p-Laplacian Operator Research Articles

Existence of positive solutions for nonlinear four-point Caputo fractional differential equation with p-Laplacian

In this article, the existence of positive solutions is considered for nonlinear four-point Caputo fractional differential equation with p-Laplacian operator. We use the monotone iterative technique to acquire the existence of positive solutions for the boundary value problem and get iterative schemes for approximating the solutions. An example is also presented to illustrate the effectiveness of the main results.

P-Laplacian equations in [formula omitted] with finite potential via truncation method, the critical case

In this paper, we consider the problem −Δpu+a(x)|u|p−2u=|u|p⁎−2u+μ|u|q−2u in RN, where 1<p<N, max⁡{p,p⁎−1}<q<p⁎=NpN−p,μ>0, Δp is the p-Laplacian operator, and the potential a(x) is positive and verifies suitable decay assumptions. We prove that the problem has infinitely many solutions by using truncation method.

Existence of positive solutions for fractional differential equation involving integral boundary conditions with p-Laplacian operator

The existence of positive solutions is considered for a fractional differential equation with p-Laplacian operator in this article. By employing the Avery-Henderson fixed point theorem, a new result is obtained for the boundary value problems. An example is also presented to illustrate the effectiveness of the main result.

Asymptotic stability for nonlinear damped Kirchhoff systems involving the fractional p-Laplacian operator

This paper is devoted to the question of global and local asymptotic stability for nonlinear damped Kirchhoff systems, with homogeneous Dirichlet boundary conditions, under fairly natural assumptions on the external force f=f(t,x,u), the distributed damping Q=Q(t,x,u,ut), the perturbation term μ|u|p−2u and the dissipative term ϱ(t)M([u]sp)|ut|p−2ut, with ϱ≥0 and in Lloc1(R0+), when the initial data are in a special region. Here u=(u1,…,uN)=u(t,x) represents the vectorial displacement, with N≥1. Particular attention is devoted to the asymptotic behavior of the solutions in the linear case specified in Section 5. Finally, the results are extended to problems where the fractional p-Laplacian is replaced by a more general elliptic nonlocal integro-differential operator. The paper extends in several directions recent theorems and covers also the so-called degenerate case, that is the case in which M is zero at zero.

Two generalized Lyapunov-type inequalities for a fractional p-Laplacian equation with fractional boundary conditions

In this paper, we investigate the existence of positive solutions for the boundary value problem of nonlinear fractional differential equation with mixed fractional derivatives and p-Laplacian operator. Then we establish two smart generalizations of Lyapunov-type inequalities. Some applications are given to demonstrate the effectiveness of the new results.

Three periodic solutions for discontinuous perturbations of the vector p-Laplacian operator

Using a variational approach we obtain the existence of at least three periodic solutions for discontinuous perturbations of the vector p-Laplacian operator .

Existence of Solutions of Boundary Value Problems for Fractional Differential Equations with p-Laplacian Operator in Banach Spaces

ABSTRACTIn this paper, we consider the existence of solutions for systems of nonlinear p-Laplacian fractional differential equations whose nonlinearity contains the first-order derivative explicityin Banach space E, where 𝜃 is the zero element of E, φp is the p-Laplacian operator, i.e., with p>1, its inverse function is denoted by φq with , and p,q satisfy , is standard Caputo derivative and f:I×E×E→E is continuous. Our main tool is the Sadovskii fixed point theorem.

Sharp estimates of radial minimizers of 𝑝–Laplace equations

We study semi-stable, radially symmetric and decreasing solutions u ∈ W 1 , p ( B 1 ) u\in W^{1,p}(B_1) of − Δ p u = g ( u ) -\Delta _p u=g(u) in B 1 ∖ { 0 } B_1\setminus \{ 0\} , where B 1 B_1 is the unit ball of R N \mathbb {R}^N , p &gt; 1 p&gt;1 , Δ p \Delta _p is the p − p- Laplace operator and g g is a general locally Lipschitz function. We establish sharp pointwise estimates for such solutions, which do not depend on the nonlinearity g g . By applying these results, sharp pointwise estimates are obtained for the extremal solution and its derivatives (up to order three) of the equation − Δ p u = λ f ( u ) -\Delta _p u=\lambda f(u) , posed in B 1 B_1 , with Dirichlet data u | ∂ B 1 = 0 u|_{\partial B_1}=0 , where the nonlinearity f f is an increasing C 1 C^1 function with f ( 0 ) &gt; 0 f(0)&gt;0 and lim t → + ∞ f ( t ) t p − 1 = + ∞ . \lim _{t\rightarrow +\infty }{\frac {f(t)}{t^{p-1}}}=+\infty .

