p-Laplacian Problems Research Articles

A bifurcation and multiplicity result for a critical growth elliptic problem

We consider a Brézis–Nirenberg type critical growth p-Laplacian problem involving a parameter μ>0 in a smooth bounded domain Ω. We prove the existence of multiple nontrivial solutions if either μ or the volume of Ω is sufficiently small. The proof is based on an abstract critical point theorem that only assumes a local (PS) condition. Our results are new even in the semilinear case p=2.

The limiting behavior of solutions to p-Laplacian problems with convection and exponential terms

We consider, for $a,l\geq1$, $b,s,\alpha> 0$, and $p> q\geq1$, the homogeneous Dirichlet problem for the equation $-\Delta_{p}u=\lambda u^{q-1}+\beta u^{a-1}\left\vert \nabla u\right\vert ^{b}+mu^{l-1}e^{\alpha u^{s}}$ in a smooth bounded domain $\Omega\subset\mathbb{R}^{N}$. We prove that under certain setting of the parameters $\lambda$, $\beta$ and $m$ the problem admits at least one positive solution. Using this result we prove that if $\lambda,\beta> 0$ are arbitrarily fixed and $m$ is sufficiently small, then the problem has a positive solution $u_{p}$, for all $p$ sufficiently large. In addition, we show that $u_{p}$ converges uniformly to the distance function to the boundary of $\Omega$, as $p\rightarrow\infty$. This convergence result is new for nonlinearities involving a convection term.

Efficient finite element strategy using enhanced high-order and second-derivative-free variants of Newton's method

In this work, we propose a stable finite element approximation by extending higher-order Newton's method to the multidimensional case for solving nonlinear systems of partial differential equations. This approach relies solely on the evaluation of Jacobian matrices and residuals, eliminating the need for computing higher-order derivatives. Achieving third and fifth-order convergence, it ensures stability and allows for significantly larger time steps compared to explicit methods. We thoroughly address accuracy and convergence, focusing on the singular p-Laplacian problem and the time-dependent lid-driven cavity benchmark. A globalized variant incorporating a continuation technique is employed to effectively handle high Reynolds number regimes. Through two-dimensional and three-dimensional numerical experiments, we demonstrate that the improved cubically convergent variant outperforms others, leading to substantial computational savings, notably halving the computational cost for the lid-driven cavity test at large Reynolds numbers.

Positive solutions for 2nth order p-Laplacian problem with Sturm–Liouville type boundary conditions

Positive solutions for 2nth order p-Laplacian problem with Sturm–Liouville type boundary conditions

Anisotropic Adaptive Finite Elements for a p-Laplacian Problem

Abstract The p-Laplacian problem - ∇ ⋅ ( ( μ + | ∇ ⁡ u | p - 2 ) ⁢ ∇ ⁡ u ) = f {-\nabla\cdot((\mu+|\nabla u|^{p-2})\nabla u)=f} is considered, where μ is a given positive number. An anisotropic a posteriori residual-based error estimator is presented. The error estimator is shown to be equivalent, up to higher order terms, to the error in a quasi-norm. The involved constants being independent of μ, the solution, the mesh size and aspect ratio. An adaptive algorithm is proposed and numerical results are presented when p = 3 {p=3} . From this model problem, we propose a simplified error estimator and use it in the framework of an industrial application, namely a nonlinear Navier–Stokes problem arising from aluminium electrolysis.

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

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

From weak to strong convergence of the gradients for Finsler [formula omitted]-Laplacian problems as [formula omitted

In this paper we investigate limits as p→∞ of solutions up to Finsler p-Laplacian problems −divF∗(x,∇up)p−1∂ξF∗(x,∇up)=f with f>0, coupled with a Dirichlet boundary condition up=g on ∂Ω. We prove that the whole sequence of solutions {up} converges to the limit function u∞strongly in W1,m(Ω) for any 1≤m<∞, provided that F∗(x,.) has some strict convexity on its unit sphere. We also characterize an explicit expression of the limit function.

Fractional p-Laplacian elliptic problems with sign changing nonlinearities via the nonlinear Rayleigh quotient

It is established existence and multiplicity of solutions to the fractional p-Laplacian problem in the whole space RN. More precisely, we consider the nonlocal elliptic problem with sign changing nonlinearities in the following form:{(−Δ)psu+V(x)|u|p−2u=λf(x)|u|q−2u+g(x)|u|r−2uinRN,u∈Ws,p(RN), where λ∈(γ⁎,0)∪(0,λ⁎),γ⁎<0, λ⁎>0 and N>ps with s∈(0,1) fixed. Furthermore, we assume that 1<q<p<r<ps⁎=Np/(N−ps). The potential V is a continuous function which is bounded from below by a positive constant. The main objective here is to consider nonlinearities f and g that can be sign changing functions. In this case, by using the nonlinear Rayleigh quotient, we prove that our main problem has at least two nontrivial solutions for each λ∈(γ⁎,0)∪(0,λ⁎). More specifically, the numbers λ⁎>0 and γ⁎<0 are sharp in order to consider the Nehari method, that is, the number λ⁎ is the largest positive number such that the Nehari method can be applied for each λ∈(0,λ⁎). The same assertion is verified for γ⁎, that is, the number γ⁎<0 is the smallest negative number such that the Nehari method can be employed for each λ∈(γ⁎,0).

