p-Laplacian Operator Research Articles

Higher order FEM for the obstacle problem of the [formula omitted]-Laplacian—A variational inequality approach

We consider higher order finite element discretizations of a nonlinear variational inequality formulation arising from an obstacle problem with the p-Laplacian differential operator for p∈(1,∞). We prove an a priori error estimate and convergence rates with respect to the mesh size h and in the polynomial degree q under assumed regularity. Moreover, we derive a general a posteriori error estimate which is valid for any uniformly bounded sequence of finite element functions. All our results contain the known results for the linear case of p=2. We present numerical results on the improved convergence rates of adaptive schemes (mesh size adaptivity with and without polynomial degree adaptation) for the singular case of p=1.5 and for the degenerated case of p=3.

English

We shall prove a weak comparison principle for quasilinear elliptic operators −div(a(x,∇u)) that includes the negative p-Laplace operator, where a: × ℝN → ℝN satisfies certain conditions frequently seen in the research of quasilinear elliptic operators. In our result, it is characteristic that functions which are compared belong to different spaces.

P-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities

Abstract This paper deals with the existence of nontrivial solutions for critical Hardy–Schrödinger–Kirchhoff systems driven by the fractional p-Laplacian operator. Existence is derived as an application of the mountain pass theorem and the Ekeland variational principle. The main features and novelty of the paper are the presence of the Hardy terms as well as critical nonlinearities.

Global solution for a thermoelastic system with [formula omitted]-Laplacian

In this work we prove global solution for the nonlinear system utt−Δpu+θ=|u|r−1uθt−Δθ=utwhere Δp is the nonlinear p-Laplacian operator, 2≤p<∞. We apply the potential well theory. The global solution is constructed by means of the Faedo–Galerkin approximations, taking into account that the initial data is in appropriated set of stability created from the Nehari manifold.

On the First Eigenvalue of the Degenerate $$\varvec{p}$$ p -Laplace Operator in Non-convex Domains

In this paper we obtain lower estimates of the first non-trivial eigenvalues of the degenerate p-Laplace operator, $$p>2$$ , in a large class of non-convex domains. This study is based on applications of the geometric theory of composition operators on Sobolev spaces that permits us to estimate constants of the Poincare–Sobolev inequalities. On this base we obtain lower estimates of the first non-trivial eigenvalues for Ahlfors-type domains (i.e. quasidiscs). This class of domains includes some snowflake-type domains with fractal boundaries.

A singular System Involving the Fractional p-Laplacian Operator via the Nehari Manifold Approach

In this work we study the fractional p-Laplacian equation with singular nonlinearity $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^s_p u = \lambda a(x)|u|^{q-2}u +\frac{1-\alpha }{2-\alpha -\beta } c(x)|u|^{-\alpha }|v|^{1-\beta }, \quad \text {in }\Omega ,\\ \\ (-\Delta )^s_p v= \mu b(x)|v|^{q-2}v +\frac{1-\beta }{2-\alpha -\beta } c(x)|u|^{1-\alpha }|v|^{-\beta }, \quad \text {in }\Omega ,\\ \\ u=v = 0 ,\quad \text{ in } \,\mathbb {R}^N{\setminus }\Omega , \end{array} \right. \end{aligned}$$ where $$0<\alpha<1,\;0<\beta <1,$$ $$2-\alpha -\beta<p<q<p^*_s,$$ $$p^*_s=\frac{N}{N-ps}$$ is the fractional Sobolev exponent, $$\lambda , \mu $$ are two parameters, $$a,\, b, \,c \in C(\overline{\Omega })$$ are non-negative weight functions with compact support in $$\Omega ,$$ and $$(-{\Delta )^{s}}_{p}$$ is the fractional p-Laplace operator. We use the Nehari manifold approach and some variational techniques in order to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter $$\lambda $$ and $$\mu $$ .

The Dirichlet problem for the [formula omitted]-fractional Laplace equation

We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order s∈(0,1) and summability growth p∈(1,∞), whose model is the fractional p-Laplacian operator with measurable coefficients. We review several recent results for the corresponding weak solutions/supersolutions, as comparison principles, a priori bounds, lower semicontinuity, boundedness, Hölder continuity up to the boundary, and many others. We then discuss the good definition of (s,p)-superharmonic functions, and the nonlocal counterpart of the Perron method in nonlinear Potential Theory, together with various related results. We briefly mention some basic results for the obstacle problem for nonlinear integro-differential equations. Finally, we present the connection amongst the fractional viscosity solutions, the weak solutions and the aforementioned (s,p)-superharmonic functions, together with other important results for this class of equations when involving general measure data, and a surprising fractional version of the Gehring lemma.We sketch the corresponding proofs of some of the results presented here, by especially underlining the development of new fractional localization techniques and other recent tools. Various open problems are listed throughout the paper.

