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EXISTENCE OF TWO POSITIVE SOLUTIONS FOR INDEFINITE KIRCHHOFF EQUATIONS IN R

In this article we study the Kirchhoff type equation − “ 1 + b Z R3 |∇u|dx ” ∆u+ u = k(x)f(u) + λh(x)u, x ∈ R, u ∈ H(R), involving a linear part −∆u + u − λh(x)u which is coercive if 0 < λ < λ1(h) and is noncoercive if λ > λ1(h), a nonlocal nonlinear term −b R R3 |∇u| 2dx∆u and a sign-changing nonlinearity of the form k(x)f(s), where b > 0, λ > 0 is a real parameter and λ1(h) is the first eigenvalue of −∆u+ u = λh(x)u. Under suitable assumptions on f and h, we obtain positives solution for λ ∈ (0, λ1(h)) and two positive solutions with a condition on k.