Fractional p-Laplacian Problem with Critical Stein–Weiss Type Term

Fractional p-Laplacian Problem with Critical Stein–Weiss Type Term

Existence and Nonexistence of Positive Solutions for Perturbations of the Anisotropic Eigenvalue Problem

We consider a Dirichlet problem, which is a perturbation of the eigenvalue problem for the anisotropic p-Laplacian. We assume that the perturbation is (p(z)−1)-sublinear, and we prove an existence and nonexistence theorem for positive solutions as the parameter λ moves on the positive semiaxis. We also show the existence of a smallest positive solution and determine the monotonicity and continuity properties of the minimal solution map.

Existence of Anti-Periodic Solutions for ϕ-Caputo Fractional p-Laplacian Problems via Topological Degree Methods

Existence of Anti-Periodic Solutions for ϕ-Caputo Fractional p-Laplacian Problems via Topological Degree Methods

Multiplicity of Solutions for Dirichlet Problem of a Superlinear Fractional p-Laplacian Equation

This paper is dedicated to studying the following fractional p-Laplacian problem where Ω⊂ℝN (N ≥ 2) is a bounded domain with C 1,1 boundary, s∈(0,1), 2 ≤ p&lt;N/s, (-Δ) s p is the fractional p-Laplacian and f is a (p-1)-superlinear Carathéodory reaction but does not satisfy the usual Ambrosetti-Rabinowitz condition. By using variational methods and Morse theory, we show that the equation has at least three nontrivial solutions: among one of them is positive and one is negative

Existence of solutions to a discrete problems for fourth order nonlinear p-Laplacian via variational method

We consider the boundary value problem for a fourth order nonlinear p-Laplacian difference equation and to prove the existence of at least two nontrivial solutions. Our approach is mainly based on the variational method and critical point theory. One example is included to illustrate the result.

On a quasilinear elliptic problem involving the 1-Laplacian operator and a discontinuous nonlinearity

In this work, we study a quasilinear elliptic problem involving the 1-Laplacian operator, with a discontinuous, superlinear and subcritical nonlinearity involving the Heaviside function $H(\cdot - \beta )$. Our approach is based on an analysis of the associated p-Laplacian problem, followed by a thorough analysis of the asymptotic behaviour or such solutions as $p \to 1^+$. We study also the asymptotic behaviour of the solutions, as $\beta \to 0^+$ and we prove that it converges to a solution of the original problem, without the discontinuity in the nonlinearity.

Strong Maximum Principle and Boundary Estimates for Nonhomogeneous Elliptic Equations

We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear nonhomogeneous degenerate elliptic equations on the formF(x,u,Du,D2u)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ F(x,u,Du,D^{2}u) = 0 $$\\end{document} under suitable assumptions allowing for non-Lipschitz growth in the gradient term. In case of smooth boundaries, we also prove a Hopf lemma, a boundary Harnack inequality, and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply, e.g., to weak solutions of an eigenvalue problem for the variable exponent p-Laplacian.

Semicontinuity of trajectory attractors with respect to exponents for p-Laplacian equation

In this paper, we are concerned with the perturbation of trajectory attractors for dissipative non autonomous p-Laplacian problems with dynamic boundary conditions and no guarantee of the uniqueness of solutions. We prove that under standard conditions, the family of trajectory attractors is upper semicontinuous in the topology of the trajectory space.

Three solutions for a fractional p-Laplacian problem

Three solutions for a fractional p-Laplacian problem

Multiple solutions for a weighted p-Laplacian problem

We prove the existence of at least three solutions for a weighted p-Laplacian operator involving Dirichlet boundary condition in a weighted Sobolev space. The main tool we use here is a three solution theorem in reflexive Banach spaces due to Bonanno and Ricceri.&#x0D; For more information see https://ejde.math.txstate.edu/conf-proc/26/k1/abstr.html

On Non Local p-Laplacian with Right Hand Side Radon Measure

The aim of this paper is to investigate the following non local p-Laplacian problem with data a bounded Radon measure ϑ∈Mb(Ω): (−Δ)psu=ϑinΩ, with vanishing conditions outside Ω, and where s∈(0,1),2−sN&lt;p≤N. An existence result is provided, and some sharp regularity has been investigated. More precisely, we prove by using some fractional isoperimetric inequalities the existence of weak solution u such that: 1. If ϑ∈Mb(Ω), then u∈W0s1,q(Ω) for all s1&lt;s and q&lt;N(p−1)N−s. 2. If ϑ belongs to the Zygmund space LLogαL(Ω),α&gt;N−sN, then the limiting regularity u∈W0s1,N(p−1)N−s(Ω) (for all s1&lt;s). 3. If ϑ∈LLogαL(Ω), and α=N−sN with p=N, then we reach the maximal regularity with respect to s and N,u∈W0s,N(Ω).

Some recent results on singularp-Laplacian equations

AbstractA short account of some recent existence, multiplicity, and uniqueness results for singularp-Laplacian problems either in bounded domains or in the whole space is performed, with a special attention to the case of convective reactions. An extensive bibliography is also provided.