Class Of Quasilinear Elliptic Systems Research Articles

Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis

Abstract This article is concerned with the stationary problem for a prey-predator model with prey-taxis/predator-taxis under homogeneous Dirichlet boundary conditions, where the interaction is governed by a Beddington-DeAngelis functional response. We make a detailed description of the global bifurcation structure of coexistence states and find the ranges of parameters for which there exist coexistence states. At the same time, some sufficient conditions for the nonexistence of coexistence states are also established. Our method of analysis uses the idea developed by Cintra et al. (Unilateral global bifurcation for a class of quasilinear elliptic systems and applications, J. Differential Equations 267 (2019), 619–657). Our results indicate that the presence of prey-taxis/predator-taxis makes mathematical analysis more difficult, and the Beddington-DeAngelis functional response leads to some different phenomena.

Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition

Abstract We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N {{\mathbb{R}}}^{N} . We assume that the nonlinear term satisfies the locally super- ( m 1 , m 2 ) \left({m}_{1},{m}_{2}) condition, that is, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ G x\in G , where G G is a domain in R N {{\mathbb{R}}}^{N} , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ R N x\in {{\mathbb{R}}}^{N} . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.

On a class of quasilinear elliptic systems

On a class of quasilinear elliptic systems

Existence of Positive Solutions for an Elliptic System Involving Nonlocal Operator

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 involving nonlocal operator.

Precise Morrey regularity of the weak solutions to a kind of quasilinear systems with discontinuous data

We consider the Dirichlet problem for a class of quasilinear elliptic systems in domain with irregular boundary. The principal part satisfies componentwise coercivity condition and the nonlinear terms are Caratheodory maps having Morrey regularity in $x$ and verifying controlled growth conditions with respect to the other variables. We have obtained boundedness of the weak solution to the problem that permits to apply an iteration procedure in order to find optimal Morrey regularity of its gradient.

Morse theory methods for a class of quasi-linear elliptic systems of higher order

We develop the local Morse theory for a class of non-twice continuously differentiable functionals on Hilbert spaces, including a new generalization of the Gromoll-Meyer's splitting theorem and a weaker Marino-Prodi perturbation type result. They are applicable to a wide range of multiple integrals with quasi-linear elliptic Euler equations and systems of higher order.

Unilateral global bifurcation for a class of quasilinear elliptic systems and applications

In this paper we establish a unilateral bifurcation result for a class of quasilinear elliptic system strongly coupled, extending the bifurcation theorem of [1]. To this aim, we use several results, such that Fredholm operator of index 0 theory, eigenvalues of elliptic operators and the Krein–Rutman theorem. Lastly, we apply this result to some particular systems arising from population dynamics and determine a region of existence of coexistence states.

Ground states for a class of quasilinear elliptic systems with critical exponent

We study the following coupled system of quasilinear equations: −Δpu+μ|u|p−2u=|u|q−2u+αλ|u|α−2u|v|β,x∈RN,−Δpv+ν|v|p−2v=|v|p∗−2v+βλ|u|α|v|β−2v,x∈RN,where N≥3, μ,ν,λ>0, α,β≥1, 2<p<q<p∗ and α+β=p. We establish some results about the existence and regularity of ground state solutions for the p-Laplacian systems by using variational methods. We also study the asymptotic behavior of solutions as the coupling parameter λ tends to zero.

Maximum principles for some quasilinear elliptic systems

We give maximum principles for solutions u:Ω→RN to a class of quasilinear elliptic systems whose prototype is −∑i=1n∂∂xi∑β=1N∑j=1nai,jα,βx,u(x)∂uβ∂xj(x)=0,x∈Ω,where α∈{1,…,N} is the equation index and Ω is an open, bounded subset of Rn. We assume that coefficients ai,jα,β(x,y) are measurable with respect to x, continuous with respect to y∈RN, bounded and elliptic. In vectorial problems, when trying to bound the solution by means of the boundary data, we need to bypass De Giorgi’s counterexample by means of some additional structure assumptions on the coefficients ai,jα,β(x,y). In this paper, we assume that off-diagonal coefficients ai,jα,β, α≠β, have support in some staircase set along the diagonal in the yα,yβ plane.

The Solvability of Concave-Convex Quasilinear Elliptic Systems Involving $p$-Laplacian and Critical Sobolev Exponent

In this work, we study the existence of non-trivial multiple solutions for a class of quasilinear elliptic systems equipped with concave-convex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and Palais-Smale condition, we prove the existence and multiplicity results of positive solutions.

