p-Laplacian Equation Research Articles

Existence of non-trivial solutions to a class of fractional p-Laplacian equations of Schrödinger-type with a combined nonlinearity

Existence of non-trivial solutions to a class of fractional p-Laplacian equations of Schrödinger-type with a combined nonlinearity

Local C0,1-Regularity for the Parabolic p-Laplacian Equation on the Group SU(3)

In this article, when 2≤p≤4, we establish the Cloc0,1-regularity of weak solutions to the degenerate parabolic p-Laplacian equation ∂tu=−∑i=16Xi*(|∇Hu|p−2Xiu) on the group SU(3) granted with horizontal vector fields X1, …, X6. Compared to the Heisenberg group, Hn, we obtained the optimal range of p; that is, 2≤p≤4.

Multiplicity and concentration of normalized solutions to p-Laplacian equations

Multiplicity and concentration of normalized solutions to p-Laplacian equations

Calderón-Zygmund estimates for nonlinear equations of differential forms with BMO coefficients

We obtain Lq-regularity estimates for weak solutions to p-Laplacian type equations of differential forms. In particular, we prove local Calderón-Zygmund type estimates for equations with discontinuous coefficients satisfying the bounded mean oscillation (BMO) condition.

On existence of positive solutions of p-Laplace equations

In this paper, we consider the existence of positive solutions of an equation with p-Laplacian where n ≥ 3, 1 < p < n, q > 0, Δ p u = div(|∇u| p−2∇u), and K(x) is a double bounded function. We will prove the following results. When 0 < q < p − 1, the equation has no solution satisfying inf Rnu = 0 for any double bounded function K(x). When q = p − 1, the equation has a positive radial singular solution for K(x) ≡ Const.. In addition, q = p − 1 is a necessary condition such that this equation (with K(x) ≡ 1) has finite energy solutions. In addition, the existence of positive solutions are also derived for the system where u, v > 0 in Rn \\ {0}, n ≥ 3, 1 < p 1, p 2 < n, q 1, q 2 > 0, and K 1(x), K 2(x) are double bounded functions.

Towards existence theorems to affine p-Laplace equations via variational approach

Towards existence theorems to affine p-Laplace equations via variational approach

P-Laplacian functional differential equations with neutral delay arguments: new oscillation theorems

Inthisstudy,weareinterestedininvestigatingtheoscillatorybehaviorofsolutionstoageneralclassoffunctionaldifferential equations. We consider a neutral-type equation with multiple delays. We first test some monotonic properties of positive solutions to the studied equation. Then, we use some techniques to obtain criteria that guarantee the oscillation of all solutions. We obtain three different forms of oscillation criteria and compare them in terms of efficiency by applying them to a special case of the studied equation. The results obtained are an extension and generalization of previous results in the literature.

A Positive Solution for a Weighted Anisotropic p-Laplace Equation Involving Vanishing Potential

A Positive Solution for a Weighted Anisotropic p-Laplace Equation Involving Vanishing Potential

Global existence for parabolic p-Laplace equations with supercritical growth in whole [formula omitted

The focus of this paper is to investigate the global existence and uniqueness of weak solutions for a class of parabolic p-Laplacian equations whose nonlinearity has a supercritical growth.

Limiting dynamics for stochastic delay p-Laplacian equation on unbounded thin domains

Limiting dynamics for stochastic delay p-Laplacian equation on unbounded thin domains

Local Second Order Sobolev Regularity for p-Laplacian Equation in Semi-Simple Lie Group

In this paper, we establish a structural inequality of the ∞-subLaplacian ▵0,∞ in a class of the semi-simple Lie group endowed with the horizontal vector fields X1,…,X2n. When 1&lt;p≤4 with n=1 and 1&lt;p&lt;3+1n−1 with n≥2, we apply the structural inequality to obtain the local horizontal W2,2-regularity of weak solutions to p-Laplacian equation in the semi-simple Lie group. Compared to Euclidean spaces R2n with n≥2, the range of this p obtained is already optimal.

A Positive Solution for a Weighted p-Laplace Equation with Hardy–Sobolev’s Critical Exponent

A Positive Solution for a Weighted p-Laplace Equation with Hardy–Sobolev’s Critical Exponent

Normalized ground states to the p-Laplacian equation with general nonlinearities

In this paper, we consider the existence of ground state solutions to the following p-Laplacian equation{−Δpu+λ|u|p−2u=f(u)inRN,∫RN|u|pdx=a>0, where 1<p<N and λ∈R. Under general assumptions on the nonlinearity f, we treat two cases. Firstly, in a Lp -subcritical framework, we show the existence of ground state solutions with negative energy and zero, which is a global minimizer. Secondly, in the at least Lp-critical growth, we establish the existence of a mountain pass solution at positive energy level by exploiting a natural constraint related to the Pohozaev identity.

