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Search: All articles in the CMB digital archive with keyword critical exponent

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1. CMB Online first

He, Yubo; Qin, Dongdong; Tang, Xianhua
 Ground state and multiple solutions for Kirchhoff type equations with critical exponent In this paper, we consider the following critical Kirchhoff type equation: \begin{align*} \left\{ \begin{array}{lll} - \left(a+b\int_{\Omega}|\nabla u|^2 \right)\Delta u=Q(x)|u|^4u + \lambda |u|^{q-1}u,~~\mbox{in}~~\Omega, \\ u=0,\quad \text{on}\quad \partial \Omega, \end{array} \right. \end{align*} By using variational methods that are constrained to the Nehari manifold, we prove that the above equation has a ground state solution for the case when $3\lt q\lt 5$. The relation between the number of maxima of $Q$ and the number of positive solutions for the problem is also investigated. Keywords:Kirchhoff type equation, variational methods, critical exponent, Nehari manifold, ground stateCategories:35J20, 35J60, 35J25

2. CMB 2012 (vol 57 pp. 113)

 A Lower Bound for the End-to-End Distance of Self-Avoiding Walk For an $N$-step self-avoiding walk on the hypercubic lattice ${\bf Z}^d$, we prove that the mean-square end-to-end distance is at least $N^{4/(3d)}$ times a constant. This implies that the associated critical exponent $\nu$ is at least $2/(3d)$, assuming that $\nu$ exists. Keywords:self-avoiding walk, critical exponentCategories:82B41, 60D05, 60K35