MULTIPLE NORMALIZED SOLUTIONS TO BIHARMONIC SCHRÖDINGER-CHOQUARD EQUATION WITH LOGARITHMIC NONLINEARITY

This paper presents the existence of multiple normalized solutions for the following nonlinear biharmonic Schrödinger-Choquard equation with logarithmic nonlinearity: $$\begin{cases}{\varepsilon}^4{\Delta}^2u+V(x)u={\lambda}u+u\,{\log}\,u^2+(I_{\alpha}{\ast}{\mid}u{\mid}^p){\mid}u{\mid}^{p-2}u+f(X){\mid}u{\mid}^{s-2}u,&{\text}x{\in}{\mathbb{R}}^N,\\\int{_{{\mathbb{R}}^N}}{\mid}u{\mid}^2dx={\alpha}^2{\varepsilon}^N,&{\text}x{\in}{\mathbb{R}}^N,\end{cases}$$ where ε, a > 0, N ⩾ 5, α ∈ (0, N), λ ∈ ℝ is a Lagrange multiplier and $p{\in}(1+{\frac{\alpha}{N}},\,1+{\frac{4+{\alpha}}{N}})$, $s{\in}(2,\,2+{\frac{s}{N}})$. Let V : ℝ^N → (-1, +∞) and f : ℝ^N → [0, +∞) be continuous functions. Our studies show that the number of normalized solutions to the equation is intricately connected to the topology of the domain where V reaches its minimum and f attains its maximum. Furthermore, our findings indicate that, when ε is sufficiently small, the count of normalized solutions is at least as large as the count of extremal points of both V and f. In addition, we introduce a novel functional space where the associated energy functional is differentiable and belongs to the C¹ class.