Solutions For Quasilinear Elliptic System Research Articles

Monotonic convergence of positive radial solutions for general quasilinear elliptic systems

We study the asymptotic behavior of positive radial solutions for quasilinear elliptic systems that have the form Δpu=c1|x|m1⋅g1v⋅|∇u|αin Rn,Δpv=c2|x|m2⋅g2v⋅g3|∇u|in Rn, where Δp denotes the p-Laplace operator, p > 1, n⩾2 , c1,c2>0 and m1,m2,α⩾0 . For a general class of functions gj which grow polynomially, we show that every non-constant positive radial solution (u, v) asymptotically approaches (u0,v0)=(Cλ|x|λ,Cμ|x|μ) for some parameters λ,μ,Cλ,Cμ>0 . In fact, the convergence is monotonic in the sense that both u/u0 and v/v0 are decreasing. We also obtain similar results for more general systems.

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.

Multiple solutions for quasilinear elliptic systems involving variable exponents

We establish the existence of multiple solutions for a nonvariational elliptic systems involving p(x)-Laplacian operator. The approach combines the methods of sub–supersolution and Leray–Schauder topological degree.

Multiple nonsemitrivial solutions for quasilinear elliptic systems via associated eigenvalue problems

In this paper, the existence, nonexistence and multiplicity of nonnegative solutions for quasilinear elliptic systems of resonant type involving $ (p,q) $-Laplacian are investigated. By using the generalized variant of the Nehari manifold and fibering method, we prove a suitable manifold with associated map for the systems and exploit the relation between them to study the existence and multiplicity of nonnegative solutions and obtain bifurcation results. Furthermore, we also apply the approach to treat a class of $ (p,q) $-Laplacian elliptic systems with concave-convex nonlinearities.

Asymptotic behavior of blowing-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids

In this paper, we deal with the following quasilinear elliptic system involving gradient terms in the form: $$ \begin{cases} \Delta_p u= v^m| \nabla u |^\alpha & \text{in}\quad \Omega\\ \Delta_p v= v^\beta| \nabla u |^q & \text{in}\quad \Omega, \end{cases} $$ where $\Omega\subset\mathbb{R}^N\ (N\geq 1)$ is either equal to $ \mathbb{R}^N $ or equal to a ball $B_R$ centered at the origin and having radius $R>0$, $1 < p < \infty$, $m,q > 0$, $\alpha\geq 0$, $0\leq \beta\leq m$ and $\delta:=(p-1-\alpha)(p-1-\beta)-qm \neq 0$. Our aim is to establish the asymptotics of the blowing-up radial solutions to the above system. Precisely, we provide the accurate asymptotic behavior at the boundary for such blowing-up radial solutions. For that, we prove a strong maximal principle for the problem of independent interest and study an auxiliary asymptotically autonomous system in $\mathbb{R}^3$.

Existence and Asymptotic Behaviors of Sign-Changing Solutions for Quasilinear Elliptic Systems

Existence and Asymptotic Behaviors of Sign-Changing Solutions for Quasilinear Elliptic Systems

Ground and bound state solutions for quasilinear elliptic systems including singular nonlinearities and indefinite potentials

It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by $ (\Phi_{1}, \Phi_{2}) $-Laplacian operator. The main feature here is to consider quasilinear elliptic systems involving both nonsingular nonlinearities combined with indefinite potentials and singular cases perturbed by superlinear and subcritical couple terms. These prevent us to use arguments based on Ambrosetti-Rabinowitz condition and variational methods for differentiable functionals. By exploring the Nehari method and doing a fine analysis on the fibering map associated, we get estimates that allow us unify the arguments to show multiplicity of semi-trivial solutions in both cases.

Existence of weak solutions for quasilinear elliptic systems in Orlicz spaces

ABSTRACTThis article deals with the existence of solutions for a quasilinear elliptic system in divergence form: in Ω, with boundary condition on , where and is a bounded open domain. The data f belongs to the dual space of and the resulting existence is proved by means of the Young measure and under mild monotonicity assumptions on σ.

Global a priori bounds for weak solutions of quasilinear elliptic systems with nonlinear boundary condition

In this paper we study quasilinear elliptic systems with nonlinear boundary condition with fully coupled perturbations even on the boundary. Under very general assumptions our main result says that each weak solution of such systems belongs to L∞(Ω‾)×L∞(Ω‾). The proof is based on Moser's iteration scheme. The results presented here can also be applied to elliptic systems with homogeneous Dirichlet boundary condition.

Solutions for quasilinear elliptic systems with vanishing potentials

In this paper, we study the following strongly coupled quasilinear elliptic system: $$ \begin{cases} -\Delta_{p} u+\lambda a(x)|u|^{p-2}u=\dfrac{\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}, &amp; x\in {\mathbb R}^{N}, \\[10pt] -\Delta_{p} v+\lambda b(x)|v|^{p-2}v=\dfrac{\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v, &amp; x\in {\mathbb R}^{N}, \\ u,v\in D^{1,p}(\R^{N}), \end{cases} $$% where $N\geq 3$, $\lambda&gt; 0$ is a parameter, $p&lt; \alpha+\beta&lt; p^{*}:={Np}/({N-p})$. Under some suitable conditions which are given in section 1, we use variational methods to obtain both the existence and multiplicity of solutions for the system on an appropriated space when the parameter $\lambda$ is sufficiently large. Moreover, we study the asymptotic behavior of these solutions when \hbox{$\lambda\rightarrow\infty$}.

