Sub-supersolution Method Research Articles

Stability Results of Positive Weak Solution for a Class of Chemically Reacting Systems

This paper aims to study the existence and non-existence results of positive weak solution to the quasilinear elliptic system: \[ \begin{cases} -\Delta_p u = \lambda a(x) \left[ f(u,v) - \dfrac{1}{u^\alpha} \right], & x \in \Omega, \\ -\Delta_q v = \lambda b(x) \left[ g(u,v) - \dfrac{1}{v^\beta} \right], & x \in \Omega, \\ u = 0 = v, & x \in \partial\Omega, \end{cases} \] where \(\Delta_r w = \operatorname{div}(|\nabla w|^{r-2} \nabla w)\) is the \(r\)-Laplacian (\(r = p, q\)), \(r > 1\), \(\alpha, \beta \in (0,1)\), \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) (\(N > 1\)) with smooth boundary \(\partial\Omega\) and \(\lambda\) is a positive parameter. Here \(f, g\) are \(C^1\) increasing functions such that \(f, g : \mathbb{R}^+ \times \mathbb{R}^+ \rightarrow \mathbb{R}^+\); \(f(\upsilon_1, \upsilon_2) > 0\), \(g(\upsilon_1, \upsilon_2) > 0\) for \(\upsilon_1, \upsilon_2 > 0\). With \(C^1\) sign-changing functions \(a(x)\), \(b(x)\) that perhaps have negative values nearby the boundary. We establish our results via the sub-supersolution method. In addition, we study the stability and instability results of positive weak solution with different choices of \(f\) and \(g\).

Existence of positive solutions for an elliptic system with sign-changing weights

Using the method of sub-super solutions and comparison principle, we study the existence of positive solutions for a class of quasilinear elliptic systems with sign-changing weights.

Existence of Positive Solutions for Non-Local Magnetic Fractional Systems

In this paper, the existence of a weak positive solution for non-local magnetic fractional systems is studied in the fractional magnetic Sobolev space through a sub-supersolution method combined with iterative techniques.

Fractional double-phase nonlocal equation in Musielak-Orlicz Sobolev space

In this paper, we analyze the existence of solutions to a double-phase fractional equation of the Kirchhoff type in Musielak-Orlicz Sobolev space with variable exponents. Our approach is mainly based on the sub-supersolution method and the mountain pass theorem.

Periodic solutions of an NPZ model with periodic delay and space heterogeneity

In this paper, we investigate a nutrient-phytoplankton-zooplankton (NPZ) model with seasonality and spatial heterogeneity, in which the immature zooplankton with size structure is described by the quasi-linear first-order partial differential equation. A three-dimensional periodic delay food chain model with initial value is derived from the original model. To establish the threshold dynamics of NPZ model, we first obtain the global attractivity of the positive periodic solution of the nutrient-phytoplankton (NP) subsystem by using the method of sub-super solutions and associated iterations. We then give the integral expression of threshold for the corresponding system, which is convenient for numerical simulations. Based on this, the threshold dynamics is discussed by appealing to the comparison arguments and the persistence theory. Finally, the analytic results are in good consistence with our numerical simulations.

Periodic fractional Ambrosetti–Prodi for one-dimensional problem with drift

We prove Ambrosetti–Prodi type results for periodic solutions of some one-dimensional nonlinear problems that can have drift term whose principal operator is the fractional Laplacian of order s∈(0,1). We establish conditions for the existence and nonexistence of solutions of those problems. The proofs of the existence results are based on the sub-supersolution method combined with topological degree type arguments. We also obtain a priori bounds in order to get multiplicity results. We also prove that the solutions are C1,α under some regularity assumptions in the nonlinearities, that is, the solutions of the mentioned equations are classical. We finish the work obtaining Ambrosetti-Prodi-type results for a problem with singular nonlinearities.

Generalized Schrödinger–Bopp–Podolsky type system with singular nonlinearity

In this paper we establish the existence, non-existence and multiplicity of solutions for a class of generalized Schrödinger–Bopp–Podolsky system with singular nonlinearity. The main techniques we use are some results on critical point theory for non-differentiable functionals and the sub-supersolution method.

A Sub-supersolution Method for Integro-differential Semilinear Elliptic Equations and Some Applications

A Sub-supersolution Method for Integro-differential Semilinear Elliptic Equations and Some Applications

Multiplicity of solutions for fractional p(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p ( z ) $\\end{document}-Kirchhoff-type equation

This work deals with the existence and multiplicity of solutions for a class of variable-exponent equations involving the Kirchhoff term in variable-exponent Sobolev spaces according to some conditions, where we used the sub-supersolutions method combined with the mountain pass theory.

Constant sign and nodal solutions for singular Gierer–Meinhardt-type system

We establish the existence of three solutions for singular semilinear elliptic system, two of which are of opposite constant sign. Under a strong singularity effect, the third solution is nodal with synchronous sign components. The approach combines sub-supersolutions method and Leray–Schauder topological degree involving perturbation argument.

Singular quasilinear convective systems involving variable exponents

The paper deals with the existence of solutions for quasilinear elliptic systems involving singular and convection terms with variable exponents. The approach combines the sub-supersolutions method and Schauder's fixed point theorem.

