Abstract multiplicity results for (p,q)(p,q)-Laplace equations with two parameters

We investigate the existence and multiplicity of abstract weak solutions of the equation $-\Delta_p u -\Delta_q u=\alpha |u|^{p-2}u + \beta |u|^{q-2}u$ in a bounded domain under zero Dirichlet boundary conditions, assuming $1<q<p$ and $\alpha,\beta \in \mathbb{R}$. We determine three generally different ranges of parameters $\alpha$ and $\beta$ for which the problem possesses a given number of distinct pairs of solutions with a prescribed sign of energy. As auxiliary results, which are also of independent interest, we provide alternative characterizations of variational eigenvalues of the $q$-Laplacian using narrower and larger constraint sets than in the standard minimax definition.Comment: 29 pages, 3 figure

Existence Results for Quasilinear Elliptic Equations with Indefinite Weight

We provide the existence of a solution for quasilinear elliptic equation −div(∞()|∇|−2∇+̃(,|∇|)∇)=()||−2+(,)+ℎ() in Ω under the Neumann boundary condition. Here, we consider the condition that ̃(,)=(−2) as →+∞ and (,)=(||−1) as ||→∞. As a special case, our result implies that the following -Laplace equation has at least one solution: −Δ=()||−2+||−2+ℎ() in Ω,/=0 on Ω for every 1<<<∞, ∈ℝ, ≠0 and ,ℎ∈∞(Ω) with ∫Ω≠0. Moreover, in the nonresonant case, that is, is not an eigenvalue of the -Laplacian with weight , we present the existence of a solution of the above -Laplace equation for every 1<<<∞, ∈ℝ and ,ℎ∈∞(Ω)

The Orthography of Saikaku gohyaku in The assimilation of Chinese words into Japanese usage and the role of ateji

Until the Muromachi era, the relationship between kanji (Chinese characters) and furigana (small‑print marginal annotations to kanji using one of the Japanese kana syllabaries) was chiefly that the latter were used as guides to the pronunciation or meaning of unusual or difficult characters. But in the early modern period, furigana came to be used more creatively as a means for expressing double meanings or overlapping significations, and in haikai from the beginning of the early modern period, furigana take on a variety of functions. But there has been little progress in research into haikai from this period. This paper aims at furthering such investigation by examining Saikaku gohyaku in. The first section demonstrates how the kanji compound 日外, which in classical Chinese meant "a place the power of the sovereign does not reach" came to be read and used in Japanese as itsuzoya. The second section looks at how the now obsolete compound 性躰 (shotai) was used in the same contexts that the present 正体 (shotai) would be employed, and seen as a legitimate usage of Chinese vocabulary. The third section considers the use of 上夫 as ateji or phonetic substitute characters for 丈夫 in order to express a double entendre. The paper thus reports on the relationship between kanji and their phonetic readings in the work Saikaku gohyaku in