Existence Of Nonnegative Solutions Research Articles

A note on quasilinear Schrödinger equations with singular or vanishing radial potentials

In this note, we complete the study of [3], where we got existence results for the quasilinear elliptic equation \begin{equation*} -\Delta w+ V\left( \left| x\right| \right) w - w \left( \Delta w^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \end{equation*} with singular or vanishing continuous radial potentials $V(r)$, $K(r)$. In [3], we assumed, for technical reasons, that $K(r)$ was vanishing as $r \rightarrow 0$, while in the present paper, we remove this obstruction. To face the problem, we apply a suitable change of variables $w=f(u)$ and we find existence of non negative solutions by the application of variational methods. Our solutions satisfy a weak formulations of the above equation, but they are in fact classical solutions in $\mathbb{R}^{N} \setminus \{0\}$. The nonlinearity $g$ has a double-power behavior, whose standard example is $g(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$ ($t > 0$), recovering the usual case of a single-power behavior when $q_1 = q_2$.

Stability and bifurcation analysis of a tri-trophic food chain model with intraguild predation

The dynamics of a three-component model for food web with intraguild predation is considered. The model is based on the collection of ordinary differential equations that describe the interactions among prey, intermediate predator and top predator. First, the model without self-limitation of the predators is studied. Boundedness of the system and existence of non-negative solutions are established. The local stability analysis of the equilibria is carried out to examine the behavior of the system. The possibility of Hopf bifurcation around non-negative equilibria with consumption rates as bifurcation parameters is studied. Center manifold theorem and the normal form theory are applied to obtain the formulas for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations support the analytical findings, which show that the extinction of one of the predators can occur under certain restrictions on the predation rate of the top predator. Subsequently, numerical analysis of the model with self-limitation of the predators is carried out. Simulations reveal that the system with intraspecific competition in the predator populations can reproduce coexistence between the three species in resource-rich environment.

Turing–Hopf Bifurcation and Its Normal Form in a Diffusive Three-Species Food Chain System with Strong Allee Effect

In this paper, we consider a diffusive three-species food chain system with strong Allee effect subject to the Neumann boundary condition. The dynamics of the local system including stability of interior equilibria and Hopf bifurcation are discussed. For the diffusive system, we study the existence of non-negative solutions, stability of a positive homogeneous steady state, diffusion-driven Turing instability and the occurrence of Turing–Hopf bifurcation. Employing the multiple scale analysis, we derive the normal form of Turing–Hopf bifurcation, which helps us classify the dynamical behaviors of the diffusive system near the Turing–Hopf bifurcation point.

On the Non-Negative Radial Solutions of the Two Dimensional Bratu Equation

In this paper, we study the boundary value problem on the unit circle for the Bratu’s equation depending on the real parameter μ. From the parameter estimate, the existence of non-negative solution is set. A numerical method is suggested to justify the theoretical result. It is a combination of the adaptation of finite difference and Gauss-Seidel method allowing us to obtain a good approximation of μc, with respect to the exact theoretical method μc = λ = 5.7831859629467.

On the non existence of non negative solutions to a critical Growth-Fragmentation Equation

A growth fragmentation equation with constant dislocation density measure is considered, in which growth and division rates balance each other. This leads to a simple example of equation where the so called Malthusian hypothesis $(M_+)$ of J. Bertoin and A. Watson (2016) is not necessarily satisfied. It is proved that when that happens, and as it was first suggested by these authors, no global non negative weak solution, satisfying some boundedness condition on several of its moments, exist. Non existence of local non negative solutions satisfying a similar condition, is proved to happen also. When a local non negative solution exists, the explicit expression is given.

Existence of Nonnegative Solutions of Linear Autonomous Functional Differential Equations

It is shown that if we exclude the existence of nontrivial small solutions, then a linear autonomous functional differential equation has a nontrivial nonnegative solution if and only if it has a nonnegative eigenfunction.

The Karcher mean of three variables and quadric surfaces

The Riemannian or Karcher mean has recently become an important tool for the averaging and study of positive definite matrices. Finding an explicit formula for the Karcher mean is problematic even for 2×2 triples. In this paper we study (1) the linear formula for the Karcher mean of 2×2 positive definite Hermitian matrices: Λ(A,B,C)=xA+yB+zC with nonnegative coefficients, where the existence of nonnegative solutions is guaranteed by Sturm's SLLN and Holbrook's no dice theorem, and (2) the quadric surface induced by the determinantal formula: det⁡(ABC)13=det⁡(xA+yB+zC). We show that the solution set forms a simplex of dimension less than equal 2 and settle the first problem for linearly dependent case. A classification of the quadric surfaces from the linear form of Karcher means is presented in terms of linear (in)dependence of A,B,C: hyperboloid of two sheets, hyperbolic cylinder, and parallel planes.

