p-Laplacian Problems Research Articles

Asymptotic Estimates for the p-Laplacian on Infinite Graphs with Decaying Initial Data

We consider the Cauchy problem for the evolutive discrete p-Laplacian in infinite graphs, with initial data decaying at infinity. We prove optimal sup and gradient bounds for nonnegative solutions, when the initial data has finite mass, and also sharp evaluation for the confinement of mass, i.e., the effective speed of propagation. We provide estimates for some moments of the solution, defined using the distance from a given vertex. Our technique relies on suitable inequalities of Faber-Krahn type, and looks at the local theory of continuous nonlinear partial differential equations. As it is known, however, not all of this approach can have a direct counterpart in graphs. A basic tool here is a result connecting the supremum of the solution at a given positive time with the measure of its level sets at previous times. We also consider the case of slowly decaying initial data, where the total mass is infinite.

Existence of solution for p-Laplacian boundary value problems with two singular and subcritical nonlinearities

We consider a boundary value problem for p-Laplacian systems with two singular and subcritical nonlinearities. We obtain one theorem which shows that there exists at least one nontrivial weak solution for these problems under some conditions. We obtain this result by variational method and critical point theory.

Existence and Nonexistence of Solutions to p-Laplacian Problems on Unbounded Domains

In this article, using a fixed point index theorem on a cone, we prove the existence and multiplicity results of positive solutions to a one-dimensional p-Laplacian problem defined on infinite intervals. We also establish the nonexistence results of nontrivial solutions to the problem.

Groundstate asymptotics for a class of singularly perturbed p-Laplacian problems in $${{\mathbb {R}}}^N$$

We study the asymptotic behavior of positive groundstate solutions to the quasilinear elliptic equation \begin{equation} -\Delta_{p} u + \varepsilon u^{p-1} - u^{q-1} +u^{\mathit{l}-1} = 0 \qquad \text{in} \quad \mathbb{R}^{N}, \end{equation} where $1<p<N $, $p<q<l<+\infty$ and $\varepsilon> 0 $ is a small parameter. For $\varepsilon\rightarrow 0$, we give a characterisation of asymptotic regimes as a function of the parameters $q$, $l$ and $N$. In particular, we show that the behavior of the groundstates is sensitive to whether $q$ is less than, equal to, or greater than the critical Sobolev exponent $p^{*} :=\frac{pN}{N-p}$.

Asymptotic boundary estimates for solutions to the p-Laplacian with infinite boundary values

In this paper, by using Karamata regular variation theory and the method of upper and lower solutions, we mainly study the second order expansion of solutions to the following p-Laplacian problems: Delta _{p} u=b(x)f(u), u>0, xin varOmega, u|_{partial varOmega }=infty , where Ω is a bounded domain with smooth boundary in mathbb{R}^{N} (Ngeq 2), p>1, b in C^{alpha }(bar{varOmega }) which is positive in Ω and may be vanishing on the boundary. The absorption term f is normalized regularly varying at infinity with index sigma >p-1. The results extend some previous findings of D. Repovš (J. Math. Anal. Appl. 395:78-85, 2012) in a certain sense.

Multiplicity results for p-Laplacian boundary value problem with jumping nonlinearities

We investigate the multiplicity of solutions for one-dimensional p-Laplacian Dirichlet boundary value problem with jumping nonlinearities. We obtain three theorems: The first states that there exists exactly one solution when nonlinearities cross no eigenvalue. The second guarantees that there exist exactly two solutions, exactly one solution and no solution, depending on the source term, when nonlinearities cross just the first eigenvalue. The third claims that there exist at least three solutions, exactly one solution and no solution, depending on the source term, when nonlinearities cross the first and second eigenvalues. We obtain the first and second theorem by considering the eigenvalues and the corresponding normalized eigenfunctions of the p-Laplacian eigenvalue problem, and the contraction mapping principle in the p-Lebesgue space (when pge 2). We obtain the third result by Leray–Schauder degree theory.

Existence and Multiplicity of Solutions for p-Laplacian Neumann Problems

In this paper, existence theorems are proved for p-Laplacian Neumann problems under the Landesman–Lazer type condition. Our results are derived from a classical saddle point theorem and the least action principle respectively. Furthermore, multiplicity of solutions is established by applying a known multiple critical points result due to H. Brezis and L. Nirenberg. The above-mentioned conclusions are based on variational methods.

Exact multiplicity and stability of solutions of a 1-dimensional, [formula omitted]-Laplacian problem with positive convex nonlinearity

We consider the p-Laplacian problem (1)−ϕp(u′)′=λf(u),on (−1,1), (2)u(±1)=0,where p>1 (p≠2), ϕp(z)≔|z|p−1sgnz, z∈R, λ⩾0, and f:R→(0,∞) is C2 and there exists C>0, q>p such that f(ξ)⩾C(1+ξq−1),f′(ξ)>0,f′′(ξ)>0,ξ>0.We show that the set of solutions (λ,u) of (1)–(2) consists of a C2 curve in (0,∞)×C1[−1,1] with end point (0,0)∈R×C1[−1,1] and which tends to {0}×∞, and has a single turning point. Thus, there exists λ∗>0 such that (1)–(2) has exactly two solutions, u−(λ)<u+(λ), when 0<λ<λ∗, and no solutions when λ>λ∗.When 0<λ<λ∗ the solutions u±(λ) are equilibria of a related time-dependent, parabolic problem, and in this context their stability is of interest. We show that the ‘lower’ solution u−(λ) is stable and the ‘upper’ solution u+(λ) is unstable, and solutions of the parabolic problem with initial values below u+(λ) converge to u−(λ), while those with initial values above u+(λ) are unbounded.

