p-Laplacian Systems Research Articles

On a parabolic p-Laplacian system with a convective term

On a parabolic p-Laplacian system with a convective term

The Uniqueness and Iterative Properties of Positive Solution for a Coupled Singular Tempered Fractional System with Different Characteristics

In this paper, we focus on the uniqueness and iterative properties of positive solution for a coupled p-Laplacian system of singular tempered fractional equations with differential order and characteristics. Firstly, the system is converted to an integral equation, and then, a coupled iterative technique and some suitable growth conditions are proposed; furthermore, some elaborate results about the uniqueness and iterative properties of positive solutions of the system are established, which include the uniqueness, the convergence analysis, the asymptotic behavior, and error estimation, as well as the convergence rate of the positive solution. The interesting points of this paper are that the order of the system of equations is different and the nonlinear terms of the system possess the opposite monotonicity and allow for stronger singularities at space variables.

Radial symmetry of positive solutions for a tempered fractional p-Laplacian system

Radial symmetry of positive solutions for a tempered fractional p-Laplacian system

The Brezis–Nirenberg Problem for Fractional p-Laplacian Systems in Unbounded Domains

The Brezis–Nirenberg Problem for Fractional p-Laplacian Systems in Unbounded Domains

Multiple positive solutions for a critical fractional p-Laplacian system with concave nonlinearities

The aim of this paper is to study the following nonlinear fractional p-Laplacian system with critical exponents: { ( − Δ ) p s u + | u | p − 2 u = λg ( x ) | u | q − 2 u + α α + β f ( x ) | u | α − 2 u | v | β in Ω , ( − Δ ) p s v + | v | p − 2 v = μh ( x ) | v | q − 2 v + β α + β f ( x ) | u | α | v | β − 2 v in Ω , u = v = 0 in R N ∖Ω , where Ω is a smooth bounded set in R N , 0<s<1, 0 $ ]]> λ , μ > 0 are two parameters, 1 < q < p < p s ∗ , N>ps, 1 $ ]]> α , β > 1 satisfy α + β = p s ∗ with p s ∗ = np n − ps is the fractional Sobolev critical exponent and ( − Δ ) p s is the fractional p-Laplacian operator. Using the Nehari manifold and Ljusternik–Schnirelmann category, we study the topology of the global maximum set Θ of f ( x ) , and show that the system has at least at least ca t Θ δ ( Θ ) + 1 distinct positive solutions.

Elliptic p-Laplacian systems with nonlinear boundary condition

In this paper we study quasilinear elliptic systems given by−Δp1u1=−|u1|p1−2u1in Ω,−Δp2u2=−|u2|p2−2u2in Ω,|∇u1|p1−2∇u1⋅ν=g1(x,u1,u2)on ∂Ω,|∇u2|p2−2∇u2⋅ν=g2(x,u1,u2)on ∂Ω, where ν(x) is the outer unit normal of Ω at x∈∂Ω, Δpi denotes the pi-Laplacian and gi:∂Ω×R×R→R are Carathéodory functions that satisfy general growth and structure conditions for i=1,2. In the first part we prove the existence of a positive minimal and a negative maximal solution based on an appropriate construction of sub- and supersolution along with a certain behavior of gi near zero related to the first eigenvalue of the pi-Laplacian with Steklov boundary condition. The second part is related to the existence of a third nontrivial solution by imposing a variational structure, that is, (g1,g2)=∇g with a smooth function (s1,s2)↦g(x,s1,s2). By using the variational characterization of the second eigenvalue of the Steklov eigenvalue problem for the pi-Laplacian together with the properties of the related truncated energy functionals, which are in general nonsmooth, we show the existence of a nontrivial solution whose components lie between the components of the positive minimal and the negative maximal solution.

Existence of Periodic Solutions for Second-Order Ordinary p-Laplacian Systems

In this paper, we study the variational principle and the existence of periodic solutions for a new class of second-order ordinary p-Laplacian systems. The variational principle is given by making use of two methods. We obtain three existence theorems of periodic solutions to this problem on various sufficient conditions on the potential function F(t,x) or nonlinearity ∇F(t,x). Four examples are presented to illustrate the feasibility and effectiveness of our results.

