Quasilinear Elliptic Systems Research Articles

Regularity and Morse Index of the Solutions to Critical Quasilinear Elliptic Systems

Regularity results and critical group estimates are studied for critical (p, r)-systems. Multiplicity results of solutions for a critical potential quasilinear system are also proved using Morse theory.

Three solutions for a class of quasilinear elliptic systems in Orlicz–Sobolev spaces

We study the solutions of the Neumann problem for a class of elliptic partial differential systems on a bounded domain and obtain three solutions under appropriate hypotheses. Our technical approach is based on the general three critical point theorem obtained by B. Ricceri in Orlicz–Sobolev spaces.

$L^p$-integrability of the gradient of solutions to quasilinear systems with discontinuous coefficients

The Dirichlet problem for a class of quasilinear elliptic systems of equations with small-BMO coefficients in a Reifenberg-flat domain $\Omega$ is considered. The lower-order terms are supposed to satisfy controlled growth conditions in ${\mathbf u}$ and $D\mathbf {u}$. $L^p$-integrability with $p>2$ of $D{\mathbf u}$ is obtained, where $p$ depends explicitly on the data. An analogous result is obtained also for the Cauchy--Dirichlet problem for quasilinear parabolic systems.

Quasilinear elliptic system in exterior domains with dependence on the gradient

In this paper, we deal with a quasilinear elliptic system in exterior domains with dependence on the gradient and coupling of the equations not only inside of the domain, but also on the boundary. We prove the existence of positive, negative or sign changing weak solutions. Our approach relies on an aproximation argument and an adequate elliptic “a priori” estimate.

Existence of positive solutions for a quasilinear elliptic system involving critical Sobolev-Hardy exponents and concave-convex nonlinearities

This paper is concerned with a quasilinear elliptic system, which involves the Caffarelli-Kohn-Nirenberg inequality and multiple critical exponents. The existence and multiplicity results of positive solutions are obtained by variational methods.

Quasilinear elliptic systems in [formula omitted] with multipower forcing terms depending on the gradient

In this paper we deal with noncoercive elliptic multipower systems of divergence type, which include p-Laplacian type operators as well as mean curvature operators and whose right hand sides depend on the product of both components of the solution and on a gradient factor. We prove that any nonnegative nontrivial entire weak solution (nonnecessarily radial) is constant. For nontrivial solutions we intend that both components are nontrivial. The paper improves former results due to Clément, Fleckinger, Mitidieri, de Thélin (2000) in [4], to Bidaut-Véron and Pohozaev (2001) in [3], where no gradient terms are considered. On the other hand the paper contains, as subcases, also some recent results of Filippucci (2011) [12] in which systems with gradient dependence for the forcing terms are treated but not of multipower type.The key ingredient of the proof technique is the method of test functions, employed by Mitidieri and Pohozaev for studying systems in Mitidieri and Pohozaev (1999) [17]. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions is required.

Quasilinear parabolic and elliptic systems with mixed quasimonotone functions

This paper deals with a class of quasilinear parabolic and elliptic systems with mixed quasimonotone reaction functions. The boundary condition in the system may be Dirichlet, nonlinear, or a combination of these two types. The elliptic operators in the system are allowed to be degenerate. The aim is to show the existence and uniqueness of a classical solution to the parabolic system, the existence of maximal and minimal solutions or quasisolutions of the elliptic system, and the asymptotic behavior of the solution of the parabolic system. This consideration leads to a global attractor of the parabolic system as well as an one-sided stability of the maximal and minimal solutions. Applications of these results are given to three models arising from biology and ecology where diffusion coefficients are density-dependent and are degenerate. These applications exhibit quite distinct dynamical behavior of the population species between degenerate density-dependent diffusion and constant diffusion.

Summability for solutions to some quasilinear elliptic systems

We prove local regularity in Lebesgue spaces for weak solutions \(u\) of quasilinear elliptic systems whose off-diagonal coefficients are small when \(|u|\) is large: the faster off-diagonal coefficients decay, the higher integrability of \(u\) becomes.

Existence of nontrivial weak solutions for a quasilinear elliptic systems with concave-convex nonlinearities

Existence of nontrivial weak solutions for a quasilinear elliptic systems with concave-convex nonlinearities

Existence of bounded solutions for a class of quasilinear elliptic systems on manifolds with boundary

We consider nonlinear elliptic partial differential equations for quasilinear operators of the formA(u)=−div(a(x,u,∇u))+A0(x,u,∇u),x∈Ω, subject to fully nonlinear boundary conditions involving boundary operators of the form, for each β⩾0,Bβ(u)=−βdivΓ(b(x,u,∇Γu))+B0(x,u,∇u,∇Γu),x∈∂Ω. The main goal of this paper is to give, under suitable assumptions on A and Bβ, an explicit L∞ estimate for bounded solutions of these elliptic boundary value problems. Then, we establish the existence of at least one solution to such problems extending the authorʼs previous work. Our methods rely on the definition of approximate problems, deducing a priori estimates for their solutions and compactness arguments in order to pass to the limit. These methods can be applied to a large class of equations involving operators of Leray–Lions type (on suitable Banach spaces) for a general class of boundary operators Bβ which are, possibly, of the same order as A. As examples, these results are shown to apply to a class of uniformly elliptic equations that occur in the theory of phase transitions, and certain elliptic systems associated with climate problems which describe the evolution of atmospheric sea-level temperatures for relatively long time scales.

