Abstract
In this article, we investigate the effect of the coefficient f(z) of the sub-critical nonlinearity. For sufficiently large λ > 0, there are at least k + 1 positive solutions of the semilinear elliptic equations where 1 ≤ q < 2 < p < 2* = 2N/(N - 2) for N ≥ 3. AMS (MOS) subject classification: 35J20; 35J25; 35J65.
Highlights
For the case q = l = 1 and f(z) ≡ 1 for all z Î RN, suppose that h is nonnegative, small, and exponential decay, Zhu [6] showed that Equation (El) admits at least two positive solutions in RN
Without the condition of exponential decay, Cao and Zhou [7] and Hirano [8] proved that Equation (El) admits at least two positive solutions in RN
For the case q = l = 1, by using the idea of category and Bahri-Li’s minimax argument, Adachi and Tanaka [9] asserted that Equation (El) admits at least four positive solutions in RN, where f(z) ≢ 1, f(z) ≥ 1 - C exp((-(2 + δ) |z|) for some C, δ > 0, and sufficiently small h H−1 > 0
Summary
We consider the existence and multiplicity of positive solutions of Equation (El) in RN. For the case q = l = 1 and f(z) ≡ 1 for all z Î RN, suppose that h is nonnegative, small, and exponential decay, Zhu [6] showed that Equation (El) admits at least two positive solutions in RN. In Hsu and Lin [10], they have studied that there are at least four positive solutions of the general case -Δu + u = f(z)vp-1 + lh(z) vq-1 in RN for sufficiently small l > 0. (II) Under assumptions (f1) - (f2) and (h1), if l is sufficiently large, Equation (El) admits at least k + 1 positive solutions. (See Theorem 4.8)
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