Existence and multiplicity of solutions for a Kirchhoff-type problem with indefinite nonlinearity in R³

In this work, we study a class of Kirchhoff-type equations with indefinite nonlinearities of the form -( 1+λ ∫ R³ |∇ u|²) Δ u = f(x)|u|γ -2u - 1 + λ ∫ R 3 | ∇ u | 2 Δ u = f ( x ) | u | γ - 2 u in R³ R 3 , where λ >0 λ > 0 is a parameter. Using variational methods and a simplified version of the concentration-compactness principle, we overcome the lack of compactness inherent to the unbounded domain. Under suitable conditions on f , we provide a precise description of the bifurcation phenomena occurring with respect to the parameter λ λ . We show that there exists an extremal parameter λ ^* λ ∗ which acts as a turning point where two branches of solutions collapse and vanish. Our results generalize previous studies by removing the boundedness condition on the domain and considering sign-changing nonlinearities.