Some results on the p(u)-Laplacian problem

The p-Laplacian problem with the exponent of nonlinearity p depending on the solution u itself is considered in this work. Both situations when p(u) is a local quantity or when p(u) is nonlocal are studied here. For the associated boundary-value local problem, we prove the existence of weak solutions by using a singular perturbation technique. We also prove the existence of weak solutions to the nonlocal version of the associated boundary-value problem. The issue of uniqueness for these problems is addressed in this work as well, in particular by working out the uniqueness for a one dimensional local problem and by showing that the uniqueness is easily lost in the nonlocal problem.

一类p-Laplace方程解的存在性问题

一类p-Laplace方程解的存在性问题

带有非负局部项的p-Laplacian问题的径向正解

带有非负局部项的p-Laplacian问题的径向正解

Existence of weak solutions for fractional p-Laplacian problem with Dirichlet-type boundary condition

Existence of weak solutions for fractional p-Laplacian problem with Dirichlet-type boundary condition

一类奇异p-Laplace方程解的存在性问题

一类奇异p-Laplace方程解的存在性问题

Positive solutions for a critical [formula omitted]-Laplacian problem with a Kirchhoff term

In this article, the following p-Laplacian problem with Kirchhoff term and critical exponent is studied −a+b∫RN|∇u|pdxΔpu=up∗−1+μh(x),x∈RN,u∈D1,p(RN),where a,μ≥0,b>0,1<p<N,p∗=NpN−p, Δpu=div(|∇u|p−2∇u), and h∈Lp∗p∗−1(RN) is nonzero and nonnegative. When μ=0, with the help of the recent result of B. Sciunzi(Adv. Math. 291(2016) 12-23), infinitely many positive solutions are obtained by some analysis techniques. While μ>0, by the variational method, some results about positive solutions are obtained.

A Hopf's lemma and the boundary regularity for the fractional p-Laplacian

We begin the paper with a Hopf's lemma for a fractional p-Laplacian problem on a half-space. Specifically speaking, we show that the derivative of the solution along the outward normal vector is strictly positive on the boundary of the half-space. Next we show that positive solutions for a fractional p-Laplacian equation possess certain Holder continuity up to the boundary.

Ground state solutions of Kirchhoff-type fractional Dirichlet problem with p-Laplacian

We consider the Kirchhoff-type p-Laplacian Dirichlet problem containing the left and right fractional derivative operators. By using the Nehari method in critical point theory, we obtain the existence theorem of ground state solutions for such Dirichlet problem.

Sign-Changing Solutions of Fractional 𝑝-Laplacian Problems

Abstract In this paper, we obtain the existence and multiplicity of sign-changing solutions of the fractional p-Laplacian problems by applying the method of invariant sets of descending flow and minimax theory. In addition, we prove that the problem admits at least one least energy sign-changing solution by combining the Nehari manifold method with the constrained variational method and Brouwer degree theory. Furthermore, the least energy of sign-changing solutions is shown to exceed twice that of the least energy solutions.

Uniqueness of positive radial solutions for infinite semipositone p-Laplacian problems in exterior domains

We prove uniqueness and asymptotic behavior of positive radial solutions to the p -Laplacian problem { − Δ p u = λ K ( | x | ) f ( u ) in | x | > r 0 , u = 0 on | x | = r 0 , u ( x ) → 0 as | x | → ∞ , where Ω = { x ∈ R n : | x | > r 0 > 0 } , n > p , f : ( 0 , ∞ ) → R is continuous, f ( u ) ∼ u q at ∞ for some q ∈ [ 0 , p − 1 ) with possible infinite semipositone structure at 0, and λ is a large parameter.

Nehari manifold and multiplicity result for elliptic equation involving p-laplacian problems

This article shows the existence and multiplicity of positive solutions of the $p$-Laplacien problem $$\displaystyle -\Delta_{p} u=\frac{1}{p^{\ast}}\frac{\partial F(x,u)}{\partial u} + \lambda a(x)|u|^{q-2}u \quad \mbox{for } x\in\Omega;\quad \quad u=0,\quad \mbox{for } x\in\partial\Omega$$ where $\Omega$ is a bounded open set in $\mathbb{R}^n$ with smooth boundary, $1&lt;q&lt;p&lt;n$, $p^{\ast}=\frac{np}{n-p}$, $\lambda \in \mathbb{R}\backslash \{0\}$ and $a$ is a smooth function which may change sign in $\overline{\Omega}$. The method is based on Nehari results on three sub-manifolds of the space $W_{0}^{1,p}$.

Positive Solutions of One-Dimensional p-Laplacian Problems with Superlinearity

We study one-dimensional p-Laplacian problems and answer the unsolved problem. Our method is to study the property of the operator, the concavity of the solutions and the continuity of the first eigenvalues. By the above study, the main difficulty is overcome and the fixed point theorem can be applied for the corresponding compact maps. An affirmative answer is given to the unsolved problem with superlinearity. A global growth condition is not imposed on the nonlinear term f. The assumptions of this paper are more general than the usual, thus the existing results cannot be utilized. Some recent results are improved from weak solutions to classical solutions and from p ≥ 2 to p ∈ ( 1 , ∞ ) .