Analysis of two-parameter capillarity Kirchhoff problem with $$\varrho (\cdot )$$-Laplacian operator by topology degree

Abstract We investigate a differential equation governed by the $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> -Hilfer derivative and the $$\varrho (\cdot )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϱ</mml:mi> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -Laplacian operator with Dirichlet boundary conditions involving two parameters. The nonlinearity of the problem, in general, does not satisfy the Ambrosetti–Rabinowitz type condition. By employing the topological degree for pseudomonotone operators within a variational approach, we establish the existence of positive solutions for a nonlinear Kirchhoff problem in suitable fractional $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> -Hilfer spaces.