Existence and Uniqueness of Positive Solutions of a Kind of Multi-point Boundary Value Problems for Nonlinear Fractional Differential Equations with p-Laplacian Operator

In this paper, we investigate the existence and uniqueness of positive solutions of a kind of multi-point boundary value problems for nonlinear fractional differential equations with p-Laplacian operator using the Banach contraction mapping principle. Furthermore, some examples are given to illustrate our results.

A Type of Brézis–Oswald Problem to the $${\Phi}$$ Φ - Laplacian Operator with Very Singular Term

In this work we consider existence and uniqueness of solutions for a quasilinear elliptic problem, which may be singular at the origin. Furthermore, we consider a comparison principle for subsolutions and supersolutions just in $${W^{1, \Phi}_{loc} (\Omega)}$$ to the problem $$\left\{\begin{array}{ll}-\Delta_{\Phi}u=f(x,u)\, {\rm in}\, \Omega,\\u > 0\, {\rm in} \, \Omega, u = 0 \,{\rm on}\, \partial\Omega,\end{array}\right.$$ where f has $${\Phi}$$ -sublinear growth. In our main results the function f(x, u) may be singular at u = 0 and the nonlinear term $${f(x, t)/t^{\ell-1}, t > 0}$$ is strictly decreasing for a suitable $${\ell > 1}$$ . Under different kind of boundary conditions we prove an improvement for the classical Brezis-Oswald and Diaz-Saa’s results in Orlicz- Sobolev framework for singular nonlinearities as well. Some results discussed here are news even for Laplacian or p-Laplacian operators.

ON EXISTENCE OF BOUNDED FEASIBLE SOLUTIONS TO NEUMANN BOUNDARY CONTROL PROBLEM FOR p-LAPLACE EQUATION WITH EXPONENTIAL TYPE OF NONLINEARITY

We study an optimal control problem for mixed Dirichlet-Neumann boundary value problem for the strongly non-linear elliptic equation with p-Laplace operator and L1-nonlinearity in its right-hand side. A distribution u acting on a part of boundary of open domain is taken as a boundary control. The optimal control problem is to minimize the discrepancy between a given distribution yd 2 L2( ) and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any admissible control. After dening a suitable functional class in which we look for solutions and assuming that this problem admits at least one feasible solution, we prove the existence of optimal pairs. We derive also conditions when the set of feasible solutions has a nonempty intersection with the space of bounded distributions L1( ).

Sharp regularity estimates for quasi-linear elliptic dead core problems and applications

In this manuscript we study geometric regularity estimates for quasi-linear elliptic equations of p-Laplace type ( $$1< p< \infty $$ ) with strong absorption condition: $$\begin{aligned} -\mathrm {div}(\Phi (x, u, \nabla u)) + \lambda _0(x) u_{+}^q(x) = 0 \quad \hbox {in} \quad \Omega \subset \mathbb {R}^N, \end{aligned}$$ where $$\Phi : \Omega \times \mathbb {R}_{+} \times \mathbb {R}^N \rightarrow \mathbb {R}^N$$ is a vector field with an appropriate p-structure, $$\lambda _0$$ is a non-negative and bounded function and $$0\le q<p-1$$ . Such a model permits existence of solutions with dead core zones, i.e, a priori unknown regions where non-negative solutions vanish identically. We establish sharp and improved $$C^{\gamma }$$ regularity estimates along free boundary points, namely $$\mathfrak {F}_0(u, \Omega ) = \partial \{u>0\} \cap \Omega $$ , where the regularity exponent is given explicitly by $$\gamma = \frac{p}{p-1-q} \gg 1$$ . Some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density and porosity of free boundary are proved. As an application, a Liouville-type result for entire solutions is established provided that their growth at infinity can be controlled in an appropriate manner. Finally, we obtain finiteness of $$(N-1)$$ -Hausdorff measure of free boundary for a particular class of dead core problems. The approach employed in this article is novel even to dead core problems governed by the p-Laplace operator $$-\Delta _p u + \lambda _0 u^q\chi _{\{u>0\}} = 0$$ for any $$\lambda _0>0$$ .

Quasilinear equations with natural growth in the gradients in spaces of Sobolev multipliers

