Let $\Omega\subset\mathbb{R}^N$ with $N\geq 2$ be a smooth bounded domain.

Define for $k>0$, $\gamma>0$, the real valued function $g_k(s):=min\{s^{-\gamma},k\}$ for $s>0$ and equals $k$ for $s\leq 0$.

Let $\psi_k$ be the primitive of $g_k$ and $v\in H_{loc}^{1}(\Omega)$ ($\geq 0$) satisfies $$\int_{\Omega}{\nabla v}\cdot\nabla\phi\,dx\geq\int_{\Omega}\frac{f(x)}{v^{\gamma}}\phi\,dx$$ for all $\phi\in C_c^{\infty}(\Omega)$ ($\geq 0$), provided $f\in L^1(\Omega)$ nonnegative.

Define $$L=\{\phi\in C_c^{\infty}(\Omega):0\leq\phi\leq v \text{ a.e in } \Omega\}.$$

Then there exist $w\in L$ such that $$ \int_{\Omega}\nabla w\cdot\nabla(\psi-w)\,dx\geq\int_{\Omega}f(x)\psi_k^{'}(x)(\psi-w)(x)\,dx $$ for $\psi\in w+(H_0^{1}(\Omega)\cap L_c^{\infty}(\Omega))$ and $0\leq\psi\leq v$, where $L_c^{\infty}(\Omega)$ is the space of $L^{\infty}$ functions with compact support in $\Omega$.

I am trying to prove the required inequality by considering the functional $J_k:L\to[-\infty,\infty]$ defined by $$J_k(\phi)=\frac{1}{2}\int_{\Omega}|\nabla\phi|^2\,dx-\int_{\Omega}f(x)\psi_k(\phi)\,dx.$$

My first attempt is to prove that there exist a minimum $w\in L$ of $J_k$ on the convex set $L$. Then I hope this minimality of $w$ will give the required inequality.

Can you please help me regarding this.

Thank you very much.


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