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Let Ω ⊂ R N , N 2, be a bounded domain. We consider the following quasilinear problem depending on a real parameter λ > 0:where f (t) is a nonlinearity that grows like e t N/N−1 as t → ∞ and behaves like t α , for some α ∈ (0, N − 1), as t → 0 + . More precisely, we require f to satisfy assumptions (A1)-(A5) in Section 1. With these assumptions we show the existence of Λ > 0 such that (P λ ) admits at least two solutions for all λ ∈ (0, Λ), one solution for λ = Λ and no solution for all λ > Λ. We also study the problem (P λ ) posed on the ball B 1 (0) ⊂ R N and show that the assumptions (A1)-(A5) are sharp for obtaining global multiplicity. We use a combination of monotonicity and variational methods to show multiplicity on general domains and asymptotic analysis of ODEs for the case of the ball.

The oligosaccharide content was determined in 12 different cultivars of black gram. The effect of various treatments such as soaking, cooking, and enzyme treatment on the raffinose family oligosaccharides of dry seeds and flour was studied. Ajugose, a higher oligosaccharide (DP 6) found in trace quantities in seeds, was shown in black gram by HPLC. The percent reduction of raffinose, stachyose, verbascose, and ajugose after soaking for 16 hr was 41.66%, 47.61%, 28.48%, and 26.82%, respectively in Local-I variety and 43.75%, 20.58%, 23.60%, and 15.88%, respectively in Local-II variety. Cooking for 60 min resulted in decrease of 100% for raffinose, 76.19% for stachyose, 36.39% for verbascose, and 60.97% for ajugose in Local-I variety and 100% for raffinose, 55.88% for stachyose, 48.52% for verbascose, and 56.07% for ajugose in Local-II variety. Thin layer chromatographic analysis of 3 hr enzyme-treated samples revealed almost complete hydrolysis of raffinose family of oligosaccharides. Among the different methods employed, enzyme treatment was found to be the most effective for removing alpha-galactosides in black gram.

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