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Let us start with the so-called Gagliardo-Nirenberg Inequality in $n$ dimensions, $$ \Vert u\Vert_{L^{n/(n-1)}(\mathbb R^n)}\le c_n\Vert \nabla u\Vert_{L^{1}(\mathbb R^n)}, $$ an inequality that can be applied to your function so that you get in two dimensions $$ \Vert u\Vert_{L^{2}(\mathbb R^n)}\le c_2\Vert \nabla u\Vert_{L^{1}(\mathbb R^n)}, $$ and applying this to $u=v^2$, you obtain $$ \Vert v\Vert_{L^{4}(\mathbb R^n)}^2\le 2c_2\Vert v\nabla v\Vert_{L^{1}(\mathbb R^n)}\lesssim\Vert v\Vert_{L^{2}(\mathbb R^n)}\Vert \nabla v\Vert_{L^{2}(\mathbb R^n)}. $$ This is true for $v\in C^1_c(\Omega)$ and consequently by density in $H^1_0(\Omega)$ (which does not contain any constant non-zero function).