Nehari manifold and existence of positive solutions to a class of ...

Nehari manifold and existence of positive solutions 15where C is a positive constant and lim ε→0 f(ε) = +∞. Let δ 2 > 0 be suchthat12N S N p − Kλ β > 0, ∀ λ ∈ (0, δ 2 ).Using the definition of I λ , we getI λ (tv ε ) ≤ tp p ‖v ε‖ p p, ∀ t ≥ 0,which implies that there exists t 0 ∈ (0, 1) satisfyingsup I λ (tv ε ) < 10≤t≤t 02N S N p − Kλ β , ∀ λ ∈ (0, δ 2 ).Analyzing the case N ≥ p 2 , we haveI λ (tv ε ) ≤ 12N S N pp−1− Cεp+ o(ε p−1p ) + O(ε N−pp)− λtqq∫v q ε, ∀ t > 0.Therefore,sup I λ (tv ε ) ≤ 1t≥t 02N S N pp−1− Cεp∫+ o(ε p−1N−pp ) + O(εp) − λtq 0qv q ε.Hence,sup I λ (tv ε ) < 1t≥t 02N S N pp−1− Cεp+ o(ε p−1p ) + O(ε N−pp)− Kλ β , ∀ λ ∈ (0, δ 3 ),whereWe fix ε > 0 such that−Cε p−1pthis is possible since N−ppobtain( tqδ 3 =− p−1p0∫vqε2Kq) 1β−1.+ o(ε p−1p ) + O(≥ (p−1)2pε N−pp)< 0,> 0. If we set λ ∗ 2 = min{δ 2 , δ 3 }, wesup I λ (tv ε ) < 1t≥0 2N S N p − Kλ β , ∀ λ ∈ (0, λ ∗ 2),