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We study a class of $p(x)$-Kirchhoff problems which is seldom studied becausethe nonlinearity has nonstandard growth and contains a bi-nonlocal term. Basedon variational methods, especially the Mountain pass theorem and Ekeland'svariational principle, we obtain the existence of two nontrivial solutions forthe problem under certain assumptions. We also apply the Symmetric mountainpass theorem and Clarke's theorem to establish the existence of infinitely manysolutions. Our results generalize and extend several existing results.

Existence and multiplicity of solutions involving the $p(x)$-Laplacian equations: On the effect of two nonlocal terms