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

Multiple solutions for the p(x)−p(x)-laplace operator with critical growth

The aim of this paper is to extend previous results regarding the multiplicity of solutions for quasilinear elliptic problems with critical growth to the variable exponent case. We prove, in the spirit of \cite{DPFBS}, the existence of at least three nontrivial solutions to the following quasilinear elliptic equation $-\Delta_{p(x)} u = |u|^{q(x)-2}u +\lambda f(x,u)$ in a smooth bounded domain $\Omega$ of $\R^N$ with homogeneous Dirichlet boundary conditions on $\partial\Omega$. We assume that $\{q(x)=p^*(x)\}\not=\emptyset$, where $p^*(x)=Np(x)/(N-p(x))$ is the critical Sobolev exponent for variable exponents and $\Delta_{p(x)} u = {div}(|\nabla u|^{p(x)-2}\nabla u)$ is the $p(x)-$laplacian. The proof is based on variational arguments and the extension of concentration compactness method for variable exponent spaces

Existence of three solutions for a first-order problem with nonlinear non-local boundary conditions

Conditions for the existence of at least three positive solutions to the nonlinear first-order problem with a nonlinear nonlocal boundary condition given by && y'(t) - p(t)y(t) = \sum_{i=1}^m f_i\big(t,y(t)\big), \quad t\in[0,1], && \lambda y(0) = y(1) + \sum_{j=1}^n \Phi_j(\tau_j,y(\tau_j)), \quad \tau_j\in[0,1], are discussed, for sufficiently large $\lambda>1$. The Leggett-Williams fixed point theorem is utilized.Comment: outline, 6 page

The p-Laplace equation in domains with multiple crack section via pencil operators

The p-Laplace equation \n \cdot (|\n u|^n \n u)=0 \whereA n>0, in a bounded domain \O \subset \re^2, with inhomogeneous Dirichlet conditions on the smooth boundary \p \O is considered. In addition, there is a finite collection of curves \Gamma = \Gamma_1\cup...\cup\Gamma_m \subset \O, \quad \{on which we assume homogeneous Dirichlet boundary conditions} \quad u=0, modeling a multiple crack formation, focusing at the origin 0 \in \O. This makes the above quasilinear elliptic problem overdetermined. Possible types of the behaviour of solution $u(x,y)$ at the tip 0 of such admissible multiple cracks, being a "singularity" point, are described, on the basis of blow-up scaling techniques and a "nonlinear eigenvalue problem". Typical types of admissible cracks are shown to be governed by nodal sets of a countable family of nonlinear eigenfunctions, which are obtained via branching from harmonic polynomials that occur for $n=0$. Using a combination of analytic and numerical methods, saddle-node bifurcations in $n$ are shown to occur for those nonlinear eigenvalues/eigenfunctions.Comment: arXiv admin note: substantial text overlap with arXiv:1310.065