Approach : Find the energy functional : $\int_\Omega \nabla u \nabla v \int_\Omega |u| u.v -\int_\Omega fu=0 \forall u \in H_0^1(\Omega) \cap L^3(\Omega)$
$\implies E(u)=\int_\Omega\frac{1}{2} |\nabla u|^2+\frac{1}{3} |u|^3 -fu dx$
If $u$ solves $\min_{u \in A} E(u)$ with $A={}u\in H_0^1( \Omega)\cap L^3(\Omega)$ then for any $v\in A$ we have
$0=\frac{d}{d\epsilon}E(u+\epsilon v)=\int\frac {d}{d\epsilon}|\nabla(u+\epsilon v)|^2 +\frac{1}{3} |u+\epsilon v|^3 -f(u+\epsilon v)dx$ $=\int(\nabla u. \nabla v +|u|v -fv) dx$
Remark: There seems to be a problem while differentiating with respect to $\epsilon$ the term $|u+\epsilon v|^3$ , how do I resolve it ?
Next : Can I say that $u$ is a weak solution now ? If I could then would proceed further this way If $A$ is not a null set , then $\exists (u_k)_{k\in \mathbb N} \subset A$
$\lim_{k\to\infty}u_k=\inf_{u\in A} E(u)$
Assume $u_k$ is not bounded in $H_0^1 $ $E(u_k)\ge \int_\Omega \frac {|\nabla u_k|^2}{2}-fu_k dx \ge \frac{1}{2} ||\nabla u||_{L^2}-C ||f||_{L^2} ||\nabla u_k||_{L^2(\Omega)}$
I am stuck now , How do I proceed further ? Thanks.