Let be $$(P)\left\{\begin{array}{ll} &-\Delta u + V(x)u=f & \text{ in }\ \Omega\\ &u=0 & \text{ on } \ \Gamma \end{array}\right.$$ with $V \in L^r(\Omega)$, for some $1<r<\infty$. I want to prove weak coercivity. Im trying to acotate this using this two inequalities:
Interpolation inequaity ($1\leq p \leq q \leq r \leq \infty$ and $u \in L^p(\Omega)\bigcap L^r(\Omega)$) $$||u||_{L^q(\Omega)}\leq ||u||_{L^p(\Omega)}^{\theta} ||u||_{L^p(\Omega)}^{1-\theta} $$ And i suspect it could be very useful this one,
Young's inequality: $1<p,q<\infty$ and $\frac{1}{p}+\frac{1}{q}=1$, then
$$ab \leq \epsilon a^p + C(\epsilon)b^q, \quad C(\epsilon)=(\epsilon p)^{-q/p} q^{-1}$$
so I could prove that $-\Delta + V(x)$ leads a weak formulation in $H_0^1(\Omega)$.
I have been fighting with this for hours, but i don't get anything. Thanks!
I thought about gettin $p=1/\theta$, $q=1/(1-\theta)$, $a=||u||_{L^p}$...
Your weak formulation is $\int\limits_{\Omega}{\nabla u\cdot\nabla vdx}+\int\limits_{\Omega}{Vuvdx}=\int\limits_{\Omega}{fvdx},\,\forall v\in H_0^1(\Omega)$. So if you denote $$a(u,v)=\int\limits_{\Omega}{\nabla u(x)\cdot\nabla v(x)dx}+\int\limits_{\Omega}{V(x)u(x)v(x)dx}$$ you have $$a(u,u)\ge C\|u\|_{H_0^1(\Omega)}+\int\limits_{\Omega}{Vu^2dx}\ge C\|u\|_{H_0^1(\Omega)}, \text{ if } V(x)\ge 0,\,a.e\,\, x\in \Omega\,\quad (*)$$ The constant $C$ comes from using Friedrich's inequality, and $(*)$ is the needed coercivity of the bilinear form, which you need in Lax-Milgram's theorem.