Asymptotic behaviour for a parabolic p-Laplacian equation with an advection force

The rescaling method is presented to allow us establishing the interface functions and the nonnegative local solutions to the evolution of nonlinear degenerate parabolic p-Laplacian process with conservation laws that are posed in one-dimensional space. This equation has a self-similar solution which represents the main feature of this work. The Cauchy problem (CP) for this equation with specific restrictions in the range of parameters and the negative advection coefficient is considered. In this work, there are several regions to discuss the qualitative analysis for the local weak solutions and the asymptotic interfaces in the irregular domains. The dominating of the advection force over the p-Laplacian type diffusion will appear clearly in these regions. Moreover, the solutions of the CP for the degenerate parabolic p-Laplacian advection equations are asymptotically equal to the solutions of advection equations under some restrictions.

Stochastic evolutionary p-Laplace equation: Large deviation principles and transportation cost inequality

We establish a large deviation principle for the strong solution of the evolutionary p-Laplace equation driven by small multiplicative Brownian noise, where the weak convergence approach plays a key role. Moreover, by using the Girsanov transformation along with the L1-contraction approach, we show the quadratic transportation cost inequality for the strong solution to the underlying problem.

Monotonicity of entire solutions to reaction-diffusion equations involving fractional p-Laplacian

Abstract We obtain the one-dimensional symmetry and monotonicity of the entire positive solutions to some reaction-diffusion equations involving fractional p-Laplacian by virtue of the sliding method. More precisely, we consider the following problem { ∂ ⁡ u ∂ ⁡ t ⁢ ( x , t ) + ( - Δ ) p s ⁢ u ⁢ ( x , t ) = f ⁢ ( t , u ⁢ ( x , t ) ) , ( x , t ) ∈ Ω × ℝ , u ⁢ ( x , t ) &gt; 0 , ( x , t ) ∈ Ω × ℝ , u ⁢ ( x , t ) = 0 , ( x , t ) ∈ Ω c × ℝ , \left\{\begin{aligned} \displaystyle{}\frac{\partial u}{\partial t}(x,t)+(-% \Delta)_{p}^{s}u(x,t)&amp;\displaystyle=f(t,u(x,t)),&amp;\hskip 10.0pt(x,t)&amp;% \displaystyle\in\Omega\times\mathbb{R},\\ \displaystyle u(x,t)&amp;\displaystyle&gt;0,&amp;\hskip 10.0pt(x,t)&amp;\displaystyle\in% \Omega\times\mathbb{R},\\ \displaystyle u(x,t)&amp;\displaystyle=0,&amp;\hskip 10.0pt(x,t)&amp;\displaystyle\in% \Omega^{c}\times\mathbb{R},\end{aligned}\right. where s ∈ ( 0 , 1 ) {s\in(0,1)} , p ≥ 2 {p\geq 2} , ( - Δ ) p s {(-\Delta)_{p}^{s}} is the fractional p-Laplacian, f ⁢ ( t , u ) {f(t,u)} is some continuous function, the domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} is unbounded and Ω c = ℝ n ∖ Ω {\Omega^{c}=\mathbb{R}^{n}\setminus\Omega} . Firstly, we establish a maximum principle involving the parabolic p-Laplacian operator. Then, under certain conditions of f, we prove the asymptotic behavior of solutions far away from the boundary uniformly in t ∈ ℝ {t\in\mathbb{R}} . Finally, the sliding method is implemented to derive the monotonicity and uniqueness of the bounded positive entire solutions. To our best knowledge, there has not been any results on the symmetry and monotonicity properties of solutions to the parabolic fractional p-Laplacian equations before.

P-Laplacian Type Equations Via Mountain Pass Theorem in Cerami Sense

p-Laplacian Type Equations Via Mountain Pass Theorem in Cerami Sense

Ground-state sign-changing homoclinic solutions for a discrete nonlinear p-Laplacian equation with logarithmic nonlinearity

By using a direct non-Nehari manifold method from (Tang and Cheng in J. Differ. Equ. 261:2384–2402, 2016), we obtain an existence result of ground-state sign-changing homoclinic solutions that only changes sign once and ground-state homoclinic solutions for a class of discrete nonlinear p-Laplacian equations with logarithmic nonlinearity. Moreover, we prove that the sign-changing ground-state energy is larger than twice the ground-state energy.

Optimality conditions for optimal control of non-well-posed p-Laplacian elliptic equations

Optimality conditions for optimal control of non-well-posed p-Laplacian elliptic equations

Symmetry and asymptotic solutions for a magnetohydrodynamics Darcy–Forchheimer flow with a p-Laplacian operator

Fluid flows under a p-Laplacian operator formulation have been considered recently in connection with the modeling of non-Newtonian fluid processes. To a certain extent, the main reason behind the interest in p-Laplacian operators is the possibility of determining experimental values for the constant p appearing in them. The goal of the present study is to introduce the analysis of solutions of a one-dimensional porous media flow arising in magnetohydrodynamics with generalized initial data under a Lebesgue integrability condition. We present a weak formulation of this problem, and we consider boundedness and uniqueness properties of solutions and also its asymptotic relation with the standard parabolic p-Laplacian equation. Then, we explore solutions arising from classical symmetries (including an explicit kink solution in the p = 3 case) along with asymptotic stationary and non-stationary solutions. The search for stationary solutions is based on a Hamiltonian approach. Finally, non-stationary solutions are obtained by using an exponential scaling resulting in a Hamilton–Jacobi type of equation.