A classical result of E. Lieb is that the functional $$\mathcal E(\phi):=\int_{\mathbb R^3}|\nabla\phi(x)|^2~dx-\int_{(\mathbb R^3)^2}\frac{|\phi(x)|^2|\phi(y)|^2}{|x-y|}~dxdy$$ for $\phi\in W^1(\mathbb R^3)$ with $\|\phi\|_2=1$ admits a unique minimizer (up to translations), and that the minimizer in question can be characterized as the unique solution of the "nonlinear Schrödinger equation" $$\Delta\phi(x)+\left(2\int_{\mathbb R^3}\frac{|\phi(y)|^2}{|x-y|}~dy\right)\phi(x)=e\phi(x)$$ for some $e>0$ that is radially symmetric and such that $\phi\in W^1(\mathbb R^3)$ with $\|\phi\|_2=1$.

Question. Does there exist a similar "theory" of unique minimizers for more general functionals of the form $$\mathcal E_V(\phi):=\int_{\mathbb R^n}|\nabla\phi(x)|^2~dx-\int_{(\mathbb R^n)^2}|\phi(x)|^2V(x-y)|\phi(y)|^2~dxdy \tag{1}$$ where $V$ is a general positive definite function or Schwartz distribution?

In the mathematical physics literature, most of the results I found consider very similar V's (e.g., the Coulomb potential $V(x):=|x|^{2-n}$ in $\mathbb R^n$ for $n\geq3$) because of the physical motivation.

That being said, it seems to me like the study of $\mathcal E_V$ in $(1)$ for more general $V$'s arises naturally in the context of large deviations. Ignoring issues of compactness and continuity for simplicity, we expect from the LDP for Brownian occupation measures and Varadhan's lemma that the exponential growth of Brownian "self-intersections" of the form $$\mathbb E\left[\exp\left(\int_{[0,t]^2}V(B_u-B_v)~dudv\right)\right]$$ is described by the maximizers of $-\mathcal E_V(\phi)$.

A [classical result][1] of E. Lieb is that the functional $$\mathcal E(\phi):=\int_{\mathbb R^3}|\nabla\phi(x)|^2~dx-\int_{(\mathbb R^3)^2}\frac{|\phi(x)|^2|\phi(y)|^2}{|x-y|}~dx~dy$$ for $\phi\in W^1(\mathbb R^3)$ with $\|\phi\|_2=1$ admits a unique minimizer (up to translations), and that the minimizer in question can be characterized as the unique solution of the "nonlinear Schrödinger equation" $$\Delta\phi(x)+\left(2\int_{\mathbb R^3}\frac{|\phi(y)|^2}{|x-y|}~dy\right)\phi(x)=e\phi(x)$$ for some $e>0$ that is radially symmetric and such that $\phi\in W^1(\mathbb R^3)$ with $\|\phi\|_2=1$.

Question. Does there exist a similar "theory" of unique minimizers for more general functionals of the form $$\mathcal E_V(\phi):=\int_{\mathbb R^n}|\nabla\phi(x)|^2~dx-\int_{(\mathbb R^n)^2}|\phi(x)|^2V(x-y)|\phi(y)|^2~dx~dy \tag{1}$$ where $V$ is a general positive definite function or Schwartz distribution?

In the mathematical physics literature, most of the results I found consider very similar V's (e.g., the Coulomb potential $V(x):=|x|^{2-n}$ in $\mathbb R^n$ for $n\geq3$) because of the physical motivation.

That being said, it seems to me like the study of $\mathcal E_V$ in $(1)$ for more general $V$'s arises naturally in the context of large deviations. Ignoring issues of compactness and continuity for simplicity, we expect from the LDP for Brownian occupation measures and Varadhan's lemma that the exponential growth of Brownian "self-intersections" of the form $$\mathbb E\left[\exp\left(\int_{[0,t]^2}V(B_u-B_v)~du~dv\right)\right]$$ is described by the maximizers of $-\mathcal E_V(\phi)$.

[1]: https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm197757293 ${}$