Show that $\frac{\int_\Omega e^{v^2}v^2}{\int_\Omega v^2} \geq \frac{\int_\Omega e^{u^2}u^2}{\int_\Omega u^2}$

Question

I would like to ask if the following statement is true or not:

Let $u,v:\Omega\subset% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{N}\rightarrow% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ~(\Omega$ is bounded)$;~v\geq u>0.$ Is it true that $$ \frac{% %TCIMACRO{\dint \limits_{\Omega}}% %BeginExpansion {\displaystyle\int\limits_{\Omega}} %EndExpansion e^{v^{2}}v^{2}}{% %TCIMACRO{\dint \limits_{\Omega}}% %BeginExpansion {\displaystyle\int\limits_{\Omega}} %EndExpansion v^{2}}\geq\frac{% %TCIMACRO{\dint \limits_{\Omega}}% %BeginExpansion {\displaystyle\int\limits_{\Omega}} %EndExpansion e^{u^{2}}u^{2}}{% %TCIMACRO{\dint \limits_{\Omega}}% %BeginExpansion {\displaystyle\int\limits_{\Omega}} %EndExpansion u^{2}}? $$

Answer 1

Following Norbert let $N=1$, $\Omega=[0,1]$ and consider the family of functions: $$ u(x;a_1,a_2) = a_1\chi_{[0,1/2]}(x) + a_2\chi_{[1/2,1]}(x). $$ Compute $$ \int_\Omega u^2 \; dx = \frac12 a_1^2 + \frac12 a_2^2 $$ and $$ \int_\Omega e^{u^2}u^2 \; dx = \frac12e^{a_1^2}a_1^2 + \frac12e^{a_2^2}a_2^2, $$ noting that $$ f(a_1,a_2) = \frac{\int_\Omega e^{u^2}u^2}{\int_\Omega u^2} = \frac{e^{a_1^2}a_1^2 + e^{a_2^2}a_2^2}{a_1^2+a_2^2}. $$ Simplifying the situation further, set $a_1=1$ and observe that $$ f(1,t) = \frac{e + e^{t^2}t^2}{1+t^2} $$ is a decreasing function on the interval $t\in(0.1,0.6)$. Therefore, $$ f(1,0.4) < f(1,0.2) $$ implying that \begin{align} u &= \chi_{[0,1/2]} + 0.2\chi_{[1/2,1]} \\ v &= \chi_{[0,1/2]} + 0.4\chi_{[1/2,1]} \end{align} are counterexamples.