Sub-supersolution Method Research Articles

Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient

In this paper, we study a nonlinear Dirichlet problem of p-Laplacian type with combined effects of nonlinear singular and convection terms. An existence theorem for positive solutions is established as well as the compactness of solution set. Our approach is based on Leray–Schauder alternative principle, method of sub-supersolution, nonlinear regularity, truncation techniques, and set-valued analysis.

Solutions to an anisotropic system via sub-supersolution method and Mountain Pass Theorem

Solutions to an anisotropic system via sub-supersolution method and Mountain Pass Theorem

A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains

We investigate the solvability of the Ambrosetti-Prodi problem for the p-Laplace operator ∆p with Venttsel' boundary conditions on a twodimensional open bounded set with Koch-type boundary, and on an open bounded three-dimensional cylinder with Koch-type fractal boundary. Using a priori estimates, regularity theory and a sub-supersolution method, we obtain a necessary condition for the non-existence of solutions (in the weak sense), and the existence of at least one globally bounded weak solution. Moreover, under additional conditions, we apply the Leray-Schauder degree theory to obtain results about multiplicity of weak solutions.

Nonlocal problems arising from the birth-jump processes

In this paper, we prove the existence and uniqueness of a positive solution for a nonlocal logistic equation arising from the birth-jump processes. For this, we establish a sub-super solution method for nonlocal elliptic equations, we perform a study of the eigenvalue problems associated with these equations and we apply these results to the nonlocal logistic equation.

Existence of Positive Solutions for a Class of Quasilinear Singular Elliptic Systems Involving Caffarelli-Kohn-Nirenberg Exponent with Sign-Changing Weight Functions

The paper deals with the existence of weak positive solutions for a new class of quasilinear singular elliptic systems involving critical Caffarelli–Kohn–Nirenberg exponent with sign-changing weight functions using the method of sub-super solutions. Our results are natural extensions from the previous ones in [3].

On the sub-supersolution principle for p(x)-Laplacian equations

This paper deals with the sub-supersolution method for the p(x) -Laplacian Dirichlet problem. A sub-supersolution principle for the Dirichlet problems involving the p(x)-Laplacian is established by using induction method.

Positive solutions of generalized nonlinear logistic equations via sub-super solutions

Let Ω be a smooth bounded domain in RN, N≥1, let m,n be two nonnegative functions defined in Ω, and let ϕ:RN→RN be a continuous and strictly monotone mapping. We consider the existence and nonexistence of positive solutions for nonlinear problems involving the ϕ-Laplacian, of the form{−divϕ(∇u)=λm(x)f(u)−n(x)g(u)in Ω,u=0on ∂Ω, where λ>0 is a real parameter, and f,g:[0,∞)→[0,∞) are continuous functions modeled by f(t)=tq and g(t)=tr with 0<q<r. The subdiffusive, equidiffusive and superdiffusive cases are all treated, and some qualitative properties of the solutions are also given. As a byproduct, we obtain results in the case of a (roughly speaking) sublinear nonlinearity f and n≡0. We point out that some of our results are new even for the usual p-Laplacian. Our main tool is the sub-supersolutions method combined with some estimates on related nonlinear problems.

Lower Bounds for the Principal Eigenvalue of the p-Laplacian on the Unit Ball

In this paper, by the sub-super solution method we provide explicit lower bounds for the principal eigenvalue of the p-Laplacian on the unit ball. We compare our results with those of earlier studies and also demonstrate that they are in good agreement with numerical results.

Study of a generalized logistic equation with nonlocal reaction term

In this paper, we consider the generalized logistic equation with nonlocal reaction term \t\t\t−Δu=u(λ+b∫Ωurdx−f(u))in Ω,u>0 in Ω,u=0 on ∂Ω.\\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}$$ -\\Delta u=u \\biggl(\\lambda +b \\int_{\\Omega }u^{r}\\,dx-f(u) \\biggr)\\quad \\text{in } \\Omega,\\qquad u>0 \\quad \\text{ in } \\Omega,\\qquad u=0 \\quad \\text{ on } \\partial \\Omega. $$\\end{document} Using the bifurcation and sub-supersolution method, we obtain the non-existence, existence, and uniqueness of positive solutions for different parameters on the nonlocal terms. Our works about the nonlocal elliptic problem improve the results in the previous literature.

