Infinitely many radial solutions for a p-Laplacian problem p-superlinear at the origin

We prove the existence of infinitely many radial solutions for a p-Laplacian Dirichlet problem which is p-superlinear at the origin. The main tool that we use is the shooting method. We extend for more general nonlinearities the results of J. Iaia in [J. Iaia, Radial solutions to a p-Laplacian Dirichlet problem, Appl. Anal. 58 (1995) 335-350]. Previous developments require a behavior of the nonlinearity at zero and infinity, while our main result only needs a condition of the nonlinearity at zero.     

Existence of radial solutions for an asymptotically linear p-Laplacian problem

We prove that an asymptotically linear Dirichlet problem which involves the p-Laplacian operator has multiple radial solutions when the nonlinearity has a positive zero and the range of the ‘p-derivative’ of the nonlinearity includes at least the first j radial eigenvalues of the p-Laplacian operator. The main tools that we use are a uniqueness result for the p-Laplacian operator and bifurcation theory.     

Up-to boundary regularity for a singular perturbation problem of p-Laplacian type

In this paper we are interested in establishing up-to boundary uniform estimates for the one phase singular perturbation problem involving a nonlinear singular/degenerate elliptic operator. Our main result states: if Ω⊂Rn is a C1,α domain, for some 0<α<1 and uε verifies

     

Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity

In this paper we deal with multiplicity of positive solutions to the p-Laplacian equation of the type

     

Nontrivial solutions of superlinear p-Laplacian equations

We consider p-Laplacian equations on a bounded domain, where the nonlinearity is superlinear but dose not satisfy the usual Ambrosetti-Rabinowitz condition near infinity, or its dual version near zero. Nontrivial solutions are obtained by computing the critical groups and Morse theory.     

On the existence of periodic solutions to p-Laplacian Rayleigh differential equation with a delay

In this paper, the authors study the existence of periodic solutions to a p-Laplacian Rayleigh differential equation with a delay as follows:

(φp(y^′(t))^)′+f(y^′(t))+g(y(t−τ(t)))=e(t),     

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.     

Three positive solutions for the one-dimensional p-Laplacian

In this paper, nonlinear two point boundary value problems with p-Laplacian operators subject to Dirichlet boundary condition and nonlinear boundary conditions are studied. We show the existence of three positive solutions by the five functionals fixed point theorem.     

Critical groups at infinity for p-Laplacian equations with indefinite nonlinearities

In this paper, by a kind of decomposition lemma and Künneth formula we study the critical groups at infinity for the associated functional of the following p-Laplacian equation with indefinite nonlinearities

     

Non-extinction of solutions to a fast diffusive p-Laplace equation with Neumann boundary conditions

In this note, the authors establish a non-extinction result for the changing sign solution with negative initial energy by discussing a suitable differential inequality. The result gives an answer to the problem unsolved in Qu, Bai, Zheng (2014) [1]. Two examples are given in the paper to show the existence of the initial datum with negative initial energy.     

Localization for the evolution p-Laplacian equation with strongly nonlinear source term

In this paper we study the strict localization for the p-Laplacian equation with strongly nonlinear source term. Let u:=u(x,t) be a solution of the Cauchy problem

     

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

     

On the structure of positive radial solutions to an equation containing a p-Laplacian with weight

Let A,B:(0,∞)?(0,∞) be two given weight functions and consider the equation

     

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.     

Global nontrivial bifurcation of homogeneous operators with an application to the p-Laplacian

We prove the existence of a global solution branch of nontrivial solutions for a class of equations by a blow-up method. In particular, positively homogeneous problems and equations with the p-Laplace operator are considered.     

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.