Global and blow-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids

We study positive radial solutions of quasilinear elliptic systems with a gradient term in the form where is either a ball or the whole space, , , , and . We first classify all the positive radial solutions in case is a ball, according to their behavior at the boundary. Then we obtain that the system has non-constant global solutions if and only if and . Finally, we describe the precise behavior at infinity for such positive global radial solutions by using properties of three component cooperative and irreducible dynamical systems.

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.

Quasilinear elliptic systems with measure data

We study the existence of solutions of quasilinear elliptic systems involving N equations and a measure on the right hand side, with the form {−∑i=1n∂∂xi(∑β=1N∑j=1nai,jα,β(x,u)∂∂xjuβ)=μα in Ω,u=0 on ∂Ω, where α∈{1,…,N} is the equation index, Ω is an open bounded subset of Rn, u:Ω→RN and μ is a finite Randon measure on Rn with values in RN. Existence of a solution is proved for two different sets of assumptions on A. Examples are provided that satisfy our conditions, but do not satisfy conditions required on previous works on this matter.

EXISTENCE OF SOLUTIONS FOR A CLASS OF QUASILINEAR ELLIPTIC SYSTEMS WITH HARDY POTENTIAL

In this paper, by using the Morse theory, we obtain the existence of nontrivial weak solutions of quasilinear elliptic systems with Hardy potential.

Infinitely Many Sign-changing Solutions for Quasilinear Elliptic Systems in ℝN

Abstract In this paper, by using a perturbation method together with the method of invariant sets of descending flow, we obtain the existence of infinitely many sign-changing solutions of quasilinear elliptic systems in ℝN.

Infinitely many singular radial solutions for quasilinear elliptic systems

We prove the existence of infinitely many singular radial positive solutions for a quasilinear elliptic system with no variational structure [Formula: see text] where [Formula: see text] is the unit ball of [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text] are non-negative functions. We separate two fundamental classes (the sublinear and superlinear class), and we use respectively the Leray–Schauder Theorem and a method of monotone iterations to obtain the existence of many solutions with a property of singularity around the origin. Finally, we give a sufficient condition for the non-existence.

Multiplicity of solutions for quasilinear elliptic systems with singularity

In this paper, we study the existence of multiple solutions for the following quasilinear elliptic system: $$\left\{ \begin{gathered} - \Delta _p u - \mu _1 \frac{{|u|^{p - 2} u}} {{|x|^p }} = \alpha _1 \frac{{u^{p*(t) - 2} }} {{|x|^t }}u + \beta _1 |v|^{\beta _2 } |u|^{\beta _1 - 2_u } ,x \in \Omega , \hfill \\ - \Delta _q v - \mu _2 \frac{{|v|^{q - 2} v}} {{|x|^q }} = \alpha _2 \frac{{v^{q*(s) - 2} }} {{|x|^s }}v + \beta _2 |u|^{\beta _1 } |v|^{\beta _2 - 2_u } ,x \in \Omega , \hfill \\ u(x) = v(x) = 0, \hfill \\ \end{gathered} \right. $$ Multiplicity of solutions for the quasilinear problem is obtained via variational method.

Existence of solutions for quasilinear elliptic systems in divergence form with variable growth

This paper is concerned with the following Dirichlet problem for a quasilinear elliptic system with variable growth: −divσ(x,u(x),Du(x))=f in Ω, u(x)=0 on ∂ Ω, where Ω⊂ R n is a bounded domain. By means of the Young measure and the theory of variable exponent Sobolev spaces, we obtain the existence of solutions in W 0 1 , p ( x ) (Ω, R m ) for each f∈ ( W 0 1 , p ( x ) ( Ω , R m ) ) ∗ .

On positive solution for a class of quasilinear elliptic systems with sign-changing weights

In this paper, we consider the problem for the existence of positive solutions of quasi- linear elliptic system ⎧ ⎪ ⎪ −Δpu = λa(x)u α v γ , x ∈ Ω, −Δqv = λb(x)u η v β , x ∈ Ω, u = v = 0, x ∈ ∂ Ω, where the λ > 0i s ap arameter,Ω is a bounded domain in RN(N > 1) with smooth bound- ary ∂ Ω ,a nd theΔpz = div(|∇z| p−2 ∇z) is the p-Laplacian operator. Here a(x) and b(x) are C 1 sign-changing functions that maybe are negative near the boundary. Using the method of sub-super solutions and comparison principle, which studied the existence of positive solutions for quasilinear elliptic system. The main results of the present paper are new and extend the previously known results.

Entire solutions of quasilinear elliptic systems on Carnot groups

We prove general a priori estimates of the solutions of a class of quasilinear elliptic systems on Carnot groups. As a consequence, we obtain several nonexistence theorems. The results are new even in the Euclidean setting.