On a \(p(x)\)-Kirchhoff equation with critical exponent and an additional nonlocal term (Q2816481)

Consider the following \(p(x)\)-Kirchhoff equation with critical exponent NEWLINE\[NEWLINE\begin{cases} -M\left(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}\right)\Delta_{p(x)} u=\lambda f(x,u)\left[\int_{\Omega}F(x,u)\right]^{r}+|u|^{q(x)-2}u, &\text{ in }\Omega, \\ u=0, &\text{ on }\partial\Omega,\end{cases}\tag{1.1}NEWLINE\]NEWLINE where \(\Omega\subset \mathbb{R}^{N}\) is a bounded domain, \(f:\overline{\Omega}\times \mathbb{R}\rightarrow \mathbb{R},M:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}\) are continuous functions with some appropriate conditions, and \(F(x,u)=\int_{0}^{u}f(x,t)dt,1<q(x)\leq\frac{Np(x)}{N-p(x)},1<p(x)<N,\lambda,r>0.\) \(\left[\int_{\Omega}F(x,u)\right]^{r}\) is a nonlocal term with \(F(x,s)\geq0\) for all \(s\in\mathbb{R}.\)NEWLINENEWLINEAssume \(M\) satisfies one of the following conditionsNEWLINENEWLINE(1)~\(M(t)=a+bt\) with \(a,b>0\),NEWLINENEWLINE(2)~\(m_{0}\leq M(t)\leq m_{1}\) where \(m_{0},m_{1}>0\) are constant,NEWLINENEWLINE(3)~\(M(t)=t^{\alpha}\) with \(\alpha>0\), when \(\lambda>0\) large enough, NEWLINENEWLINEthe existence of nontrivial solutions for problem (1.1) is obtained in \(W_{0}^{1,p(x)}(\Omega)\) by the concentration-compactness principle.