A free boundary problem for p-Laplacian in the plane

We consider the following free boundary problem in an unbounded domain Ω in two dimensions: Δpu=0 in Ω, on J₀, on J₁, where ∂Ω=J₀∪J₁. We prove that if 0<u<1 in Ω, Ji is the graph of a function in and gi is a constant for each i=0,1, then the free boundary ∂Ω must be two parallel straight lines and the solution u must be a linear function. The proof is based on maximum principle.     

Existence of positive solutions for the p-Laplacian with p-gradient term

In this paper, we study the existence of positive solutions for the p-Laplacian involving a p-gradient term. Due to the non-variational structure and the fact that the nonlinearity may be critical or supercritical, the variational method is no longer valid. Taking advantage of global C1,α estimates and the Liouville type theorems, we employ the blow-up argument to obtain the a priori estimates on solutions, and finally obtain the existence result based on the Krasnoselskii fixed point theorem.     

p-Laplacian problems with jumping nonlinearities

We consider the p-Laplacian boundary value problem

(1)     

Multiple positive solutions of the one-dimensional p-Laplacian

In this work we investigate the existence of positive solutions of the p-Laplacian, using the quadrature method. We prove the existence of multiple solutions of the one-dimensional p-Laplacian for α?0, and determine their exact number for α=0.     

Bounds for the positive eigenvalues of the p-Laplacian with decaying potential

An upper bound is obtained for the positive eigenvalues of the p-Laplacian with decaying potential on [0,∞). The bound is expressed in terms of the potential and is shown to be the best possible of its kind.     

An eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities

A multiplicity result for an eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities is obtained. The proof is based on a three critical points theorem for nondifferentiable functionals.     

Nontrivial solutions for p-Laplacian systems

The paper deals with the existence and nonexistence of nontrivial nonnegative solutions for the sublinear quasilinear system

     

Global attractors for p-Laplacian equation

The existence of a -global attractor is proved for the p-Laplacian equation ut−div(|∇u|^p−2∇u)+f(u)=g on a bounded domain Ω⊂Rn(n?3) with Dirichlet boundary condition, where p?2. The nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c₁q|u|−k?f(u)u?c₂q|u|+k and f^′(u)?−l, where q?2 is arbitrary. There is no other restriction on p and q. The asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method, called asymptotic a priori estimate.     

Uniqueness results for the one-dimensional p-Laplacian

This paper is concerned with positive solutions of the boundary value problem ^′(|y^′|^p−2y^′)+f(y)=0, y(−b)=0=y(b) where p>1, b is a positive parameter. Assume that f is continuous on (0,+∞), changes sign from nonpositive to positive, and f(y)/yp−1 is nondecreasing in the interval of f>0. The uniqueness results are proved using a time-mapping analysis.     

Eigenvalues of the p-Laplacian in fractal strings with indefinite weights

In this paper we study the spectral counting function of the weighted p-Laplacian in fractal strings, where the weight is allowed to change sign. We obtain error estimates related to the interior Minkowski dimension of the boundary. We also find the asymptotic behavior of eigenvalues.     

Periodic solutions of the evolutionary p-Laplacian with nonlinear sources

This paper is concerned with the evolutionary p-Laplacian with nonlinear and periodic sources. We will give a rather complete characterization, in terms of the parameter p and the exponent q of the source, of whether or not the positive periodic solutions exist.     

Landesman-Lazer type conditions for a system of p-Laplacian like operators

We study the existence of periodic solutions for a nonlinear second order system of ordinary differential equations of p-Laplacian type. Assuming suitable Nagumo and Landesman-Lazer type conditions we prove the existence of at least one solution applying topological degree methods. We extend a celebrated result by Nirenberg for resonant systems.     

Resonance problems for the p-Laplacian systems

Two existence theorems of the solutions are obtained for the p-Laplacian systems at resonance under a Landesman-Lazer-type condition by critical point theory.     

Multiplicity results for a Neumann problem involving the p-Laplacian

In this paper, we establish some multiplicity results for the following Neumann problem:

     

On a class of singular p-Laplacian boundary value problems

We prove the existence and nonexistence of positive solutions for the boundary value problem

     

Multiple solutions to p-Laplacian problems with concave nonlinearities

In this paper we study the p-Laplacian type elliptic problems with concave nonlinearities. Using some asymptotic behavior of f at zero and infinity, three nontrivial solutions are established.     

Periodic solutions of second-order differential inclusions systems with p-Laplacian

Some existence results are obtained for periodic solutions of nonautonomous second-order differential inclusions systems with p-Laplacian.     

Positive solutions for one-dimensional p-Laplacian boundary value problems with dependence on the first order derivative

This paper presents sufficient conditions for the existence and multiplicity of positive solutions to the one-dimensional p-Laplacian differential equation (?p^′(u^′))+a(t)f(u,u^′)=0, subject to some boundary conditions. We show that it has at least one or two or three positive solutions under some assumptions by applying the fixed point theorem.     

Positive solutions of the p-Laplace equation with singular nonlinearity

For a given bounded domain Ω in Rn with C1,? boundary for some 0<?<1, and a possibly singular nonlinearity f on Ω×(0,∞), we give sufficient conditions on f so that the p-Laplace equation −Δpu=f(x,u) admits a solution . On the basis of a comparison principle we will give a sufficient condition under which such a problem admits a unique solution.