p-Laplacian Problems Research Articles

An exact estimate result for singular [formula omitted]-Laplacian equations with sign-changing potential

In this paper, we study the multiplicity of positive solutions for a singular p-Laplacian problem with sign-changing potential. By the Nehari method and some analysis techniques, we obtain two positive solutions and point out one of the two positive solutions is a ground state solution. Moreover, we provide an exact estimate to ensure the singular p-Laplacian problem has two positive solutions.

Asymmetric critical p-Laplacian problems

We obtain nontrivial solutions for two types of asymmetric critical p-Laplacian problems with Ambrosetti–Prodi type nonlinearities in a smooth bounded domain in $${\mathbb {R}}^N,\, N \ge 2$$ . For $$1< p < N$$ , we consider an asymmetric problem involving the critical Sobolev exponent $$p^*= Np/(N - p)$$ . In the borderline case $$p = N$$ , we consider an asymmetric critical exponential nonlinearity of the Trudinger–Moser type. In the absence of a suitable direct sum decomposition, we use a linking theorem based on the $${\mathbb {Z}}_2$$ -cohomological index to prove existence of solutions.

Weighted Beckmann problem with boundary costs

We show that a solution to a variant of the Beckmann problem can be obtained by studying the limit of some weighted p − p- Laplacian problems. More precisely, we find a solution to the following minimization problem: min { ∫ Ω k d | w | + ∫ ∂ Ω g − d ν − − ∫ ∂ Ω g + d ν + : w ∈ M d ( Ω ) , ν ∈ M ( ∂ Ω ) , − ∇ ⋅ w = f + ν } , \begin{equation*} \min \bigg \{\int _\Omega k \,\mathrm {d}|w| + \int _{\partial \Omega } g^-\,\mathrm {d}\nu ^- - \int _{\partial \Omega } g^+\,\mathrm {d}\nu ^+\,:\,w \in \mathcal {M}^d(\Omega ),\,\nu \in \mathcal {M}(\partial \Omega ),\,-\nabla \cdot w =f + \nu \bigg \}, \end{equation*} where f , k f,\,k , and g ± g^\pm are given. In addition, we connect this problem to a formulation with Kantorovich potentials with Dirichlet boundary conditions.

$$\varvec{p}$$-Laplacian problems involving critical Hardy–Sobolev exponents

We prove existence, multiplicity, and bifurcation results for p-Laplacian problems involving critical Hardy–Sobolev exponents. Our results are mainly for the case $$\lambda \ge \lambda _1$$ and extend results in the literature for $$0< \lambda < \lambda _1$$ . In the absence of a direct sum decomposition, we use critical point theorems based on a cohomological index and a related pseudo-index.

Decay estimates for Wolff potentials in [formula omitted] and gradient-dependent quasilinear elliptic equations

We prove decay estimates of positive weak solutions u∈D1,p(RN)∩C1(RN) and their gradients |∇u| of the p-Laplacian problem (1<p<N):u∈D1,p(RN):−Δpu=a(x), where the measurable function a satisfies the decay estimate (α>0):0<a(x)≲11+|x|N+α,∀x∈RN. The positive Borel measure on RN generated by the right-hand side function a and its associated Wolff potential is known to be the main tool for pointwise estimates of u(x)>0 and |∇u(x)|. Therefore, one of the main goal of this paper is to prove decay estimates for corresponding Wolff potentials in RN. The obtained decay estimates in conjunction with a sub-supersolution method, which is developed here for quasilinear elliptic equations, are then applied to prove the existence of positive solutions for classes of gradient-dependent quasilinear equations in the formu∈D1,p(RN):−Δpu=a(x)f(u,∇u).

Resonant p-Laplacian problems with functional boundary conditions

By constructing a suitable projection scheme and using the extension of Mawhin’s continuation theorem, the existence of solution for functional p-Laplacian boundary value problems at resonance is studied. The paper is a generalization of some current results to a fully nonlinear case.

Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient

We consider the boundary value problem where $$\Omega \subset \mathbb {R}^{N}$$ , $$N \ge 2$$ , is a bounded domain with smooth boundary. We assume $$c,\, h \in L^q(\Omega )$$ for some $$q > \max \{N/p,1\}$$ with $$c \gneqq 0$$ and $$\mu \in L^{\infty }(\Omega )$$ . We prove existence and uniqueness results in the coercive case $$ \lambda \le 0$$ and existence and multiplicity results in the non-coercive case $$ \lambda >0$$ . Also, considering stronger assumptions on the coefficients, we clarify the structure of the set of solutions in the non-coercive case.

