Let $\delta,k>0$ and define a function $g_k:\mathbb{R}\to\mathbb{R}$ by $g_k(t)=\min\{t^{-\delta},k\}$ if $t\geq 0$ and $g_k(t)=0$ if $t<0$. Let $G_k$ be the primitive of $g_k$. Then for a bounded smooth domian $\Omega\subset\mathbb{R}^N$ and $1<p<\infty$, we define the energy functional $I:W_0^{1,p}(\Omega)\to[-\infty,+\infty]$ by $$ I(u):=\frac{1}{p}\int_{\Omega}|\nabla u|^p\,dx-\int_{\Omega}f(x)G_k(u)\,dx, $$ where $f\in L^1(\Omega)$ is a nonnegative function. Let $v\in W_{\text{loc}}^{1,p}(\Omega)$ be positive in $\Omega$ and consider the closed and convex subset $K$ of $W_0^{1,p}(\Omega)$ by $$ K:=\{\phi\in W_0^{1,p}(\Omega):0\leq\phi\leq v\text{ in }\Omega\}. $$ Then $I$ has a minimum in $K$. I tried in the following way: $I$ is weakly lower semicontinuous, coercive over $K$. Thus it attains a minimum, say $z\in K$.
To prove the weak lower semicontinuity of $I$ over $K$, we use the result that if $I$ is convex and lower semicontinuous in the strong topology, then $I$ is weakly lower semicontinuous. Then we prove the coercivity of $I$ over $K$. Thus the coercivity and w.l.s.continuity will give the existence of a minimum $z\in K$ of $I$.
To prove the convexity, observe that since $p>1$ and $|\cdot|^p$ is convex, the first integral in $I$ is convex. The second derivative of the second integral in $I$ is nonpositive and thus it is concave. As a consequence $I$ is convex.
To prove the strong lower semicontinuity, let $u_n,u\in K$ be such that $u_n\to u$ strongly in $W_0^{1,p}(\Omega)$. Then $$ \int_{\Omega}|\nabla u_n|^p\,dx\to\int_{\Omega}|\nabla u|^p\,dx. $$ Also we get $u_n\to u$ pointwise almost everywhere in $\Omega$ and so $G_k(u_n)\to G_k(u)$ almost everywhere in $\Omega$. Note that $0\leq u_n,u\leq v$ in $\Omega$. Then I am unable to proceed to pass the limit in the second integral over $I$.
Also, I am stuck to get the coercivity of $I$ over $K$. One thing I am confused about here is $I$ finite over the set $K$? (Do we really need this finiteness to prove the existence of the minimum?)
Can anyone please help me?
Thanks.