Existence of Periodic Solutions for a Class of p-Laplacian Systems with Delay

Existence of Periodic Solutions for a Class of p-Laplacian Systems with Delay

Existence and convergence of solutions for p-Laplacian systems with homogeneous nonlinearities on graphs

Existence and convergence of solutions for p-Laplacian systems with homogeneous nonlinearities on graphs

Nonexistence result for p-Laplacian systems in a ball

We consider the \(p\)-Laplacian system $$ \displaylines{ -\Delta_p u = \lambda f(v) \quad \text{in } \Omega; \cr -\Delta_p v = \lambda g(u) \quad \text{in } \Omega; \cr u = v=0 \quad \text{on }\partial \Omega, }$$ where \(\lambda &gt;0\) is a parameter, \(\Delta_p u:= \operatorname{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian operator for \(p &gt; 1\) and \(\Omega\) is a unit ball in \(\mathbb{R}^N\) (\(N \geq 2)\). The nonlinearities \(f, g: [0,\infty) \to \mathbb{R}\) are assumed to be \(C^1\) non-decreasing semipositone functions (\(f(0)&lt; 0\) and \(g(0)&lt;0\)) that are \(p\)-superlinear at infinity. By analyzing the solution in the interior of the unit ball as well as near the boundary, we prove that the system has no positive radially symmetric and radially decreasing solution for \(\lambda\) large. See also https://ejde.math.txstate.edu/special/02/a1/abstr.html

Positive solution for a class of p-Laplacian systems with critical homogeneous nonlinearity

Positive solution for a class of p-Laplacian systems with critical homogeneous nonlinearity

Global Calderón–Zygmund theory for parabolic p-Laplacian system: the case [formula omitted

The aim of this paper is to establish global Calderón–Zygmund theory to parabolic p-Laplacian system:ut−div(|∇u|p−2∇u)=div(|F|p−2F)inΩ×(0,T)⊂Rn+1, proving thatF∈Lq⇒∇u∈Lq, for any q>max⁡{p,n(2−p)2} and p>1. Acerbi and Mingione [2] proved this estimate in the case p>2nn+2. In this article we settle the case 1<p≤2nn+2. We also treat systems with discontinuous coefficients having small BMO (bounded mean oscillation) norm.

Multiple solutions to quasi-linear elliptic Robin systems

Two opposite constant-sign solutions to a non-variational p-Laplacian system with Robin boundary conditions are obtained via sub-super-solution techniques. A third nontrivial one comes out by means of topological degree arguments.

Existence of weak solutions to a p-Laplacian system on the Sierpiński gasket on $${\mathbb {R}}^2$$

Existence of weak solutions to a p-Laplacian system on the Sierpiński gasket on $${\mathbb {R}}^2$$

Monotonicity and symmetry of solutions to fractional p-laplacian systems

Monotonicity and symmetry of solutions to fractional p-laplacian systems

Left Riemann–Liouville Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales

First, we show the equivalence of two definitions of the left Riemann–Liouville fractional integral on time scales. Then, we establish and characterize fractional Sobolev space with the help of the notion of left Riemann–Liouville fractional derivative on time scales. At the same time, we define weak left fractional derivatives and demonstrate that they coincide with the left Riemann–Liouville ones on time scales. Next, we prove the equivalence of two kinds of norms in the introduced space and derive its completeness, reflexivity, separability, and some embedding. Finally, as an application, by constructing an appropriate variational setting, using the mountain pass theorem and the genus properties, the existence of weak solutions for a class of Kirchhoff-type fractional p-Laplacian systems on time scales with boundary conditions is studied, and three results of the existence of weak solutions for this problem is obtained.

A direct method of moving planes for the fractional p-Laplacian system with negative powers

A direct method of moving planes for the fractional p-Laplacian system with negative powers

EXISTENCE RESULTS FOR FRACTIONAL P-LAPLACIAN SYSTEMS VIA YOUNG MEASURES

In this paper, we show the existence result of the following fractional p-Laplacian system (−∆)spu = f(x,u) in Ω, u = 0 in Rn\Ω, for a given datum f. The existence of weak solutions is obtained by using the theory of Young measures.

The existence of positive solutions for the Neumann problem of p-Laplacian elliptic systems with Sobolev critical exponent

The paper aims to consider a class of p-Laplacian elliptic systems with a double Sobolev critical exponent. We obtain the existence result of the above problem under the Neumann boundary for some suitable range of the parameters in the systems.

Existence of solutions for a p-Laplacian system with a nonresonance condition between the first and the second eigenvalues

In this article, we study the existence of positive solutions for the quasilinear elliptic system −∆_p u(x) = f_1(x, v(x)) + h_1(x) in Ω,−∆_p v(x) = f_2(x, u(x)) + h_2(x) in Ω,u = v = 0 on ∂Ω,where f_i(x, s), (i = 1, 2) locates between the first and the second eigenvalues of the p-Laplacian. To prove the existence of solutions, we use a topological method the Leray-Schauder degree.