Existence of three weak solutions for some singular elliptic problems

In this paper, we are interested in applying a three critical points theorem by Ricceri, to establish the existence of three weak solutions for some singular elliptic problems involving a p-Laplace operator, subject to Dirichlet boundary conditions in a smooth bounded domain in .

Existence of positive solutions for integral boundary value problems of fractional differential equations with p-Laplacian

This paper is concerned with the existence of positive solutions for integral boundary value problems of Caputo fractional differential equations with p-Laplacian operator. By means of the properties of the Green’s function, Avery-Peterson fixed point theorems, we establish conditions ensuring the existence of positive solutions for the problem. As an application, an example is given to demonstrate the main result.

EIGENVALUES VARIATION OF THE P-LAPLACIAN UNDER THE RICCI FLOW ON SM

Let (M,F) be a compact Finsler manifold. Studying the eigenvalues and eigenfunctions for the linear and nonlinear geometric operators is a known problem. In this paper we will consider the eigenvalue problem for the p-laplace operator for Sasakian metric acting on the space of functions on SM. We find the first variation formula for the eigenvalues of p-Laplacian on SM evolving by the Ricci flow on M and give some examples.DOI : http://dx.doi.org/10.22342/jims.22.2.215.157-177

Stationary Solutions and Connecting Orbits for p-Laplace Equation

We deal with one dimensional p-Laplace equation of the form ut=(|ux|p-2ux)x+f(x,u),x∈(0,l),t>0,\\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}$$\\begin{aligned} u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \\ x\\in (0,l), \\ t>0, \\end{aligned}$$\\end{document}under Dirichlet boundary condition, where p>2 and f:[0,l]times {mathbb {R}}rightarrow {mathbb {R}} is a continuous function with f(x,0)=0. We will prove that if there is at least one eigenvalue of the p-Laplace operator between lim _{urightarrow 0} f(x,u)/|u|^{p-2}u and lim _{|u|rightarrow +infty } f(x,u)/|u|^{p-2}u, then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are based on Conley index and detect stationary states even when those based on fixed point theory do not apply. In order to compute the Conley index for nonlinear semiflows deformation along p is used.

Nonlocal problems with singular nonlinearity

We investigate existence and uniqueness of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian operator and singular nonlinearities.

Positive solutions of boundary value problems for p-Laplacian fractional differential equations

This work is devoted to the existence of positive solutions for nonlinear fractional differential equations with p-Laplacian operator. By using five functionals fixed point theorem, the existence of at least three positive solutions are obtained. As an application, an example is presented to demonstrate our main result.

Positive Solutions of Fractional Differential Equations with p-Laplacian

The multiplicity of positive solution for a new class of four-point boundary value problem of fractional differential equations with p-Laplacian operator is investigated. By the use of the Leggett-Williams fixed-point theorem, the multiplicity results of positive solution are obtained. An example is given to illustrate the main results.

Multiple positive solutions for nonlocal boundary value problems with p-Laplacian operator

Multiple positive solutions for nonlocal boundary value problems with p-Laplacian operator

A problem involving the p-Laplacian operator

Using a variational technique we guarantee the existence of a solution to the \emph{resonant Lane-Emden} problem $-\Delta_p u=\lambda |u|^{q-2}u$, $u|_{\partial\Omega}=0$ if and only if a solution to $-\Delta_p u=\lambda |u|^{q-2}u+f$, $u|_{\partial\Omega}=0$, $f\in L^{p'}(\Omega)$ ($p'$ being the conjugate of $p$), exists for $q\in (1,p)\bigcup (p,p^{*})$ under a certain condition for both the cases, i.e., $1<q<p<p^{*}$ and $1< p < q < p^{*}$ - the sub-linear and the super-linear cases.

Existence and non-existence of positive solutions for integral boundary value problems of high-order fractional differential equations with generalised p-Laplacian operator

In this paper, we investigate the non-linear integral boundary value problem of fractional differential equations with generalised p-Laplacian operator. By means of the monotone iteration method, we obtain the existence of positive solutions and establish the iterative sequence for approximating the solutions. Moreover, the non-existence of positive solution is also considered. An example is given to show the applicability of our main results.

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)

Existence and non-existence of positive solutions for integral boundary value problems of high-order fractional differential equations with generalised p-Laplacian operator

In this paper, we investigate the non-linear integral boundary value problem of fractional differential equations with generalised p-Laplacian operator. By means of the monotone iteration method, we obtain the existence of positive solutions and establish the iterative sequence for approximating the solutions. Moreover, the non-existence of positive solution is also considered. An example is given to show the applicability of our main results.