We study the existence problem for a class of nonlinear elliptic equations whose prototype is of the form $$-\Delta _p u = |\nabla u|^p + \sigma $$ in a bounded domain $$\Omega \subset \mathbb {R}^n$$ . Here $$\Delta _p$$ , $$p>1$$ , is the standard p-Laplacian operator defined by $$\Delta _p u=\mathrm{div}\, (|\nabla u|^{p-2}\nabla u)$$ , and the datum $$\sigma $$ is a signed distribution in $$\Omega $$ . The class of solutions that we are interested in consists of functions $$u\in W^{1,p}_0(\Omega )$$ such that $$|\nabla u|\in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))$$ , a space pointwise Sobolev multipliers consisting of functions $$f\in L^{p}(\Omega )$$ such that $$\begin{aligned} \int _{\Omega } |f|^{p} |\varphi |^p dx \le C \int _{\Omega } (|\nabla \varphi |^p + |\varphi |^p) dx \quad \forall \varphi \in C^\infty (\Omega ), \end{aligned}$$ for some $$C>0$$ . This is a natural class of solutions at least when the distribution $$\sigma $$ is nonnegative and compactly supported in $$\Omega $$ . We show essentially that, with only a gap in the smallness constants, the above equation has a solution in this class if and only if one can write $$\sigma =\mathrm{div}\, F$$ for a vector field F such that $$|F|^{\frac{1}{p-1}}\in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))$$ . As an important application, via the exponential transformation $$u\mapsto v=e^{\frac{u}{p-1}}$$ , we obtain an existence result for the quasilinear equation of Schrodinger type $$-\Delta _p v = \sigma \, v^{p-1}$$ , $$v\ge 0$$ in $$\Omega $$ , and $$v=1$$ on $$\partial \Omega $$ , which is interesting in its own right.

Efficient algorithm for principal eigenpair of discrete p-Laplacian

This paper is a continuation of the author’s previous papers [Front. Math. China, 2016, 11(6): 1379–1418; 2017, 12(5): 1023–1043], where the linear case was studied. A shifted inverse iteration algorithm is introduced, as an acceleration of the inverse iteration which is often used in the non-linear context (the p-Laplacian operators for instance). Even though the algorithm is formally similar to the Rayleigh quotient iteration which is well-known in the linear situation, but they are essentially different. The point is that the standard Rayleigh quotient cannot be used as a shift in the non-linear setup. We have to employ a different quantity which has been obtained only recently. As a surprised gift, the explicit formulas for the algorithm restricted to the linear case (p = 2) is obtained, which improves the author’s approximating procedure for the leading eigenvalues in different context, appeared in a group of publications. The paper begins with p-Laplacian, and is closed by the non-linear operators corresponding to the well-known Hardy-type inequalities.

Positive solution for a fractional singular boundary value problem with p-Laplacian operator

In this paper, we consider a fractional singular three-point boundary value problem with p-Laplacian operator. The nonlinearity f(t,u) may be singular at t = 0,1 and u = 0. Some properties of the associated Green function are obtained. By using the upper and lower solutions method and a fixed point theorem, the existence result of positive solution is established.

Space quasiconformal mappings and Neumann eigenvalues in fractal type domains

Abstract We study the variation of Neumann eigenvalues of the p-Laplace operator under quasiconformal perturbations of space domains. This study allows us to obtain the lower estimates of Neumann eigenvalues in fractal type domains. The proposed approach is based on the geometric theory of composition operators in connection with the quasiconformal mapping theory.

Infinitely Many Solutions for Fractional p-Kirchhoff Equations

In this paper we consider the existence of infinitely many weak solutions for fractional Schrodinger–Kirchhoff problems. Precisely speaking, we investigate $$\begin{aligned} \left\{ \begin{array}{cl} M\left( \int _{\mathbb {R}^{2n}}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\mathrm{d}x\mathrm{d}y\right) (-\triangle )_p^su+V(x)|u|^{p-2}u=f(x,u), &{}\quad \mathrm{in}~\Omega ,\\ u=0, &{}\quad \mathrm{in}~\mathbb {R}^n\setminus \Omega , \end{array}\right. \end{aligned}$$ where $$\Omega $$ is a bounded subset with Lipshcitz boundary $$\partial \Omega $$ , $$0<s<1$$ is fixed, and $$1<p<n$$ , $$(-\triangle )_p^s$$ is the fractional p-Laplacian operator. Kirchhoff function M, potential function V and nonlinearity f satisfy some suitable assumptions. Under those conditions, some new multiplicity results are obtained by applying the fountain theorem and the dual fountain theorem.

Upper Bounds of the Eigenvalues Related to a Weighted Fractional p-Laplacian on Metric Graphs

The main result of the article states explicit upper bounds of eigenvalues for a weighted fractional p-Laplacian operator on a connected metric graph with a finite total length.

Entire nodal solutions to the pure critical exponent problem for the p-Laplacian

We establish the existence of multiple sign-changing solutions to the quasilinear critical problem−Δpu=|u|p⁎−2u,u∈D1,p(RN), for N≥4, where Δpu:=div(|∇u|p−2∇u) is the p-Laplace operator, 1<p<N and p⁎:=NpN−p is the critical Sobolev exponent.

A Result of the Solutions to a Hamiltonian System with p-Laplace Operator

We focus on a Hamiltonian system with p-Laplacian. By some technique we obtain the results of the solutions to our problem. Our main difficulty in our work is the p-laplace operator in our system.

The Construction of Iterative Solutions to A kind of BVP

We focus on the positive iterative solution to the problem of a kind of BVP with the p-Laplace operator. We will study its positive solutions. The specific methods used in our work is the combination of fixed point theorem and monotone iterative technique. What’s more, at the end of this work, an example is presented to show that this method can be well used to get the main results.