# Inequality for the Laplace operator

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.