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Approach : Find the energy functional :
$int_Omega nabla u nabla v int_Omega |u| u.v -int_Omega fu=0 forall u in H_0^1(Omega) cap L^3(Omega)$

$implies E(u)=int_Omegafrac{1}{2} |nabla u|^2+frac{1}{3} |u|^3 -fu dx$

If $u$ solves $min_{u in A} E(u)$ with $A={}uin H_0^1( Omega)cap L^3(Omega)$ then for any $vin A$ we have

$0=frac{d}{depsilon}E(u+epsilon v)=intfrac {d}{depsilon}|nabla(u+epsilon v)|^2 +frac{1}{3} |u+epsilon v|^3 -f(u+epsilon v)dx$
$$=int(nabla u. nabla v +|u|v -fv) dx$$

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 $exists (u_k)_{kin mathbb N} subset A$

$$lim_{ktoinfty}u_k=inf_{uin A} E(u)$$

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.

Concerning the differentiation of $epsilonmapsto |u+epsilon v|^3$. The function $g(x)=|x|^3$ is in $C^2(mathbb R)$ with $g,'(x)=3x|x|$ and $g,''(x)=6|x|$. Notably, $g$ is convex, which makes the entire functional $E$ convex; we need this for its lower semicontinuity with respect to the weak convergence in $A$.

But to begin with, we want to show that $E$ is bounded from below on $A$; otherwise the entire approach is doomed. The only question is what to do with $int-fu$: suppressing the urge to use Holder's inequality, I would write $-fuge -frac12f^2-frac12 u^2$. Here the integral of $f^2$ is a finite constant and $u^2le frac12|u|^3+8$ pointwise. It follows that $E$ is bounded from below: $E(u)ge int (frac12|nabla u|^2+frac{1}{12}|u|^3-frac12f^2-8)$ or something of the kind. Note that I bounded $u^2$ by a small multiple of $|u|^3$ in order to have some of the cubic term left over. This will be useful in a moment.

Pick a minimizing sequence $u_kin A$ as you've done already. The lower bound for $E(u)$ in the previous paragraph tells us that $int|nabla u_k|^2$ and $int|u_k|^3$ are uniformly bounded. Thus, the sequences is bounded in $H_0^1$ and in $L^3$. Pick a weakly convergent subsequence in $H_0^1$; extract from it a weakly convergent subsequence in $L^3$. The lower semicontinuity of $E$ implies $E(u)=min_A E$ where $u$ is the weak limit.

And yes, having $int (nabla ucdotnabla v +frac13 u|u|v-fv)=0$ for all test functions $v$ (you only need to consider $C^infty$-smooth $v$ with compact support) means that $u$ is a weak solution; this is what weak solution means. The boundary condition $u=0$ is automatically fulfilled since we work in $H_0^1$ all the time.