Analyzing the nonlinear p-Laplacian problem with the improved element-free Galerkin method

In this paper, the improved element-free Galerkin (IEFG) method is developed for the numerical analysis of the nonlinear p-Laplacian problem. In this method, the improved moving least squares approximation is adopted to form meshless shape functions, the Galerkin weak form is used to derive the nonlinear system of algebraic equations, and the subroutine ‘fsolve’ of MATLAB is adopted to deal with the strong nonlinearity. The computational formulas of the IEFG method for the p-Laplacian problem are obtained in detail. Numerical results show that the IEFG method for the p-Laplacian problem is convergent and has high computational accuracy.

Global existence and blow-up results for p-Laplacian parabolic problems under nonlinear boundary conditions

This paper is devoted to studying the global existence and blow-up results for the following p-Laplacian parabolic problems: \t\t\t{(h(u))t=∇⋅(|∇u|p−2∇u)+f(u)in D×(0,t∗),∂u∂n=g(u)on ∂D×(0,t∗),u(x,0)=u0(x)≥0in D‾.\\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}$$\\textstyle\\begin{cases} (h(u) )_{t} =\\nabla\\cdot (|\\nabla u|^{p-2}\\nabla u )+f(u) &\\mbox{in } D\\times(0,t^{*}), \\\\ \\frac{\\partial u}{\\partial n}=g(u) &\\mbox{on } \\partial D\\times (0,t^{*}), \\\\ u(x,0)=u_{0}(x)\\geq0 & \\mbox{in } \\overline{D}. \\end{cases} $$\\end{document} Here p>2, the spatial region D in mathbb{R}^{N} (Ngeq2) is bounded, and ∂D is smooth. We set up conditions to ensure that the solution must be a global solution or blows up in some finite time. Moreover, we dedicate upper estimates of the global solution and the blow-up rate. An upper bound for the blow-up time is also specified. Our research relies mainly on constructing some auxiliary functions and using the parabolic maximum principles and the differential inequality technique.

Existence of positive radial solutions for a superlinear semipositone p-Laplacian problem on the exterior of a ball

We prove the existence of positive radial solutions to a class of semipositone p-Laplacian problems on the exterior of a ball subject to Dirichlet and nonlinear boundary conditions. Using variational methods we prove the existence of a solution, and then use a priori estimates to prove the positivity of the solution.

On branches of positive solutions for 𝑝-Laplacian problems at the extreme value of the Nehari manifold method

This paper is concerned with variational continuation of branches of solutions for nonlinear boundary value problems, which involve the p p -Laplacian, an indefinite nonlinearity, and depend on a real parameter λ \lambda . A special focus is given to the extreme value λ ∗ \lambda ^* of the Nehari manifold that determines the threshold of applicability of the Nehari manifold method. In the main result the existence of two branches of positive solutions for the cases where the parameter λ \lambda lies above the threshold λ ∗ \lambda ^* is obtained.

Multiplicity and concentration of positive solutions for fractional [formula omitted]-Laplacian problem involving concave–convex nonlinearity

In this paper, we discuss a fractional p-Laplacian problem. Under suitable assumptions on nonlinearity and weight, the existence and multiplicity of positive solutions are obtained via variational method. Moreover, the global maximum point of the solution concentrates at a local minimum point of the weight function.

Existence and uniqueness of solutions for a fractional p-Laplacian problem in \mathbb{R}^N

Existence and uniqueness of solutions for a fractional p-Laplacian problem in \mathbb{R}^N

Influence of Gradient Terms on the Existence of Solutions to the Dirichlet Problem for p-Laplacian

We study the Dirichlet problem for the inhomogeneous p-Laplace equation with a nonlinear source and a gradient term. We investigate the influence of the gradient term on the existence of radially symmetric solutions. Sufficient conditions for the existence of solutions are explicetly given in terms of the data of the problem.

Existence of Solution for a p-Laplacian Multi-point Boundary Value Problem at Resonance

In this paper, we are concerned with the p-Laplacian multi-point boundary value problem $$\begin{aligned} (\phi _{p}(x''(t)))'= & {} f(t,x(t),x'(t),x''(t)),\,t\in (0, 1),\\ \phi _{p}(x''(0))= & {} \sum _{i=1}^{m}\alpha _{i}\phi _{p}(x''(\xi _{i})),\\ ~~x'(1)= & {} \sum _{j=1}^{n}\beta _{j}x'(\eta _{j}), ~~x''(1)=0, \end{aligned}$$ where \(\phi _p(s)=|s|^{p-2}s,~p>1, \phi _{q}=\phi _{p}^{-1}, \frac{1}{p}+\frac{1}{q}=1, f: [0, 1]\times R^3\rightarrow R\) is a continuous function, \(0<\xi _{1}<\xi _{2}<\cdots<\xi _{m}<1, \alpha _{i}\in R, i=1,2,\ldots , m, m\ge 2\) and \(0<\eta _{1}<\cdots<\eta _{n}<1, \beta _{j}\in R, j=1,\ldots , n, n\ge 1\). Based on the extension of Mawhin’s continuation theorem, a new general existence result of the p-Laplacian problem is established in the resonance case.

