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Consider the following 2nd order nonlinear elliptic equation on $\mathbb{R}^n$: $$-\Delta \varphi + \sum_i a_i(x, \varepsilon)\partial_i \varphi + \varphi = N(\varphi),$$ where $N$ is a smooth function which gives the nonlinearity, $a$ is smooth, and $\varepsilon$ is a parameter (one can also think of the above pde as a one-parameter family of equations). Now, let us consider solutions $\varphi_\varepsilon \in H^1(\mathbb{R}^n)$, and suppose we know that if $\varphi_\varepsilon$ varies continuously with $\varepsilon$ in the $L^p(\mathbb{R}^n)$-norm, where $p \in [2, \frac{2n}{n - 2})$ (the range of Sobolev embedding). Let us also assume, if need be, nice decay properties on $\varphi_\varepsilon$, like vanishing at infinity. Can we somehow conclude from this that $\varphi_\varepsilon$ varies continuously with $\varepsilon$ in the $L^r(\mathbb{R}^n)$-norm, for some $r \geq \frac{2n}{n - 2}$? Any ideas would be appreciated.

Consider the following 2nd order nonlinear elliptic equation on $\mathbb{R}^n$: $$-\Delta \varphi_\varepsilon + \sum_i a_i(x, \varepsilon)\partial_i \varphi_\varepsilon + \varphi_\varepsilon = N(\varphi_\varepsilon),$$ where $N$ is a smooth function which gives the nonlinearity, $a$ is smooth, and $\varepsilon$ is a parameter (one can also think of the above pde as a one-parameter family of equations). Now, let us consider solutions $\varphi_\varepsilon \in H^1(\mathbb{R}^n)$, and suppose we know that if $\varphi_\varepsilon$ varies continuously with $\varepsilon$ in the $L^p(\mathbb{R}^n)$-norm, where $p \in [2, \frac{2n}{n - 2})$ (the range of Sobolev embedding). Let us also assume, if need be, nice decay properties on $\varphi_\varepsilon$, like vanishing at infinity. Can we somehow conclude from this that $\varphi_\varepsilon$ varies continuously with $\varepsilon$ in the $L^r(\mathbb{R}^n)$-norm, for some $r \geq \frac{2n}{n - 2}$? Any ideas would be appreciated.