Existence and multiplicity of solutions for a Kirchhoff-type problem with indefinite nonlinearity in $$\mathbb {R}^3$$

Abstract In this work, we study a class of Kirchhoff-type equations with indefinite nonlinearities of the form $$-\left( 1+\lambda \int _{\mathbb {R}^3} |\nabla u|^2\right) \Delta u = f(x)|u|^{\gamma -2}u$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mfenced> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:msub> <mml:mo>∫</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:msub> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>∇</mml:mi> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mfenced> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> </mml:mrow> </mml:math> in $$\mathbb {R}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> , where $$\lambda &gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> 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 $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> . We show that there exists an extremal parameter $$\lambda ^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>λ</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> 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.