-homogeneous, i.e.(0.1)0⩽limt→+∞f(x,t)tp−1=αf<λ1(Ω) (where λ1(Ω) is the first eigenvalue of −Δp in Ω with homogeneous Dirichlet boundary conditions) and u0∈L∞(Ω)∩W01,p(Ω), satisfying a cone condition defined below. Then, for any δ∈(0,2+1p−1), we prove the existence and the uniqueness of a weak solution u∈V(QT) to (Pt). Furthermore, u∈C([0,T],W01,p(Ω)) and the restriction δ<2+1p−1 is sharp. The proof relies on a semi-discretization in time with implicit Euler method and on the study of the stationary problem. The key points in the proof is to show that u belongs to the cone C defined below and by the weak comparison principle that 1uδ∈L∞(0,T;W−1,p′(Ω)) and u1−δ∈L∞(0,T;L1(Ω)). When t→f(x,t)tp−1 is nonincreasing for a.e. x∈Ω, we show that u(t)→u∞ in L∞(Ω) as t→∞, where u∞ is the unique solution to the stationary problem. This stabilization property is proved by using the accretivity of a suitable operator in L∞(Ω).Finally, in the last section we analyze the case p=2. Using the interpolation spaces theory and the semigroup theory, we prove the existence and the uniqueness of weak solutions to (Pt) for any δ>0 in C([0,T],L2(Ω))∩L∞(QT) and under suitable assumptions on the initial data we give additional regularity results. Finally, we describe their asymptotic behaviour in L∞(Ω)∩H01(Ω) when δ<3.