Existence and multiplicity of solutions for a Kirchhoff-type problem with indefinite nonlinearity in $$\mathbb {R}^3$$
Year: 2026
Type: article
Abstract: In this work, we study a class of Kirchhoff-type equations with indefinite nonlinearities of the form $$-\left( 1+\lambda \int _{\mathbb {R}^3} |\nabla u|^2\right) \Delta u = f(x)|u|^{\gamma -2}u$$ - 1 + λ ∫ R 3 | ∇ u | 2 Δ u = f ( x ) | u | γ - 2 u in $$\mathbb {R}^3$$ R 3 , where $$\lambda >0$$ 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 $$\lambda $$ λ . We show that there exists an extremal parameter $$\lambda ^*$$ 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.
Author: Steffânio Moreno de Sousa
Institution: Universidade Federal do Amazonas
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,390
Funder: Universidade Federal do Amazonas
OpenAlex ID: W7161940700