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Using the variational method, we prove the existence and multiplicity of solutions for a Steklov problem involving the $p(x)$-Laplacian-like operator, originated from a capillary phenomena. Especially, an existence criterion for infinite many pairs of solutions for the problem is obtained.

We study the existence and non-existence of positive solutions for $(p,q)$-Laplacian Steklov problem with two parameters. The main result of our research is the construction of a continuous curve in plane, which becomes a threshold between the existence and non-existence of positive solutions.

In the presentp aper, we study the existence and non-existence results of a positive solution for the Steklov eigenvalue problem driven by nonhomogeneous operator $(p,q)$-Laplacian with indefinite weights. We also prove that in the case where $\mu>0$ and with $10$ there exists an interval of principal eigenvalues for our Steklov eigenvalue problem.