Multiplicity results for a Neumann problem involving the p-Laplacian

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

     

The critical Neumann problem for semilinear elliptic equations with concave perturbations

We consider the semilinear Neumann problem involving the critical Sobolev exponent with an indefinite weight function and a concave purturbation. We prove the existence of two distinct solutions.      

The Neumann problem for some degenerate elliptic equations

In the paper we study the equation L _u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.     

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.     

On the nonlinear Neumann problem involving the critical Sobolev exponent on the boundary

In this paper we investigate the solvability of the Neumann problem (1.1) involving the critical Sobolev exponents on the right-hand side of the equation and in the boundary condition. It is assumed that the coefficients Q and P are smooth. We examine the common effect of the mean curvature of the boundary ∂Ω and the shape of the graph of the coefficients Q and P on the existence of solutions of problem (1.1).     

On Coron's problem for the p-Laplacian

We prove that the critical problem for the p-Laplacian operator admits a nontrivial solution in annular shaped domains with sufficiently small inner hole. This extends Coron's result [4] to a class of quasilinear problems.     

On an eigenvalue problem involving the one-dimensional p-Laplacian

We apply general results on operator equations in ordered spaces and properties of the principal eigenvalues for weighted semi-linear equations to prove the existence of a global continua of positive solutions and eigenvalue intervals to the problem (?(x′))′+λf(t,x,x′)=0 in (0,1), x(0)=x(1)=0, where ?(x)=|x|^p−2x, p>1, λ>0.     

On a superlinear elliptic p-Laplacian equation

We prove the existence of four solutions for the p-Laplacian equation

     

Existence and multiplicity for solutions of Neumann problem for semilinear elliptic equations

The existence and multiplicity results are obtained for solutions of Neumann problem for semilinear elliptic equations by the least action principle and the minimax methods respectively.     

Multiple solutions for a system of equations with p-Laplacian

In the paper, by using of the Limit Index, we prove a theorem applying to get multiple critical values of some strongly indefinite nonsmooth functionals, and then we apply it to a system of equations involving the p-Laplacian.     

On the Cauchy problem for the evolution p-Laplacian equations with gradient term and source

In this paper, the authors study the equation ut=div(|Du|^p−2Du)+|u|^q−1u−λl|Du| in RN with p>2. We first prove that for 1?l?p−1, the solution exists at least for a short time; then for , the existence and nonexistence of global (in time) solutions are studied in various situations.     

A multiplicity theorem for the Neumann problem

Here is a particular case of the main result of this paper: Let be a bounded domain, with a boundary of class , and let be two continuous functions, , with 0$">, , with n$">. If

and if the set of all global minima of the function has at least connected components, then, for each 0$"> small enough, the Neumann problem

admits at least strong solutions in .

     

On a critical Neumann problem with a perturbation of lower order

We investigate the solvability of the Neumann problem （1.1） involving the critical Sobolev nonlinearity and a term of lower order. We allow a coefficient of u in equation （1.1） to be unbounded. We prove the existence of a solution in a weighted Sobolev space.     

Exact structure of positive solutions for some p-Laplacian equations

This paper establishes the exact multiplicities and properties of positive solutions for some second order differential equations involving p-Laplacian operator.     

Multiple solutions for semilinear elliptic equations with Neumann boundary condition and jumping nonlinearities

We obtain nonconstant solutions of semilinear elliptic Neumann boundary value problems with jumping nonlinearities when the asymptotic limits of the nonlinearity fall in the type (Il), l>2 and (IIl), l?1 regions formed by the curves of the Fucik spectrum. Furthermore, we have at least two nonconstant solutions in every order interval under resonance case. In this paper, we apply the sub-sup solution method, Fucik spectrum, mountain pass theorem in order intervals, degree theory and Morse theory to get the conclusions.     

Remarks on a multiplicity theorem for a perturbed Neumann problem

The aim of this article is to prove that, given two potential functionals Ψ₁, Ψ₂ on W1,2(Ω) which coincide on a set of the type , then under suitable summability conditions, certain local minima of Ψ₁ are local minima for Ψ₂ as well. An application of this result allows us to obtain a multiplicity theorem for a Neumann problem where we impose a less restrictive oscillating behavior on the nonlinearity than the one required in an analogous result recently established by B. Ricceri.     

Resonant equations with the Neumann p-Laplacian plus an indefinite potential

We consider a nonlinear Neumann problem driven by the p  -Laplacian plus an indefinite potential and a Carathéodory reaction which at ±∞ is resonant with respect to any nonprincipal variational eigenvalue of the differential operator. Using critical point theory and Morse theory (critical groups), we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign. In the process we prove some results of independent interest concerning the unique continuation property of eigenfunctions and the critical groups at infinity of a C¹$C^1$-functionals.