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

We consider one-dimensional (p, q)-Laplace problems: {-(phi(u'))' = lambda h(t) f(u), t is an element of (0, 1). u(0) = 0 = au'(1) + g(lambda, u(1))u(1), where lambda > 0, a >= 0, phi(s) := vertical bar s vertical bar(p-2) s+ vertical bar s vertical bar(q-2)s, 1 < p < q < infinity, h is an element of C((0, 1), (0, infinity)), f is an element of C((0, infinity), R) with lim(s -> 0+) f (s) is an element of (-infinity,0) U {-infinity}, and g is an element of C((O, infinity) x [0, infinity), (0, infinity)) such that g(r, s)s is nondecreasing with respect to s is an element of [0, infinity). 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 ->infinity )f(s)/s(r-1) is an element of (0, infinity), 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. (C) 2020 Elsevier Inc. All rights reserved.