| Abstract: |
| We consider the Dirichlet problem on a bounded smooth domain $\Omega\subset \mathbb{R}^4$, with $0$-boundary data, for the nonlinear critical Kirchhoff equation
\begin{eqnarray*}
-\left(a+b\int_\Omega|\nabla u|^2dx\right)\Delta u=u^3+\lambda u^{p-1}, \ \ u>0, \ \ {\rm in} \ \ \Omega
\end{eqnarray*}
where $a,b,\lambda>0$ and $p\in (1,4)$. The main feature of this problem is that the critical exponent for the embedding
$W_0^{1,2}(\Omega)\hookrightarrow L^m(\Omega)$ and the exponent of the Kirchhoff term involved in the associated energy functional $I_\lambda$
are both equals to 4. This gives rise to some difficulties in applying variational methods to find solutions. For instance,
working with $I_\lambda$, it is not clear how to obtain the boundedness of Palais-Smale sequences or the boundedness of minimizing sequences. We
present an approach based on an approximation process that allows to obtain precise constraints on the parameters $b,\lambda$ ensuring the existence
of solutions. In particular, we improve previous results where such constraints were not made explicit. |
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