Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional p-Laplacian

AbstractWe consider the boundary value problems: (ϕp(x′(t)))′+q(t)f(t,x(t),x(t−1),x′(t))=0, ϕp(s)=|s|p−2s, p>1, t∈(0,1), subject to some boundary conditions. By using a generalization of the Leggett–Williams fixed-point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of at least three positive solutions to the above problems

Multiple Positive solutions of a (p1,p2)(p_1,p_2)-Laplacian system with nonlinear BCs

Using the theory of fixed point index, we discuss existence, non-existence, localization and multiplicity of positive solutions for a $(p_1,p_2)$-Laplacian system with nonlinear Robin and/or Dirichlet type boundary conditions. We give an example to illustrate our theory.Comment: arXiv admin note: text overlap with arXiv:1408.017

Existence results to a nonlinear p(k)-Laplacian difference equation

In the present paper, by using variational method, the existence of non-trivial solutions to an anisotropic discrete non-linear problem involving p(k)-Laplacian operator with Dirichlet boundary condition is investigated. The main technical tools applied here are the two local minimum theorems for differentiable functionals given by Bonanno.Comment: The final version of this paper will be published in Journal of Difference Equations and Applications in 201

Existence and multiplicity results for resonant fractional boundary value problems

We study a Dirichlet-type boundary value problem for a pseudo-differential equation driven by the fractional Laplacian, with a non-linear reaction term which is resonant at infinity between two non-principal eigenvalues: for such equation we prove existence of a non-trivial solution. Under further assumptions on the behavior of the reaction at zero, we detect at least three non-trivial solutions (one positive, one negative, and one of undetermined sign). All results are based on the properties of weighted fractional eigenvalues, and on Morse theory

A note on the Dancer-Fucik spectra of the fractional p-Laplacian and Laplacian operators

We study the Dancer-Fucik spectrum of the fractional p-Laplacian operator. We construct an unbounded sequence of decreasing curves in the spectrum using a suitable minimax scheme. For p=2, we present a very accurate local analysis. We construct the minimal and maximal curves of the spectrum locally near the points where it intersects the main diagonal of the plane. We give a sufficient condition for the region between them to be nonempty, and show that it is free of the spectrum in the case of a simple eigenvalue. Finally we compute the critical groups in various regions separated by these curves. We compute them precisely in certain regions, and prove a shifting theorem that gives a finite-dimensional reduction in certain other regions. This allows us to obtain nontrivial solutions of perturbed problems with nonlinearities crossing a curve of the spectrum via a comparison of the critical groups at zero and infinity.Comment: 13 pages, typos correcte