Blow-up phenomena for p-Laplacian parabolic problems with Neumann boundary conditions

In this paper, we deal with the blow-up and global solutions of the following p-Laplacian parabolic problems with Neumann boundary conditions: \t\t\t{(g(u))t=∇⋅(|∇u|p−2∇u)+k(t)f(u)in Ω×(0,T),∂u∂n=0on ∂Ω×(0,T),u(x,0)=u0(x)≥0in Ω‾,\\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}$$\\textstyle\\begin{cases} (g(u) )_{t} =\\nabla\\cdot ( {|\\nabla u|^{p-2}}\\nabla u )+k(t)f(u) & \\mbox{in } \\Omega\\times(0,T), \\\\ \\frac{\\partial{u}}{\\partial n}=0 &\\mbox{on } \\partial\\Omega\\times (0,T), \\\\ u(x,0)=u_{0}(x)\\geq0 & \\mbox{in } \\overline{\\Omega}, \\end{cases} $$\\end{document} where p>2 and Ω is a bounded domain in mathbb{R}^{n} (ngeq 2) with smooth boundary ∂Ω. By introducing some appropriate auxiliary functions and technically using maximum principles, we establish conditions to guarantee that the solution blows up in some finite time or remains global. In addition, the upper estimates of blow-up rate and global solution are specified. We also obtain an upper bound of blow-up time.

Existence and multiplicity of solutions for critical Kirchhoff-type p-Laplacian problems

In this paper, we deal with the existence, multiplicity and asymptotic behavior of solutions for the following stationary Kirchhoff problems involving the p-Laplacian:−M(∫Ω|∇u|pdx)Δpu=λf(x,u)+|u|p⁎−2uinΩ,u=0 on∂Ω, where Ω is a bounded smooth domain of RN, M:R0+→R0+ is a continuous function satisfying some extra assumptions. By using the mountain pass theorem, the existence of solutions is obtained. By applying an abstract critical point theorem based on the cohomological index, we prove the existence of multiple solutions for the above problem, when parameter λ belongs to some left neighborhood of the eigenvalue of the nonlinear operator −‖∇u‖(θ−1)pΔp, where p<N<θp<p⁎. We also investigate the asymptotic behavior of solutions when λ converges to infinity. The main feature, as well as the main difficulty, of our problem is the fact that the Kirchhoff term M could be zero at zero, i.e. the problem is degenerate.

Multiple positive solutions for some P-Laplacian nonlinear problem at infinity

In recent years, boundary value problem of second-order has received a lot of ‎attention.‎ In this paper, I study the existence of positive solutions for a class of p-Laplacian boundary value ‎problem at infinity. The fixed point theorems in cones are our main tools to prove the existence ‎of solutions.‎ I provide sufficient conditions under which this system has a solution. I establish some propositions ‎to prove the existence of positive solutions for these equations‎.

Variational approach for a p-Laplacian boundary value problem on time scales

ABSTRACTIn this paper, we study a second-order p-Laplacian dynamic equation on a periodic time scale subject to certain boundary value conditions. The variational structure of the above-mentioned boundary value problem is presented. Some new results on the existence of at least one or three distinct periodic solutions are established by means of the saddle point theorem, the least action principle as well as the three critical point theorem.

The Gelfand problem for the 1-homogeneous p-Laplacian

Abstract In this paper, we study the existence of viscosity solutions to the Gelfand problem for the 1-homogeneous p-Laplacian in a bounded domain {\Omega\subset\mathbb{R}^{N}} , that is, we deal with -\frac{1}{p-1}\lvert\nabla u\rvert^{2-p}\operatorname{div}(\lvert\nabla u% \rvert^{p-2}\nabla u)=\lambda e^{u} in Ω with {u=0} on {\partial\Omega} . For this problem we show that, for {p\in[2,\infty]} , there exists a positive critical value {\lambda^{*}=\lambda^{*}(\Omega,N,p)} such that the following holds: • If {\lambda&lt;\lambda^{*}} , the problem admits a minimal positive solution {w_{\lambda}} . • If {\lambda&gt;\lambda^{*}} , the problem admits no solution. Moreover, the branch of minimal solutions {\{w_{\lambda}\}} is increasing with λ. In addition, using degree theory, for fixed p we show that there exists an unbounded continuum of solutions that emanates from the trivial solution {u=0} with {\lambda=0} , and for a small fixed λ we also obtain a continuum of solutions with {p\in[2,\infty]} .