Radial equivalence and study of self-similarity for two very fast diffusion equations

In this paper we continue the study of the radial equivalence between the porous medium equation and the evolution p-Laplacian equation, begun in a previous work. We treat the cases m<0 and p<1. We perform an exhaustive study of self-similar solutions for both equations, based on a phase-plane analysis and the correspondences we discover. We also obtain special correspondence relations and self-maps for the limit case m=−1, p=0, which is particularly important in applications in image processing. We also find self-similar solutions for the very fast p-Laplacian equation that have finite mass and, in particular, some of them that conserve mass, while this phenomenon is not true for the very fast diffusion equation.     

Gradient blowup rate for a viscous Hamilton–Jacobi equation with degenerate diffusion

This paper is concerned with the gradient blowup rate for the one-dimensional p-Laplacian parabolic equation ${u_t=(|u_x|^{p-2} u_x)_x +|u_x|^q}$ with q > p > 2, for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish the blowup rate estimates of lower and upper bounds and show that in this case the blowup rate does not match the self-similar one.     

Multiple positive solutions for nonlinear periodic problems

We consider a nonlinear periodic equation driven by the scalar p-Laplacian and with a Caratheodory asymptotically (p−1)-linear nonlinearity. Using variational methods coupled with suitable truncation techniques, we show that the problem has at least two positive solutions. For the semilinear case (p=2), using Morse theory we show that the problem has three distinct positive solutions.     

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.     

Generalized convexity and the existence of finite time blow-up solutions for an evolutionary problem

In this paper we study a class of nonlinearities for which a nonlocal parabolic equation with Neumann-Robin boundary conditions, for p-Laplacian, has finite time blow-up solutions.     

Sign-changing and constant-sign solutions for p-Laplacian problems with jumping nonlinearities

By variational methods, we prove the existence of a sign-changing solution for the p-Laplacian equation under Dirichlet boundary condition with jumping nonlinearity having relation to the Fu?ík spectrum of p-Laplacian. We also provide the multiple existence results for the p-Laplacian problems.     

The existence of multiple equilibrium points in a global attractor for some p-Laplacian equation

In this paper, we are mainly concerned with some properties of the global attractor for some p-Laplacian equation with a Lyapunov function in a Banach space. Under some suitable assumptions, we prove the existence of multiple equilibrium points in a global attractor for some p-Laplacian equation.     

Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian

We consider a nonlinear elliptic equation driven by the p-Laplacian with Dirichlet boundary conditions. Using variational techniques combined with the method of upper-lower solutions and suitable truncation arguments, we establish the existence of at least five nontrivial solutions. Two positive, two negative and a nodal (sign-changing) solution. Our framework of analysis incorporates both coercive and p-superlinear problems. Also the result on multiple constant sign solutions incorporates the case of concave-convex nonlinearities.     

On p-superlinear equations with a nonhomogeneous differential operator

We consider a nonlinear elliptic equation driven by a nonhomogeneous differential operator and with a (p?1)-superlinear Carathéodory reaction. Our formulation incorporates as a special case equations monitored by the p-Laplacian. Using variational methods coupled with truncation techniques and comparison principles, we show that the problem has at least five nontrivial smooth solutions.     

Novel existence and uniqueness criteria for periodic solutions of a Duffing type p-Laplacian equation

In this study, we investigate a Duffing type p-Laplacian equation. Some new criteria for guaranteeing the existence and uniqueness of periodic solutions of this equation are given by using the Manásevich–Mawhin continuation theorem and some analysis techniques. Our results improve and extend some known results from the literature.     

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.     

On a quasilinear Schrödinger-Poisson system

In this paper we study a quasilinear elliptic system coupled by a Schrödinger equation with p-Laplacian operator and a Poisson equation. Some scaling transformation and ingenious methods are applied to produce the bounded Palais-Smale sequences and the existence of nontrivial solutions for the system is obtained by the mountain pass theorem.     

Positive solutions for nonlinear hemivariational inequalities

In this paper we study the existence of positive solutions for nonlinear problems driven by the p-Laplacian or more generally, by multivalued p-Laplacian-like operators. Both problems have a nonsmooth locally Lipschitz potential (hemivariational inequalities). Using variational methods based on the nonsmooth critical point theory, we prove two existence results with the p-Laplacian and multivalued p-Laplacian-like operators.     

Multiple solutions for resonant nonlinear periodic equations

We consider a nonlinear periodic problem driven by the scalar p-Laplacian, with an asymptotically (p?1)-linear nonlinearity. We permit resonance with respect to the second positive eigenvalue of the negative periodic scalar p-Laplacian and we assume nonuniform nonresonance with respect to the first positive eigenvalue. Using a combination of variational methods, with truncation techniques and Morse theory, we show that the problem has at least three nontrivial solutions.     

Hadamard variational formulas for <Emphasis Type="Italic">p</Emphasis>-torsion and <Emphasis Type="Italic">p</Emphasis>-eigenvalue with applications

In this paper, we pose two kinds of Minkowski problems involving the p-Laplacian operator. The Hadamard variational formulas for some p-Laplacian functionals are obtained. A good application is to prove symmetry results for solutions to some overdetermined problems of p-Laplacian equations.     

Existence and uniqueness of positive solutions for the Neumann p-Laplacian

We consider a nonlinear Neumann problem driven by the p-Laplacian and with a Carathéodory reaction which satisfies only a unilateral growth restriction. Using the principal eigenvalue of an eigenvalue problem involving the Neumann p-Laplacian plus an indefinite potential, we produce necessary and sufficient conditions for the existence and uniqueness of positive smooth solutions.     

A multiplicity theorem for <Emphasis Type="Italic">p</Emphasis>-superlinear <Emphasis Type="Italic">p</Emphasis>-Laplacian equations using critical groups and morse theory

We study a nonlinear elliptic equation driven by the Dirichlet p-Laplacian and with a Carathéodory nonlinearity. We assume that the nonlinearity exhibits a p-superlinear growth near infinity but need not satisfy the Ambrosetti–Rabinowitz condition. Using truncation techniques, minimax methods and Morse theory, we show that the problem admits at least three nontrivial solutions, two of which have constant sign (one positive, the other negative).     

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.     

Principal frequency and existence of solutions for quasilinear elliptic obstacle problems

In this paper, we are interested in the first eigenvalue of p-Laplacian and the relation between the first eigenvalue and the existence (or nonexistence) of nontrivial (positive) solution for quasilinear elliptic obstacle problems. Utilizing the fact that obstacle problem have consanguineous relations with corresponding equation, we get a simple approach to study the properties of solutions of obstacle problems, such as existence and nonexistence, regularity and stability, etc. In this paper we are mainly concerned with the existence and nonexistence.     

Periodic solutions for some nonautonomous p(t)-Laplacian Hamiltonian systems

In this paper, we deal with the existence of periodic solutions of the p(t)-Laplacian Hamiltonian system . Some new existence theorems are obtained by using the least action principle and minimax methods in critical point theory, and our results generalize and improve some existence theorems.