Nonlinear p-Laplacian Equation Research Articles

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.

Liouville Type Theorems for Nonlinear p-Laplacian Equation on Complete Noncompact Riemannian Manifolds

Liouville Type Theorems for Nonlinear p-Laplacian Equation on Complete Noncompact Riemannian Manifolds

P-Laplacian Equations on Locally Finite Graphs

This paper is mainly concerned with the following nonlinear p-Laplacian equation $$ - {\Delta _p}u(x) + (\lambda a(x) + 1){\left| u \right|^{p - 2}}(x)u(x) = f(x,u(x)),\;\;\;{\rm{in}}\;V$$ on a locally finite graph G =(V, E) with more general nonlinear term, where Δp is the discrete p-Laplacian on graphs, p ≥ 2. Under some suitable conditions on f and a(x), we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλ via the method of Nehari manifold, for any λ > 1. In addition, as λ → + ∞, we prove that the solution uλ converge to a solution of the following Dirichlet problem $$\left\{ {\matrix{ { - {\Delta _p}u(x) + {{\left| u \right|}^{p - 2}}(x)u(x) = f(x,u(x)),} \;\;\;\;\;\;\;\; {{\rm{in}}\;\Omega ,} \hfill \cr {u(x) = 0,} \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\; {{\rm{on}}\;\partial \Omega ,} \hfill \cr } } \right.$$ where Ω = {x ∈ V: a(x) = 0} is the potential well and ∂Ω denotes the the boundary of Ω.

An efficient meshless radial point collocation method for nonlinear p-Laplacian equation

This paper considered the spectral meshless radial point interpolation (SMRPI) method to unravel for the nonlinear p-Laplacian equation with mixed Dirichlet and Neumann boundary conditions. Through this assessment, which includes meshless methods and collocation techniques based on radial point interpolation, we construct the shape functions, with the Kronecker delta function property, as basis functions in the framework of spectral collocation methods. Studies in this regard require one to evaluate the high-order derivatives without any kind of integration locally over the small quadrature domains. Finally, some examples are given to illustrate the low computing costs and high enough accuracy and efficiency of this method to solve a p-Laplacian equation and it would be of great help to fulfill the implementation related to the element-free Galerkin (EFG) method. Both the SMRPI and the EFG methods have been compared by similar numerical examples to show their application in strongly nonlinear problems.

Anisotropic and p, q-nonlinear partial differential equations

Anisotropic partial differential equations recently received a large interest in the mathematical literature, due to their applications to double and multiphase variational energies, as well as to anisotropic energies in integral form. More specifically, from a mathematical point of view, we need to consider a generalization of the classical Laplacian elliptic and parabolic partial differential equations, as well as the nonlinear p-Laplacian equations, which naturally arises new and interesting mathematical questions, nowadays only partially solved in the context of anisotropic p, q-growth nonlinear elliptic and parabolic partial differential equations. In this context, we describe some “mathematical pathologies”; more precisely, some singularities in the potential generated by some anisotropic energies. The singularities appear if the anisotropy of the energy is too large in some directions, while these singularities do not appear, not only if the energy is isotropic with respect to all directions, but also even if we allow an energy integral with a mild anisotropy.

The element-free Galerkin method for the nonlinear p-Laplacian equation

The element-free Galerkin (EFG) method is developed in this paper for solving the nonlinear p-Laplacian equation. The moving least squares approximation is used to generate meshless shape functions, the penalty approach is adopted to enforce the Dirichlet boundary condition, the Galerkin weak form is employed to obtain the system of discrete equations, and two iterative procedures are developed to deal with the strong nonlinearity. Then, the computational formulas of the EFG method for the p-Laplacian equation are established. Numerical results are finally given to verify the convergence and high computational precision of the method.

Coiflet solution of strongly nonlinear p-Laplacian equations

A new boundary extension technique based on the Lagrange interpolating polynomial is proposed and used to solve the function approximation defined on an interval by a series of scaling Coiflet functions, where the coefficients are used as the single-point samplings. The obtained approximation formula can exactly represent any polynomials defined on the interval with the order up to one third of the length of the compact support of the adopted Coiflet function. Based on the Galerkin method, a Coiflet-based solution procedure is established for general two-dimensional p-Laplacian equations, following which the equations can be discretized into a concise matrix form. As examples of applications, the proposed modified wavelet Galerkin method is applied to three typical p-Laplacian equations with strong nonlinearity. The numerical results justify the efficiency and accuracy of the method.

On the existence of sign-changing radial solutions to nonlinear [formula omitted]-Laplacian equations in [formula omitted

A class of nonlinear equations −(rn−1|u′|p−2u′)′=rn−1w(r)f(u)on [0,1], where 1<p<∞, is considered. We study the existence of sign-changing solutions to this problem. Some sufficient conditions for such solutions with the prescribed number of zeros are developed. We generalize the result of Naito and Tanaka (2008) to the radial solutions of a class of nonlinear p-Laplacian equations in Rn. Some related issues on the half-line will be also discussed.

Sum of weighted Lebesgue spaces and nonlinear elliptic equations

We study the sum of weighted Lebesgue spaces, by considering an abstract measure space \({(\Omega ,\mathcal{A},\mu)}\) and investigating the main properties of both the Banach space $$L\left( \Omega \right) =\left\{u_{1}+u_{2}:u_{1} \in L^{q_{1}} \left(\Omega \right),u_{2} \in L^{q_{2}} \left( \Omega \right) \right\}, L^{q_{i}} \left( \Omega \right) :=L^{q_{i}} \left( \Omega ,d\mu \right),$$and the Nemytskiĭ operator defined on it. Then we apply our general results to prove existence and multiplicity of solutions to a class of nonlinear p-Laplacian equations of the form $$-\triangle _{p}u+V\left( \left| x\right| \right) \left| u\right| ^{p-2}u=f\left( \left| x\right| ,u\right) \quad {\rm in} \mathbb{R}^{N}$$where V is a nonnegative measurable potential, possibly singular and vanishing at infinity, and f is a Carathéodory function satisfying a double-power growth condition in u.

Localized radial solutions for a nonlinear p-Laplacian equation inRN

Localized radial solutions for a nonlinear p-Laplacian equation in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>R</mml:mi><mml:mi>N</mml:mi></mml:msup></mml:math>

Asymptotic behavior of positive solutions of some quasilinear elliptic problems

We discuss the asymptotic behavior of positive solutions of the quasilinear elliptic problem −Δpu = a up−1−b(x) uq, u|∂ Ω = 0, as q → p − 1 + 0 and as q → ∞, via a scale argument. Here Δp is the p-Laplacian with 1 < p ∞ and q > p − 1. If p = 2, such problems arise in population dynamics. Our main results generalize the results for p = 2, but some technical difficulties arising from the nonlinear degenerate operator −Δp are successfully overcome. As a by-product, we can solve a free boundary problem for a nonlinear p-Laplacian equation.

Positive solutions of nonlinear p-Laplacian equations in [formula omitted

This paper studies the existence of positive solutions of the nonlinear p-Laplacian equation − Δ p u = f ( x , u ) in R n . Existence of infinitely many positive solutions is established via super–sub-solution methods. Decaying conditions on f ( x , u ) at | x | = ∞ are imposed. Necessity of the conditions and nonexistence of positive solutions are illustrated through decay–growth estimates of solutions at infinity. Special situation and examples are checked in detail.