Existence and Uniqueness of a Chemotaxis System Influenced by Cancer Cells

We present a mathematical analysis of a reaction-diffusion model in a bounded open domain which describes vascular endothelial growth factor(VEGF), endothelial cells and oxygen. We use the parabolic theory to prove the existence of the solution in the function space under the homogeneous Neumann conditions. Then we get the existence of nonnegative solution in by using the global Schauder estimation.

Stieltjes Differential in Impulse Nonlinear Problems

An impulse nonlinear problem admitting discontinuous solutions that are functions of bounded variation is studied. This problem models the deformation of a discontinuous string (chains of strings fastened together by springs) with elastic supports in the form of linear and nonlinear springs (for example, springs with different turns, whose deformations do not obey Hooke’s law). The model is described by a second-order differential equation with derivatives in special measures and Dirichlet boundary conditions. Existence theorems are proved, and conditions for the existence of nonnegative solutions are obtained.

Existence of Nonnegative Solutions for a Fractional Parabolic Equation in the Entire Space

We study the existence of nonnegative solutions of a parabolic problem $$ \frac{\partial u}{\partial t}=-{\left(-\Delta \right)}^{\frac{\alpha }{2}}u+\frac{c}{{\left|x\right|}^{\alpha }}u $$ in ℝd × (0, T). Here, 0 < α < min(2, d), $$ {\left(-\Delta \right)}^{\frac{\alpha }{2}} $$ is the fractional Laplacian on ℝd and u0 ∈ L2(ℝd).

Fractional Kirchhoff problems with critical Trudinger\u2013Moser nonlinearity

This paper is concerned with the existence of solutions for a class of fractional Kirchhoff-type problems with Trudinger–Moser nonlinearity: M∬R2N|u(x)-u(y)|N/s|x-y|2Ndxdy(-Δ)N/ssu=f(x,u)inΩ,u=0inRN\\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}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} M\\left( \\displaystyle \\iint _{{\\mathbb {R}}^{2N}}\\frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}dxdy\\right) (-\\Delta )^{s}_{N/s}u=f(x,u)\\,\\, \\ &{}\\quad \\mathrm{in}\\ \\Omega ,\\\\ u=0\\ \\ \\ \\ &{}\\quad \\mathrm{in}\\ {\\mathbb {R}}^N{\\setminus } \\Omega , \\end{array}\\right. } \\end{aligned}$$\\end{document}where (-Delta )^{s}_{N/s} is the fractional N / s-Laplacian operator, Nge 1, sin (0,1), Omega subset {mathbb {R}}^N is a bounded domain with Lipschitz boundary, M:{mathbb {R}}^+_0rightarrow {mathbb {R}}^+_0 is a continuous function, and f:Omega times {mathbb {R}}rightarrow {mathbb {R}} is a continuous function behaving like exp (alpha t^{2}) as trightarrow infty for some alpha >0. We first obtain the existence of a ground state solution with positive energy by using minimax techniques combined with the fractional Trudinger–Moser inequality. Next, the existence of nonnegative solutions with negative energy is established by using Ekeland’s variational principle. The main feature of this paper consists in the presence of a (possibly degenerate) Kirchhoff model, combined with a critical Trudinger–Moser nonlinearity.

A MICROBIAL CONTINUOUS CULTURE SYSTEM WITH DIFFUSION AND DIVERSIFIED GROWTH

A reaction-diffusion model is presented to describe the microbial continuous culture with diversified growth. The existence of nonnegative solutions and attractors for the system is obtained, the stability of steady states and the steady state bifurcation are studied under three growth conditions. In the case of no growth inhibition or only product inhibition, the system admits one positive constant steady state which is stable; in the case of growth inhibition only by substrate, the system can have two positive constant steady states, explicit conditions of the stability and the steady state bifurcation are also determined. In addition, numerical simulations are given to exhibit the theoretical results.

Complete quenching phenomenon for a parabolic p-Laplacian equation with a weighted absorption

Throughout this paper, we mainly consider the parabolic p-Laplacian equation with a weighted absorption u_{t}-operatorname{div} (|nabla u|^{p-2}nabla u )=-lambda |x|^{alpha} {chi}_{{u>0}}u^{-beta} in a bounded domain Omegasubseteqmathbb{R}^{n} (ngeq1) with Lipschitz continuous boundary subject to homogeneous Dirichlet boundary condition. Here lambda>0 and alpha>-n are parameters, and betain(0,1) is a given constant. Under the assumptions u_{0}in W_{0}^{1,p}(Omega)cap L^{infty}(Omega), u_{0}geq0 a.e. in Ω, we can establish conditions of local and global in time existence of nonnegative solutions, and show that every global solution completely quenches in finite time a.e. in Ω. Moreover, we give some numerical experiments to illustrate the theoretical results.