The Sub-Supersolution Method and Extremal Solutions of Quasilinear Elliptic Equations in Orlicz-Sobolev Spaces

We prove the existence of extremal solutions of the following quasilinear elliptic problem -∑i=1N∂/∂xiai(x,u(x),Du(x))+g(x,u(x),Du(x))=0 under Dirichlet boundary condition in Orlicz-Sobolev spaces W01LM(Ω) and give the enclosure of solutions. The differential part is driven by a Leray-Lions operator in Orlicz-Sobolev spaces, while the nonlinear term g:Ω×R×RN→R is a Carathéodory function satisfying a growth condition. Our approach relies on the method of linear functional analysis theory and the sub-supersolution method.

Structure of the set of positive solutions of a non-linear Schrödinger equation

In this paper we study the existence, uniqueness and multiplicity of positive solutions to a non-linear Schr¨odinger equation. We describe the set of positive solutions. We use mainly the sub-supersolution method, bifurcation and variational arguments to obtain the results.

Positive solutions of indefinite semipositone problems via sub-super solutions

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq1$, be a smooth bounded domain, and let $m:\Omega\rightarrow\mathbb{R}$ be a possibly sign-changing function. We investigate the existence of positive solutions for the semipositone problem \[ \left\{ \begin{array} [c]{lll} -\Delta u=\lambda m(x)(f(u)-k) & \mathrm{in} & \Omega,\\ u=0 & \mathrm{on} & \partial\Omega, \end{array} \right. \] where $\lambda,k>0$ and $f$ is either sublinear at infinity with $f(0)=0$, or $f$ has a singularity at $0$. We prove the existence of a positive solution for certain ranges of $\lambda$ provided that the negative part of $m$ is suitably small. Our main tool is the sub-supersolutions method, combined with some rescaling properties.

Spatial decay and stability of traveling fronts for degenerate Fisher type equations in cylinder

This paper is concerned with the spatial decay and asymptotic stability of multidimensional traveling fronts for the degenerate Fisher type equation in an unbounded cylinder. Firstly, by applying the moving plane argument and the generalized center manifold theorem, we obtain the uniqueness and exponential decay of the traveling front with the critical speed and the non-exponential decay of traveling fronts with non-critical speeds, especially for the p-degree Fisher equation we get the precise algebraic decaying rates and the higher order expansion of traveling fronts with non-critical speeds. Secondly, by applying the spectral analysis and sub-super solution method we prove the nonlinear exponential stability of all traveling fronts in some exponentially weighted spaces and the Lyapunov stability of traveling fronts with non-critical speeds in some polynomially weighted spaces. Finally, by combining the sub-super solution method with the nonlinear stability results, we get the asymptotic behavior and the asymptotic spreading speeds of the solution for more general initial data, which are proved to be determined by the spatial decay of the initial data at one end.

Some remarks on the comparison principle in Kirchhoff equations

In this paper we study the validity of the comparison principle and the sub-supersolution method for Kirchhoff type equations. We show that these principles do not work when the Kirchhoff function is increasing, contradicting some previous results. We give an alternative sub-supersolution method and apply it to some models.

Decay estimates for Wolff potentials in [formula omitted] and gradient-dependent quasilinear elliptic equations

We prove decay estimates of positive weak solutions u∈D1,p(RN)∩C1(RN) and their gradients |∇u| of the p-Laplacian problem (1<p<N):u∈D1,p(RN):−Δpu=a(x), where the measurable function a satisfies the decay estimate (α>0):0<a(x)≲11+|x|N+α,∀x∈RN. The positive Borel measure on RN generated by the right-hand side function a and its associated Wolff potential is known to be the main tool for pointwise estimates of u(x)>0 and |∇u(x)|. Therefore, one of the main goal of this paper is to prove decay estimates for corresponding Wolff potentials in RN. The obtained decay estimates in conjunction with a sub-supersolution method, which is developed here for quasilinear elliptic equations, are then applied to prove the existence of positive solutions for classes of gradient-dependent quasilinear equations in the formu∈D1,p(RN):−Δpu=a(x)f(u,∇u).

