Normalized solutions to polyharmonic equations with Hardy-type potentials and exponential critical nonlinearities

TL;DR

The paper addresses the existence of normalized (mass-constrained) solutions to the polyharmonic equation $(-\Delta)^m u + \frac{\mu}{|x|^{2m}}u + \lambda u = \eta u^3 + g(u)$ in $\mathbb{R}^{2m}$ with exponential critical growth nonlinearities. It develops a constrained variational framework on the spaces $X^m$ and $X^m_{\mathrm{rad}}$, defining an energy functional $J$ and a constrained manifold $\mathcal{M}$ together with the mass constraint sets, and proves the existence of a minimizer $u$ at level $c_\beta(\rho)$ under (A0)--(A4) and smallness conditions (e.g., $\eta C_4^4 \rho<2$ and a bound on $\beta$). The main result guarantees a pair $(\lambda,u)$ with $\lambda>0$ and $u\in \mathcal{S}\cap\mathcal{M}$ solving the equation, with $J(u)$ attaining the infimum on the constrained sets; in the 1D case ($m=1$) and with odd $g$, the ground state can be taken nonnegative. The work leverages Moser–Trudinger type inequalities, Pohožaev and Nehari identities, and radial-compactness arguments to obtain a robust variational construction of normalized solutions in the presence of a Hardy-type potential and exponential critical nonlinearities.

Abstract

Via a constrained minimization, we find a solution $(λ,u)$ to the problem \begin{equation*} \begin{cases} (-Δ)^m u+\fracμ{|x|^{2m}}u + λu = ηu^3 + g(u)\\ \int_{\mathbb{R}^{2m}} u^2 \, dx = ρ\end{cases} \end{equation*} with $1 \le m \in \mathbb{N}$, $μ,η\ge 0$, $ρ> 0$, and $g$ having exponential critical growth at infinity and mass supercritical growth at zero.

Theorems & Definitions (28)

• Theorem 1.1
• Remark 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• Proposition 2.3: Pohožaev identity
• Lemma 2.4
• Lemma 3.1
• proof
• Remark 3.2