Liouville‐type results for a class of quasilinear elliptic systems and applications

In this paper, we study the following quasilinear elliptic systems − Δ m u = f 1 ( u , v ) i n R N , − Δ m v = f 2 ( u , v ) i n R N . Here Δ m is the well-known m-Laplacian operator, m > 1 . Under appropriate conditions on the functions f 1 and f 2 , we prove some Liouville-type nonexistence theorems. The Liouville-type results can be applied to non-cooperative quasilinear Schrödinger-type systems, and we obtain a priori estimates, singularity and decay estimates for the non-negative solutions of the non-cooperative quasilinear Schrödinger-type systems.

Existence of three solutions for a class of quasilinear elliptic systems involving the $p(x)$-Laplace operator

The aim of this paper is to obtain three weak solutions for the Dirichlet quasilinear elliptic systems on a bonded domain. Our technical approach is based on the general three critical points theorem obtained by Ricceri.

Quasilinear elliptic systems convex-concave singular terms and $\Phi$-Laplacian operator

This paper deals with the existence of positive solutions for a class of quasilinear elliptic systems involving the $\Phi$-Laplacian operator and convex-concave singular terms. Our approach is based on the generalized Galerkin Method along with perturbation techniques and comparison arguments in the setting of Orlicz-Sobolev spaces.

Global Existence Structure of Parameters for Positive Solutions of a Singular $$(p_1,p_2)$$ ( p 1 , p 2 ) -Laplacian System

We study the homogeneous Dirichlet boundary value problem of a singular \((p_1,p_2)\)-Laplacian system with multiparameters. The global structure of multiparameters for existence, nonexistence and multiplicity of positive solutions can be obtained. Proofs are mainly based on the upper and lower solution method, the fixed point index theory in a cone, generalized Picone identity, global continuum theorem and degree arguments. As an application, we can establish the global result of the positive radial and decaying solutions for a class of quasilinear elliptic systems in exterior domains.

Existence of ground state solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces

In this paper, we investigate the following nonlinear and non-homogeneous elliptic system: \t\t\t{−div(a1(|∇u|)∇u)+V1(x)a1(|u|)u=Fu(x,u,v)in RN,−div(a2(|∇v|)∇v)+V2(x)a2(|v|)v=Fv(x,u,v)in RN,(u,v)∈W1,Φ1(RN)×W1,Φ2(RN),\\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} $$\\begin{aligned} \\textstyle\\begin{cases} {-}\\operatorname{div}(a_{1}( \\vert \\nabla{u} \\vert )\\nabla{u})+V_{1}(x)a_{1}( \\vert u \\vert )u=F_{u}(x,u,v)\\quad \\mbox{in } \\mathbb{R}^{N},\\\\ {-}\\operatorname{div}(a_{2}( \\vert \\nabla{v} \\vert )\\nabla{v})+V_{2}(x)a_{2}( \\vert v \\vert )v=F_{v}(x,u,v) \\quad\\mbox{in } \\mathbb{R}^{N},\\\\ (u, v)\\in W^{1,\\Phi_{1}}(\\mathbb{R}^{N})\\times W^{1, \\Phi_{2}}(\\mathbb{R}^{N}), \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} where phi_{i}(t)=a_{i}( vert t vert )t (i=1,2) are two increasing homeomorphisms from mathbb{R} onto mathbb{R}, functions V_{i}(i=1,2) and F are 1-periodic in x, and F satisfies some (phi_{1},phi_{2})-superlinear Orlicz-Sobolev conditions. By using a variant mountain pass lemma, we obtain that the system has a ground state.

Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces

Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces

Nonlinear elliptic systems and mean-field games

We consider a class of quasilinear elliptic systems of PDEs consisting of N Hamilton–Jacobi–Bellman equations coupled with N divergence form equations, generalising to N > 1 populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. We provide a wide range of sufficient conditions for the existence of solutions to these systems: either the Hamiltonians are required to behave at most linearly for large gradients, as it occurs when the controls of the agents are bounded, or they must grow faster than linearly and not oscillate too much in the space variables, in a suitable sense. We show the connection of these systems with the classical strongly coupled systems of Hamilton–Jacobi–Bellman equations of the theory of N-person stochastic differential games studied by Bensoussan and Frehse. We also prove the existence of Nash equilibria in feedback form for some N-person games.

Positive solutions for infinite semipositone/positone quasilinear elliptic systems with singular and superlinear terms

We establish existence and regularity of positive solutions for a class of quasilinear elliptic systems with singular and superlinear terms. The approach is based on sub-supersolution methods for systems of quasilinear singular equations and the Schauder's fixed point Theorem.

A comparison type principle for a class of quasilinear elliptic systems: critical and subcritical cases

A comparison type principle for a class of quasilinear elliptic systems: critical and subcritical cases

Existence of weak solutions for a class of quasilinear elliptic systems

Existence of weak solutions for a class of nonlinear elliptic systems is obtained under the certain Landesman-Lazer-type conditions by variational method.