Positive solutions of superlinear indefinite prescribed mean curvature problems

This paper analyzes the superlinear indefinite prescribed mean curvature problem -div u/1 + |u|2 = λa(x)h(u)in ω,u = 0on ℓω, where ω is a bounded domain in N with a regular boundary ℓω, h C0() satisfies h(s) - sp, as s → 0+, p > 1 being an exponent with p < N+2 N-2 if N ≥ 3, λ > 0 represents a parameter, and a C0(ω¯) is a sign-changing function. The main result establishes the existence of positive regular solutions when λ is sufficiently large, providing as well some information on the structure of the solution set. The existence of positive bounded variation solutions for λ small is further discussed assuming that h satisfies h(s) - sq as s → +∞, q > 0 being such that q < 1 N-1 if N ≥ 2; thus, in dimension N ≥ 2, the function h is not superlinear at + ∞, although its potential H(s) =0sh(t)dt is. Imposing such different degrees of homogeneity of h at 0 and at + ∞ is dictated by the specific features of the mean curvature operator.