On a class of quasilinear elliptic systems with critical growth

A class of quasilinear elliptic systems with critical growth are considered in a bounded domain. Employing critical point theory and concentration estimates, we establish the existence of infinitely many solutions in higher values of dimension N>p2+p, 1<p<N.

Radial solutions for a quasilinear elliptic system of Schrödinger type

In this paper we analyze the existence of entire radially symmetric solutions for the Schrödinger system Δpiui+hi(r)|∇ui|pi−1=ai(r)fi(u1,…,ud)fori=1,…,d on RN, where pi>1,d∈{1,2,3,…},hi and ai are nonnegative radial continuous functions and fi are nonnegative non-decreasing continuous functions on [0,∞)d in all variables.

Existence and boundedness of solutions for a singular cooperative quasilinear elliptic system

This article sets forth results on the existence, a priori estimates and boundedness of positive solutions of a singular quasilinear system of elliptic equations. The systems studied here have in the principal part different p-Laplacians with Dirichlet boundary condition on a bounded domain. The approach is based on the sub-supersolution methods for systems of quasilinear equations combined with perturbation arguments involving singular terms. The structure of the singular terms in the system is essentially used to construct the sub-supersolution. A Moser iteration procedure is also obtained for our class of singular systems establishing a priori estimates and boundedness of solutions.

The Existence and Nonexistence of Entire Positive Radial Solutions of Quasilinear Elliptic Systems with Gradient Term

The Existence and Nonexistence of Entire Positive Radial Solutions of Quasilinear Elliptic Systems with Gradient Term

Multiple Solutions for a Singular Quasilinear Elliptic System

We consider the multiplicity of nontrivial solutions of the following quasilinear elliptic system −div(|x|−ap|∇u|p−2∇u) + f 1(x)|u|p−2 u = (α/(α + β))g(x)|u|α−2 u|v|β + λh 1(x)|u|γ−2 u + l 1(x), −div(|x|−ap|∇v|p−2∇v) + f 2(x)|v|p−2 v = (β/(α + β))g(x)|v|β−2 v|u|α + μh 2(x)|v|γ−2 v + l 2(x), u(x) > 0, v(x) > 0, x ∈ ℝN, where λ, μ > 0, 1 < p < N, 1 < γ < p < α + β < p* = Np/(N − pd), 0 ≤ a < (N − p)/p, a ≤ b < a + 1, d = a + 1 − b > 0. The functions f 1(x), f 2(x), g(x), h 1(x), h 2(x), l 1(x), and l 2(x) satisfy some suitable conditions. We will prove that the problem has at least two nontrivial solutions by using Mountain Pass Theorem and Ekeland's variational principle.

Multiple Solutions for a Class of Concave-Convex Quasilinear Elliptic Systems with Nonlinear Boundary Condition

In this paper, a quasilinear elliptic system is investigated, which involves concave-convex nonlinearities and nonlinear boundary condition. By Nehari manifold, fibering method and analytic techniques, the existence of multiple nontrivial nonnegative solutions to this equation is verified.

A variational approach to a singular elliptic system involving critical Sobolev-Hardy exponents and concave-convex nonlinearities

This paper is concerned with a quasilinear elliptic system, which involves the Caffarelli-Kohn-Nirenberg inequality and multiple critical exponents. The existence and multiplicity results of positive solutions are obtained by variational methods.

Regularity Result for Quasilinear Elliptic Systems with Super Quadratic Natural Growth Condition

We consider boundary regularity for weak solutions of second-order quasilinear elliptic systems under natural growth condition with super quadratic growth and obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. Combined with existing results on interior partial regularity, this result yields an upper bound on the Hausdorff dimension of the singular set at the boundary.

Quasilinear and singular elliptic systems

In this paper, we investigate a general quasilinear elliptic and singular system. By monotonicity methods, we give some existence and uniqueness results. Next, we give some applications to biological models.

Gradient system with dependence on the gradient at nonlinearity

In this paper we establish existence of solutions for quasilinear elliptic system of gradient form with dependence on the gradient at nonlinearity.