Dirichlet Boundary Value Problems for Second Order pp-Laplacian Difference Equations

In this paper, the solutions to second order Dirichlet boundary value problems of $p$-Laplacian difference equations are investigated. By using critical point theory, existence and multiplicity results are obtained. The proof is based on the Mountain Pass Lemma in combination with variational techniques

Mechanistic examination of causes for narrow distribution in an endangered shrub: a comparison of its responses to drought stress with a widespread congeneric species

Although deep rooting is usually considered a drought-tolerant trait, we found that Syringapinnatifolia, a deep rooting and hydrotropic shrub, has a limited distribution in arid areas. To elucidate the mechanisms for its narrow distribution, we conducted two experiments to examine the physiological and morphological responses to water availability and heterogeneity in S. pinnatifolia and a widespread congeneric species, S. oblata. We measured gas exchange, water use efficiency, and plasticity index in plants of these two species grown at different levels of soil water regimes and in containers with patched water distribution. Our results showed that high photosynthetic capacity in the narrowly distributed S. pinnatifolia was an important factor enabling its survival in the harsh sub-alpine environment. High photosynthetic capacity in S. pinnatifolia, however, was obtained at the expense of high transpiratory water loss, resulting in lower integrative water use efficiency. Biomass allocation to roots in S. pinnatifolia increased by 73 % when soil water increased from 75 to 95 % field capacity, suggesting that S. pinnatifolia could be less competitive for above-ground resources under favorable water regimes. The horizontal root hydrotropism and vertical root hydrotropism of S. pinnatifolia in soil with patched water patterns were likely related to compensation for leaf water loss at low soil water level, indicating a limited capacity for homeostasis within the plant for water conservation and lower level of inherent drought-tolerance. In summary, greater degree of morphological plasticity but lower degree of physiological adjustment may be the main causes for the hydrotropism and narrow distribution of S. pinnatifolia in the sub-alpine habitats