Existence of nonnegative solutions for parabolic problem on Dirichlet forms

ABSTRACTIn this work, we investigate perturbed parabolic problem on measure spaces. For strongly local Dirichlet form, we obtain some results on existence and uniqueness of nonnegative solution. We also presented some examples as an applications.

Fractional Schrödinger–Poisson–Kirchhoff type systems involving critical nonlinearities

The paper is concerned with existence, multiplicity and asymptotic behavior of nonnegative solutions for a fractional Schrödinger–Poisson–Kirchhoff type system. As a consequence, the results can be applied to the special case (a+b‖u‖2)[(−Δ)su+V(x)u]+ϕk(x)|u|p−2u=λh(x)|u|q−2u+|u|2s∗−2uinR3,(−Δ)tϕ=k(x)|u|pinR3,‖u‖=∫R3∫R3|u(x)−u(y)|2|x−y|3+2sdxdy+∫R3V(x)|u|2dx1∕2,where a,b≥0 are two numbers, with a+b>0, 1<p<2s,t∗=3+2t3−2s, λ>0 is a parameter, s,t∈(0,1), (−Δ)s is the fractional Laplacian, k∈L63+2t−p(3−2s)(R3) may change sign, V:R3→[0,∞) is a potential function, 2s∗=6∕(3−2s) is the critical Sobolev exponent, 1<q<2s∗ and h∈L2s∗2s∗−q(R3). First, when θ<p<2s∗∕2, 2p≤q<2s∗ and λ is large enough, existence of nonnegative solutions is obtained by the mountain pass theorem. Moreover, we obtain that limλ→∞‖uλ‖=0. Then, via the Ekeland variational principle, existence of nonnegative solutions is investigated when θ<p<2s∗∕2, 1<q<2 and λ is small enough, and we obtain that limλ→0‖uλ‖=0. Finally, we consider the system with double critical exponents, that is, p=2s,t∗, and obtain two nontrivial and nonnegative solutions in which one is least energy solution and another is mountain pass solution. The paper covers a novel feature of Kirchhoff problems, which is the fact that the parameter a can be zero. Hence the results of the paper are new even for the standard stationary Schrödinger–Poisson–Kirchhoff system.

Global Existence of Solutions of the Gierer-Meinhardt System with Mixed Boundary Conditions

We study the global (in time) existence of nonnegative solutions of the Gierer-Meinhardt system with mixed boundary conditions. In the research, the Robin boundary and Neumann boundary conditions were used on the activator and the inhibitor conditions respectively. Based on the priori estimates of solutions, the considerable results were obtained.

Existence of solutions for a parabolic equation in bounded domain

In this paper, we study the existence of nonnegative solutions for the parabolic problemwhere D is a smooth domain of , (), the function is such that and is the normalized positive eigenfunction corresponding to the first eigenvalue of and V belongs to a class of time-dependant potentials which can be written as a nonlinear combination of derivatives of a function.

Existence of solutions for fractional p-Laplacian problems via Leray-Schauder’s nonlinear alternative

In this paper, we are concerned with the existence of solutions for a class of quasilinear elliptic problems driven by a nonlocal integro-differential operator with homogeneous Dirichlet boundary data. As a particular case, we study the following problem: $$\begin{aligned}& (-\Delta)_{p}^{s} u =f (x,u )\quad \mbox{in } \Omega, \\& u=0 \quad \mbox{in } \mathbb{R}^{N}\setminus\Omega, \end{aligned}$$ where $(-\Delta)_{p}^{s}$ is the fractional p-Laplace operator, Ω is an open bounded subset of $\mathbb{R}^{N}$ with Lipschitz boundary, and $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Caratheodory function. The existence of nonnegative solutions is obtained by using Leray-Schauder’s nonlinear alternative.

On the existence of nonnegative solutions to the Dirichlet boundary value problem for the p-Laplace equation in the presence of exterior mass forces

We consider the Dirichlet problem for an inhomogeneous p-Laplace equation with nonlinear source in the presence of exterior mass forces.We obtain new sufficient conditions for the existence of a weak nonnegative bounded solution. The sufficient conditions are written in explicit form through the data of the problem.

Well-posedness for a class of biological diffusion models with hysteresis effect

This paper is concerned with a class of biological models which consist of nonlinear diffusion equations and a hysteresis operator describing the relationship between some variables of the equations. The existence of solutions to the analogous problem was ever considered by Aiki and Minchev (SIAM J Math Anal 36:2020–2032, 2005) under some assumptions including the global Lipschitz continuity of reaction terms. We show the existence of nonnegative solutions to the problem under consideration using the approximation method when the reaction terms are locally Lipschitz continuous. Moreover, we discuss the continuous dependence of solutions on initial data.