Asymptotics of solutions for fully nonlinear elliptic problems at nearly critical growth (Q1607608)

The present paper is devoted to study the general class of Euler-Lagrange equations with nearly critical growth \[ \begin{cases} -\text{div} \bigl(\nabla_\xi L(x,u,\nabla u)\bigr)+ D_sL(x,u,\nabla u)=|u|^{p^*-2-\varepsilon} u\quad &\text{in }\Omega\\ u=0\quad &\text{on }\partial \Omega, \end{cases} \tag{1} \] associated with the functional \(f_\varepsilon: W_0^{ 1,p} (\Omega)\to \mathbb{R}\) given by \[ f_\varepsilon (u)= \int_\Omega L(x,u, \nabla u)dx- {1\over p^*-\varepsilon} \int_\Omega|u |^{p^*- \varepsilon}dx. \] Let \(u_\varepsilon\) be a solution of (1). The main goal of this paper is to prove that if the weak limit of \((|\nabla u_\varepsilon |^p)_{\varepsilon >0}\) has no blow-up points in \(\Omega\) then the limit problem \[ \begin{cases} -\text{div} \bigl(\nabla_\xi L(x,u,\nabla u) \bigr) +D_sL(x,u, \nabla u)=|u|^{p^*-2}u\quad &\text{in }\Omega\\ u=0\quad &\text{on }\partial \Omega\end{cases} \tag{2} \] has a nontrivial solution, provided that \(f_\varepsilon\) satisfies some natural conditions.