10,474 research outputs found

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

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    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 −Δp(x)u=∣u∣q(x)−2u+λf(x,u)-\Delta_{p(x)} u = |u|^{q(x)-2}u +\lambda f(x,u) in a smooth bounded domain Ω\Omega of RN\R^N with homogeneous Dirichlet boundary conditions on ∂Ω\partial\Omega. We assume that {q(x)=p∗(x)}≠∅\{q(x)=p^*(x)\}\not=\emptyset, where p∗(x)=Np(x)/(N−p(x))p^*(x)=Np(x)/(N-p(x)) is the critical Sobolev exponent for variable exponents and Δp(x)u=div(∣∇u∣p(x)−2∇u)\Delta_{p(x)} u = {div}(|\nabla u|^{p(x)-2}\nabla u) is the p(x)−p(x)-laplacian. The proof is based on variational arguments and the extension of concentration compactness method for variable exponent spaces

    Hamiltonian elliptic systems: a guide to variational frameworks

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    Consider a Hamiltonian system of type −Δu=Hv(u,v), −Δv=Hu(u,v)   in Ω,u,v=0 on ∂Ω -\Delta u=H_{v}(u,v),\ -\Delta v=H_{u}(u,v) \ \ \text{ in } \Omega, \qquad u,v=0 \text{ on } \partial \Omega where HH is a power-type nonlinearity, for instance H(u,v)=∣u∣p/p+∣v∣q/qH(u,v)= |u|^p/p+|v|^q/q, having subcritical growth, and Ω\Omega is a bounded domain of RN\mathbb{R}^N, N≥1N\geq 1. The aim of this paper is to give an overview of the several variational frameworks that can be used to treat such a system. Within each approach, we address existence of solutions, and in particular of ground state solutions. Some of the available frameworks are more adequate to derive certain qualitative properties; we illustrate this in the second half of this survey, where we also review some of the most recent literature dealing mainly with symmetry, concentration, and multiplicity results. This paper contains some original results as well as new proofs and approaches to known facts.Comment: 78 pages, 7 figures. This corresponds to the second version of this paper. With respect to the original version, this one contains additional references, and some misprints were correcte

    Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities

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    We study the stability and exact multiplicity of periodic solutions of the Duffing equation with cubic nonlinearities. We obtain sharp bounds for h such that the equation has exactly three ordered T-periodic solutions. Moreover, when h is within these bounds, one of the three solutions is negative, while the other two are positive. The middle solution is asymptotically stable, and the remaining two are unstable.Comment: Keywords: Duffing equation; Periodic solution; Stabilit
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