Positive solutions to classes of infinite semipositone (p,q)-Laplace problems with nonlinear boundary conditions

We consider one-dimensional ( p , q ) -Laplace problems: { − ( φ ( u ′ ) ) ′ = λ h ( t ) f ( u ) , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 = a u ′ ( 1 ) + g ( λ , u ( 1 ) ) u ( 1 ) , where λ > 0 , a ≥ 0 , φ ( s ) : = | s | p − 2 s + | s | q − 2 s , 1 p q ∞ , h ∈ C ( ( 0 , 1 ) , ( 0 , ∞ ) ) , f ∈ C ( ( 0 , ∞ ) , R ) with lim s → 0 + ⁡ f ( s ) ∈ ( − ∞ , 0 ) ∪ { − ∞ } , and g ∈ C ( ( 0 , ∞ ) × [ 0 , ∞ ) , ( 0 , ∞ ) ) such that g ( r , s ) s is nondecreasing with respect to s ∈ [ 0 , ∞ ) . Classifying the behaviors of f near infinity, we establish the existence, multiplicity and nonexistence of positive solutions. In particular, we provide a sufficient condition on f to obtain a multiplicity result for the case when lim s → ∞ ⁡ f ( s ) s r − 1 ∈ ( 0 , ∞ ) , 1 r q , which is new even in semilinear problems ( p = q = 2 ). The proofs are based on a Krasnoselskii type fixed point theorem which is fit to overcome a lack of homogeneity.