A paradigmatic superlinear indefinite problem

As they are of class C2+ν , the open sets Ω+, Ω− and Ω0 at most have finitely many components. Figure 9.1 shows an admissible nodal configuration of a(x) with Ω0 non-empty. The problem (9.1) is superlinear because a > 0 somewhere, though of indefinite type because simultaneously it is of sublinear type in Ω−. Indeed, for every u > 0,

λu+ a(x)|u|pu

 < λu if a(x) < 0,

= λu if a(x) = 0,

> λu if a(x) > 0.