Existence and Multiplicity of Solutions for a Class of Kirchhoff–Boussinesq-Type Problems with Logarithmic Growth

In this paper, two problems related to the following class of elliptic Kirchhoff–Boussinesq-type models are analyzed in the subcritical (\(\beta =0\)) and critical (\(\beta =1\)) cases:

$$\begin{aligned} \Delta ^{2} u \!- \!\Delta _p u \!=\! \tau |u|^{q-2} u{\ln |u|}\!+\!\beta |u|^{2_{**}-2}u\ \text{ in } \ \Omega \ \ \text{ and } \ {\Delta u=u=0} \ \text{ on } \ \ \partial \Omega , \end{aligned}$$

where \(\tau >0\), \(2< p< 2^{*}= \frac{2N}{N-2}\) for \( N\ge 3\) and \(2_{**}= \infty \) for \(N=3\), \(N=4\), \(2_{**}= \frac{2N}{N-4}\) for \(N\ge 5\). The first one is concerned with the existence of a nontrivial ground-state solution via variational methods. As for the second problem, we prove the multiplicity of such a solution using the Mountain Pass Theorem.