Quasilinear Elliptic Systems Research Articles

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.

Existence and a priori estimates of solutions for quasilinear singular elliptic systems with variable exponents

This article sets forth results on the existence, a priori estimates and boundedness of positive solutions of a singular quasilinear systems of elliptic equations involving variable exponents. The approach is based on Schauder’s fixed point Theorem. A Moser iteration procedure is also obtained for singular quasilinear systems involving variable exponents establishing a priori estimates and boundedness of solutions.

Quasilinear elliptic systems in divergence form associated to general nonlinearities

Abstract The paper is concerned with a priori estimates of positive solutions of quasilinear elliptic systems of equations or inequalities in an open set of Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} associated to general continuous nonlinearities satisfying a local assumption near zero. As a consequence, in the case Ω = ℝ N {\Omega=\mathbb{R}^{N}} , we obtain nonexistence theorems of positive solutions. No hypotheses on the solutions at infinity are assumed.

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.

On the extremal parameters curve of a quasilinear elliptic system of differential equations

We consider a system of quasilinear elliptic equations, with indefinite super-linear nonlinearity, depending on two real parameters $$\lambda ,\mu $$ . By using the Nehari manifold and the notion of extremal parameter, we extend some results concerning existence of positive solutions.

Multiple Solutions for Nonvariational Quasilinear Elliptic Systems

We establish the existence of at least three nontrivial solutions for a nonvariational quasilinear elliptic system with homogeneous Dirichlet boundary condition. Two of these solutions are of opposite constant sign and the third one is nodal in an appropriate sense provided that a suitable location occurs. The approach combines the methods of sub-supersolution and Leray–Schauder topological degree.

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

3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler–Poisson system

We address the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler–Poisson system in a cylinder supplemented with non-small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl (= angular momentum density) component. With the newly introduced decomposition, a quasilinear elliptic system of second order is derived from the elliptic modes in Euler–Poisson system for subsonic flows. Due to the nonzero swirl, the main difficulties lie in the solvability of a singular elliptic equation which concerns the angular component of the vorticity in its cylindrical representation, and in analysis of streamlines near the axis r=0.

Positive symmetric results for a weighted quasilinear elliptic system with multiple critical exponents in mathbb{R}^{N}

This work focuses on the symmetric solutions for a weighted quasilinear elliptic system involving multiple critical exponents in mathbb{R}^{N}. Based upon the Caffarelli-Kohn-Nirenberg inequality and the symmetric criticality principle due to Palais, we prove a variety of symmetric results under certain appropriate hypotheses on the singular potentials and the parameters.

Positive solutions for singular elliptic systems with convection term

We prove the existence and regularity of solutions for a quasi-linear elliptic system with convection terms that can be singular in the solution and its gradient. Comparison properties and a priori estimates are also obtained. Our approach relies on invariance, regularity, strong maximum principle, and fixed-point arguments.

THREE NONTRIVIAL NONNEGATIVE SOLUTIONS FOR SOME CRITICAL p-LAPLACIAN SYSTEMS WITH LOWER-ORDER NEGATIVE PERTURBATIONS

Three nontrivial nonnegative solutions for some critical quasilinear elliptic systems with lower-order negative perturbations are obtained by using the Ekeland's variational principle and the mountain pass theorem.

Symmetry-Breaking Global Bifurcation in a Surface Continuum Phase-Field Model for Lipid Bilayer Vesicles

We study a model for lipid bilayer membrane vesicles exhibiting phase separation, incorporating a phase field together with membrane fluidity and bending elasticity. We prove the existence of a plethora of equilibria in the large, corresponding to symmetry-breaking solutions of the Euler--Lagrange equations, via global bifurcation from the spherical state. To the best of our knowledge, this constitutes the first rigorous existence results for this class of problems. We overcome several difficulties in carrying this out. Due to inherent surface fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. This is resolved via a singularity-free radial-map description, which effectively eliminates the grossly underdetermined midplane deformation. The resulting governing equations comprise a quasi-linear elliptic system with nonlinear constraints. We show the equivalence of the problem to that of findin...

Entire solutions of quasilinear symmetric systems

We study the following quasilinear elliptic system for all i = 1,··· ,m −div(� ' (|∇ui| 2 )∇ui) = Hi(u) in R n where u = (ui) m=1 : R n → R m and the nonlinearity Hi(u) ∈ C 1 (R m ) → R is a general nonlinearity. Several celebrated operators such as the prescribed mean curvature, the Laplacian and the p-Laplacian operators fit in the above form, for appropriate �. We establish a Hamiltonian identity of the following form for all xn ∈ R Z Rn−1 m X i=1 � 1 2 � |∇u i| 2 � − � ' |∇ui| 2 � |@xnui| 2 � − ˜ H(u)

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.

Existence of classical solutions of quasi-linear non-cooperative elliptic systems

Method of sub- and super-solutions is applied in investigation of solvability in classical \(C^2(\Omega ) \bigcap C(\overline{\Omega })\) sense of quasi-linear non-cooperative weakly coupled systems of elliptic second-order PDE. The result is illustrated on a model example arising in differential geometry.

Multiple symmetric results for singular quasilinear elliptic systems with critical homogeneous nonlinearity

This paper deals with the existence and multiplicity of symmetric solutions for a class of singular quasilinear elliptic systems with critical homogeneous nonlinearity in a bounded symmetric domain. Applying variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of G‐symmetric solutions under some appropriate assumptions on the weighted functions and the parameters. Copyright © 2016 John Wiley & Sons, Ltd.