The sub-supersolution method for variable exponent double phase systems with nonlinear boundary conditions

In this paper, we study quasilinear elliptic systems driven by variable exponent double phase operators involving fully coupled right-hand sides and nonlinear boundary conditions. The aim of our work is to establish an enclosure and existence result for such systems by means of trapping regions formed by pairs of sub- and supersolutions. Under very general assumptions on the data, we then apply our result to get infinitely many solutions. Moreover, we discuss the case when we have homogeneous Dirichlet boundary conditions and present some existence results for this kind of problem.

Optimal investment and consumption strategies for an investor with stochastic economic factor in a defaultable market

This paper considers the issue of optimal investment and consumption strategies for an investor with stochastic economic factor in a defaultable market. In our model, the price process is composed of a money market account and a default-free risky asset, assuming they rely on a stochastic economic factor described by a diffusion process. A defaultable perpetual bond is depicted by the reduced-form model, and both the default risk premium and the default intensity of it rely on the stochastic economic factor. Our goal is to maximize the infinite horizon expected discounted power utility of the consumption. Applying the dynamic programming principle, we derive the Hamilton–Jacobi–Bellman (HJB) equations and analyze them using the so-called sub-super solution method to prove the existence and uniqueness of their classical solutions. Next, we use a verification theorem to derive the explicit formula for optimal investment and consumption strategies. Finally, we provide a sensitivity analysis.

Fractional Kirchhoff-Type and Method of Sub-supersolutions

Fractional Kirchhoff-Type and Method of Sub-supersolutions

Existence of the positive solutions for a class of nonlinear elasticity problems involving Orlicz spaces via sub-supersolution method

In this paper, we study nonlinear elastic problem of the type{−ΔΦu+K(x)u−α|u|LΛγ=λup|u|LΨq,inΩ,u=0,on∂Ω, where q,γ are positive constants and p,α∈(0,1). |⋅|LΨ, |⋅|LΛ and Φ are defined later. Based on the pseudomonotone operators theory, we establish some new sub-supersolution theorems, that generalize the results in [12]. In addition, based on the obtained theorems, we prove the existence of positive solutions to the above mentioned under certain assumptions about the parameters. Our work improves the results of some past researches.

On positive periodic solutions of the time–space periodic Lotka–Volterra cooperating system in multi-dimensional media

This work is devoted to the study of the time–space periodic reaction–diffusion–advection Lotka–Volterra cooperating system in multi-dimensional media. By using the method of sub-super solutions and its associated iterations, we prove the existence and uniqueness of the positive periodic solution under appropriate conditions. Finally, we are able to derive the asymptotic behavior of the solutions to the associated Cauchy problem.

An exact bifurcation diagram for ap-qLaplacian boundary value problem

We study positive solutions to thep–qLaplacian two-point boundary value problem:{−μ[(u′)p−1]′−[(u′)q−1]′=λu(1−u)on (0,1)u(0)=0=u(1)whenp=4andq=2. Hereλ>0is a parameter andμ≥0is a weight parameter influencing the higher-order diffusion term. Whenμ=0(the Laplacian case) the exact bifurcation diagram for a positive solution is well-known, namely, whenλ≤π2there are no positive solutions, and forλ>π2there exists a unique positive solutionuλ,μsuch that‖uλ,μ‖∞→0asλ→π2and‖uλ,μ‖∞→1asλ→∞. Here, we will prove that for allμ>0similar bifurcation diagrams preserve, and they all bifurcate from(λ,u)=(π2,0). Our results are established via the method of sub-super solutions and a quadrature method. We also present computational evaluations of these bifurcation diagrams for various values ofμand illustrate how they evolve whenμvaries.

On the ratio of total masses of species to resources for a logistic equation with Dirichlet boundary condition

We consider the stationary problem for a diffusive logistic equation with the homogeneous Dirichlet boundary condition. Concerning the corresponding Neumann problem, Wei-Ming Ni proposed a question as follows: Maximizing the ratio of the total masses of species to resources. For this question, Bai, He and Li [1] showed that the supremum of the ratio is 3 in the one dimensional case, and the author and Kuto [14] showed that the supremum is infinity in the multi-dimensional ball. In this paper, we prove that the same results still hold true for the Dirichlet problem, and our proof is based on the sub-super solution method. Moreover, we show the differences of the solutions corresponding to the maximizing sequence between the one- and two-dimensional domains.

Multiplicity for a strongly singular quasilinear problem via bifurcation theory

A [Formula: see text]-Laplacian elliptic problem in the presence of both strongly singular and [Formula: see text]-superlinear nonlinearities is considered. We employ bifurcation theory, approximation techniques and sub-supersolution method to establish the existence of an unbounded branch of positive solutions, which is bounded in positive [Formula: see text]-direction and bifurcates from infinity at [Formula: see text]. As consequence of the bifurcation result, we determine intervals of existence, nonexistence and in particular cases, global multiplicity.

Existence and nonexistence of positive solutions of some elliptic problems with variable exponents

In this paper, we study the existence and the nonexistence of some positive solutions for nonlinear equations involving variable exponent Laplace operator and concave–convex second term: under Robin boundary condition in a regular open‐bounded domain of . The ‐Laplacian is given by where and . Our proofs are based on the subsupersolutions method mixed with variational arguments.