p-Laplacian Equation Research Articles

Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs

We develop a functional framework suitable for the treatment of partial differential equations and variational problems on evolving families of Banach spaces. We propose a definition for the weak time derivative that does not rely on the availability of a Hilbertian structure and explore conditions under which spaces of weakly differentiable functions (with values in an evolving Banach space) relate to classical Sobolev–Bochner spaces. An Aubin–Lions compactness result is proved. We analyse concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we explicitly provide the definition of the time derivative and verify isomorphism properties with the aforementioned Sobolev–Bochner spaces. We conclude with the proof of well posedness for a class of nonlinear monotone problems on an abstract evolving space (generalising the evolutionary p-Laplace equation on a moving domain or surface) and identify some additional problems that can be formulated with the setting developed in this work.

Study on a new [formula omitted]-Laplacian fractional differential model generated by instantaneous and non-instantaneous impulsive effects

In this paper, we discuss a new p-Laplacian fractional differential equation involving instantaneous and non-instantaneous impulses, supplemented with Sturm–Liouville boundary conditions. To study the stated problem, we introduce a new norm for fractional space, denoted by Eα,p. Under this framework, using the critical point theorem proposed by Bonanno and Marano, we prove the existence result of multiple solutions. Finally, an illustrative example is presented.

Oblique derivative problem in a plane sector for strong quasi-linear elliptic equation with p-Laplacian

ABSTRACT We study the behavior near the boundary corner point of weak bounded solutions to the oblique derivative problem for singular p-Laplacian equation with strong nonlinearities singular absorption term in a bounded plane sector. However, no work has been done for investigating the behavior of solutions to the oblique problem for singular p-Laplacian equation in a corner.

Existence and multiplicity of solutions to p-Laplacian equations on graphs

Existence and multiplicity of solutions to p-Laplacian equations on graphs

Random Attractor, Invariant Measures, and Ergodicity of Lattice p-Laplacian Equations Driven by Superlinear Noise

Random Attractor, Invariant Measures, and Ergodicity of Lattice p-Laplacian Equations Driven by Superlinear Noise

On the p-Laplacian type equation with logarithmic nonlinearity: existence, decay and blow up

This work is deal with a problem of wave equation with p-Laplacian, strong damping and logarithmic source terms under initial-boundary conditions. The global existence of weak solution was proved for related to the equation. Global existence results of solutions are obtained using the potential well method, Galerkin method and compactness approach corresponding to the logarithmic source term. Besides, we established the energy functional decaying polynomially to zero as the time goes to infinity due to Nakao?s inequality and some precise priori estimates on logarithmic nonlinearity. For suitable conditions we proved the finite time blow up results of solutions. The proof is based on the concavity method, perturbation energy method and differential-integral inequality technique. Additionally, under suitable assumptions on initial data, the infinite time blow up result is investigated with negative initial energy.

Existence of positive solutions for a p-Laplacian type fractional differential equation with boundary conditions

Existence of positive solutions for a p-Laplacian type fractional differential equation with boundary conditions

一类p-Laplace方程基态解的存在性

一类p-Laplace方程基态解的存在性

On the radial solutions of a p-Laplace equation with the Hardy potential

In this paper, we study the asymptotic behavior of radial solutions of the following quasi-linear equation with the Hardy potential Δpu+h(|x|)|u|p−2u=0, x∈RN−{0}, where 2<p<N, h is a radial function on RN−{0} such that h(|x|)=γ|x|−p, γ>0 and Δpu=div(|∇u|p−2∇u) is the p-Laplacian operator. The study strongly depends on the sign of γ−(σ/p∗)p where σ=(N−p)/(p−1) and p∗=p/(p−1).

更高分数阶p-Laplacian方程的特征值问题

更高分数阶p-Laplacian方程的特征值问题

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<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.

Heteroclinic solutions for damped p-Laplacian difference equations

"In this paper, we investigate the existence of heteroclinic solutions for a class of p-Laplacian difference equations with a parameter. The proof of the main theorem is variational and based on the use of the Mountain Pass Theorem. Our results successfully improve recent ones in the literature and partially answer an open problem proposed by Cabada and Tersian in [23]."

Periodic solutions for a higher-order p-Laplacian neutral differential equation with multiple deviating arguments

In this article, we consider the following high-order p-Laplacian neutral differential equation with multiple deviating arguments:$$(\varphi_{p}(x(t)-cx(t-r))^{(m)}(t)))^{(m)}= f(x(t))x'(t)+g(t,x(t),x(t-\tau_{1}(t)),...,x(t-\tau_{k}(t)))+e(t).$$By appling the continuation theorem, theory of Fourier series, Bernoulli numbers theory and some analytic techniques, sufficient conditions for the existence of periodic solutions are established.

Homogenization of the [formula omitted]-Laplace equation in a periodic setting with a local defect

In this paper, we consider the homogenization of the p-Laplace equation with a periodic coefficient that is perturbed by a local defect. This setting has been introduced in Blanc et al. (2012, 2015) in the linear setting p=2. We construct the correctors and we derive convergence results to the homogenized solution in the case p>2 under the assumption that the periodic correctors are non degenerate.

Pointwise A Priori Estimates for Solutions to Some p-Laplacian Equations

In this article, we apply blow-up analysis to study pointwise a priori estimates for some p-Laplacian equations based on Liouville type theorems. With newly developed analysis techniques, we first extend the classical results of interior gradient estimates for the harmonic function to that for the p-harmonic function, i.e., the solution of Δpu = 0, x ∈ Ω. We then obtain singularity and decay estimates of the sign-changing solution of Lane-Emden-Fowler type p-Laplacian equation −Δpu = |u|λ − 1u, x ∈ Ω, which are then extended to the equation with general right hand term f(x, u) with certain asymptotic properties. In addition, pointwise estimates for higher order derivatives of the solution to Lane-Emden type p-Laplacian equation, in a case of p = 2, are also discussed.

Generalized Pohozhaev’s identity for radial solutions of p-Laplace equations

Abstract We derive a generalized Pohozhaev’s identity for radial solutions of p-Laplace equations, by using the approach in [P. Korman, Global Solution Curves for Semilinear Elliptic Equations, World Scientific Publishing, Hackensack, 2012], thus extending the work [H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 1983, 4, 437–477], where this identity was implicitly used for the Laplace equation.

Regularity for Quasi-Linear p-Laplacian Type Non-Homogeneous Equations in the Heisenberg Group

When 2−1/Q<p≤2, we establish the Cloc0,1 and Cloc1,α-regularities of weak solutions to quasi-linear p-Laplacian type non-homogeneous equations in the Heisenberg group Hn, where Q=2n+2 is the homogeneous dimension of Hn.

Dissipation enhancement by transport noise for stochastic p-Laplace equations

The stochastic $p$-Laplace equation with multiplicative transport noise is studied on the torus $\mathbb T^d\, (d\geq 2)$. It is shown that the dissipation is enhanced by transport noise in both the averaged sense and the pathwise sense.

A Bourgain-Brezis-Mironescu formula for anisotropic fractional Sobolev spaces and applications to anisotropic fractional differential equations

In this paper we prove Bourgain-Brezis-Mironescu's type results (cf. [4]) (BBM for short) for an energy functional which is strongly related to the pseudo anisotropic p-Laplacian. We also provide with the analogous of Maz'ya-Shaposhnikova (see [23]) type results for these energies and finally we apply these results to analyze the stability of solutions to pseudo anisotropic p-Laplace equations.