Existence of positive weak solutions for a class of Kirrchoff elliptic systems with multiple parameters

In this paper, by using subsuper solutions method, we study the existence of weak positive solution for a class of Kirrchoff elliptic systems in bounded domains with multiple parameters.

Existence of Three Positive Solutions for a Nonlocal Singular Dirichlet Boundary Problem

Abstract In this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem: { ( - Δ ) s ⁢ u = λ ⁢ f ⁢ ( u ) u q , u &gt; 0 ⁢ in ⁢ Ω , u = 0 in ⁢ ℝ n ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&amp;\displaystyle=\lambda\frac{% f(u)}{u^{q}},&amp;&amp;\displaystyle u&gt;0\text{ in }\Omega,\\ \displaystyle u&amp;\displaystyle=0&amp;&amp;\displaystyle\phantom{}\text{in }\mathbb{R}^{% n}\setminus\Omega,\end{aligned}\right. where ( - Δ ) s {(-\Delta)^{s}} denotes the fractional Laplace operator for s ∈ ( 0 , 1 ) {s\in(0,1)} , n &gt; 2 ⁢ s {n&gt;2s} , q ∈ ( 0 , 1 ) {q\in(0,1)} , λ &gt; 0 {\lambda&gt;0} and Ω is a smooth bounded domain in ℝ n {\mathbb{R}^{n}} . Here f : [ 0 , ∞ ) → [ 0 , ∞ ) {f:[0,\infty)\to[0,\infty)} is a continuous nondecreasing map satisfying lim u → ∞ ⁡ f ⁢ ( u ) u q + 1 = 0 . \lim_{u\to\infty}\frac{f(u)}{u^{q+1}}=0. We show that under certain additional assumptions on f, the above problem possesses at least three distinct solutions for a certain range of λ. We use the method of sub-supersolutions and a critical point theorem by Amann [H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 1976, 4, 620–709] to prove our results. Moreover, we prove a new existence result for a suitable infinite semipositone nonlocal problem which played a crucial role to obtain our main result and is of independent interest.

Boundary Blow-up Solutions to Nonlocal Elliptic Systems of Cooperative Type

In this paper, we consider the boundary blow-up solutions to the nonlocal elliptic systems of cooperative type. By introducing the boundary measure, the boundary blow-up problem becomes a Cauchy problem. Then using the super-subsolution method, we obtain the existence and nonexistence of positive solutions. Moreover, we study the stability of the minimal solution to the Cauchy problem.

The existence of positive solutions for Kirchhoff-type problems via the sub-supersolution method

Abstract In this paper we discuss the existence of a solution between wellordered subsolution and supersolution of the Kirchhoff equation. Using the sub-supersolution method together with a Rabinowitz-type global bifurcation theory, we establish the existence of positive solutions for Kirchhoff-type problems when the nonlinearity is singular or sign-changing. Moreover, we obtain some necessary and sufficient conditions for the existence of positive solutions for the problem when N = 1.

Dynamics of a Nonlocal Dispersal Model with a Nonlocal Reaction Term

In this paper, we study a class of nonlocal dispersal problem with a nonlocal term arising in population dynamics: [Formula: see text] where [Formula: see text] is a bounded domain, [Formula: see text], [Formula: see text] represents the nonlocal dispersal operator with continuous and non-negative dispersal kernel. The kernel [Formula: see text] is assumed to be non-negative and is allowed to have a degeneracy in a smooth subdomain [Formula: see text] of [Formula: see text]. When [Formula: see text] is either positive or vanishes in a subdomain, we respectively investigate the existence, multiplicity and asymptotical stability of positive steady states under the local/global variation of parameter by means of sub-supersolution method, Lyapunov–Schmidt reduction, and bifurcation theory.