The Choquard equation is a partial differential equation that has gained significant interest and attention in recent decades. It is a nonlinear equation that combines elements of both the Laplace and Schrödinger operators, and it arises frequently in the study of numerous physical phenomena, from condensed matter physics to nonlinear optics.

In particular, the steady states of the Choquard equation were thoroughly investigated using a variational functional acting on the wave functions.

In this article, we introduce a dual formulation for the variational functional in terms of the potential induced by the wave function, and use it to explore the existence of steady states of a multi-state version the Choquard equation in critical and sub-critical cases.

This paper is concerned with the static Choquard equation

$$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u=\bigg(\frac{1}{|x|^{N-\unicode[STIX]{x1D6FC}}}\ast |u|^{p}\bigg)|u|^{p-2}u\quad \text{in }\mathbb{R}^{N},\end{eqnarray}$$

$N,p>2$

$\max \{0,N-4\}<\unicode[STIX]{x1D6FC}<N$

$u\in C^{1}(\mathbb{R}^{N})$

$u\equiv 0$

$\unicode[STIX]{x1D6FC}\neq 2$

We are concerned with the semi-classical states for the Choquard equation

$$-{\epsilon }^2\Delta v + Vv = {\epsilon }^{-\alpha }(I_\alpha *|v|^p)|v|^{p-2}v,\quad v\in H^1({\mathbb R}^N),$$

For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation,

$$-\Delta u+V_{\mu, \nu} u=(I_\alpha\ast \vert u \vert ^{({N+\alpha})/{N}}){ \vert u \vert }^{{\alpha}/{N}-1}u,\quad {\rm in} \ {\open R}^N, $$

$N\ges 3$

$I_\alpha : {\open R}^N \to 0$

$N \ges 4$

$\ker (-\Delta + V_{\mu ,\nu }) = \{ 0\} $

We study the multiplicity and concentration behaviour of positive solutions for a quasi-linear Choquard equation

where Δp is the p-Laplacian operator, 1 < p < N, V is a continuous real function on ℝN, 0 < μ < N, F(s) is the primitive function of f(s), ε is a positive parameter and * represents the convolution between two functions. The question of the existence of semiclassical solutions for the semilinear case p = 2 has recently been posed by Ambrosetti and Malchiodi. We suppose that the potential satisfies the condition introduced by del Pino and Felmer, i.e.V has a local minimum. We prove the existence, multiplicity and concentration of solutions for the equation by the penalization method and Lyusternik–Schnirelmann theory and even show novel results for the semilinear case p = 2.