Existence and Concentration of Multiple Positive Solutions for a Logarithmic Fractional Schrödinger--Poisson System

Abstract: We study a logarithmic fractional Schrödinger--Poisson system in (\R^{3}): \begin{equation*} \begin{cases} \varepsilon^{2α}(-Δ)^αu+V(x)u+φu=u\log u^{2}+|u|^{p-2}u, & \text{in }\R^{3},\ \varepsilon^{2α}(-Δ)^αφ=u^{2}, & \text{in }\R^{3}. \end{cases} \end{equation*} Here (α\in\bigl(\frac34,1\bigr)), (4<p\<2_α^{*}=\frac{6}{3-2α}\), and \(V\) satisfies a global potential condition. Using a suitable Orlicz-type Banach space, we establish a \(C^{1}\) variational framework for the problem and combine the Nehari manifold method with Lusternik--Schnirelmann category theory. We then prove that, for every fixed \(δ\>0) and all sufficiently small (\varepsilon>0), the system admits at least (\operatorname{cat}{Mδ}(M)) distinct positive solutions. Moreover, the maximum points of these solutions concentrate near the global minimum set of (V) as (\varepsilon\to0).

• The paper establishes that for sufficiently small ε, the system admits multiple positive solutions that concentrate near the minimum of the potential.
• It overcomes challenges from the logarithmic nonlinearity by splitting the term and utilizing Orlicz–Sobolev spaces in a refined variational framework.
• Lusternik–Schnirelmann theory is applied to connect the number of solutions directly with the topology of the potential's global minimum set.

Existence and Multiplicity of Concentrating Positive Solutions in a Logarithmic Fractional Schrödinger–Poisson System

Introduction and Problem Statement

The paper "Existence and Concentration of Multiple Positive Solutions for a Logarithmic Fractional Schrödinger–Poisson System" (2604.04148) addresses the existence, multiplicity, and concentration phenomena for positive solutions of the nonlocal system in $\mathbb{R}^3$: $$\begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u + V(x)u + \phi u = u \log u^{2} + |u|^{p-2} u,\ \varepsilon^{2\alpha}(-\Delta)^{\alpha}\phi = u^{2}. \end{cases}$$ Here, $0 < \varepsilon \ll 1$, $\alpha \in (\frac{3}{4}, 1)$, $4 < p < 2_\alpha^* = \frac{6}{3-2\alpha}$, and $V$ is a $C^{1}$ potential satisfying $$V_\infty := \lim_{|x|\to\infty} V(x) > V_0 := \inf_{\mathbb{R}^3} V(x) > -1$$, where $M := \{x \in \mathbb{R}^3 : V(x) = V_0\}$ is nonempty and compact.

The coupling introduces two major nonlocalities: the fractional Laplacian of order $\alpha$ and the Poisson interaction via $$\begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u + V(x)u + \phi u = u \log u^{2} + |u|^{p-2} u,\ \varepsilon^{2\alpha}(-\Delta)^{\alpha}\phi = u^{2}. \end{cases}$$0. In parallel, the nonlinear response combines both logarithmic and power-type terms, with the former inducing strong technical challenges due to its critical growth and lack of sufficient regularity in standard Sobolev spaces. This work extends multiplicity and concentration results for nonlinear Schrödinger–Poisson systems into the logarithmic and fractional nonlocal regime, building on techniques from critical point theory and nonlinear analysis.

Variational Structure and Function Space Setting

A fundamental technical aspect is the correct functional analytic treatment of the logarithmic nonlinearity. Due to the lack of $$\begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u + V(x)u + \phi u = u \log u^{2} + |u|^{p-2} u,\ \varepsilon^{2\alpha}(-\Delta)^{\alpha}\phi = u^{2}. \end{cases}$$1 regularity of the associated energy on the standard $$\begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u + V(x)u + \phi u = u \log u^{2} + |u|^{p-2} u,\ \varepsilon^{2\alpha}(-\Delta)^{\alpha}\phi = u^{2}. \end{cases}$$2 space, the authors utilize a splitting of the primitive of the logarithmic term into a convex and a smooth part, $$\begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u + V(x)u + \phi u = u \log u^{2} + |u|^{p-2} u,\ \varepsilon^{2\alpha}(-\Delta)^{\alpha}\phi = u^{2}. \end{cases}$$3 and $$\begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u + V(x)u + \phi u = u \log u^{2} + |u|^{p-2} u,\ \varepsilon^{2\alpha}(-\Delta)^{\alpha}\phi = u^{2}. \end{cases}$$4, with $$\begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u + V(x)u + \phi u = u \log u^{2} + |u|^{p-2} u,\ \varepsilon^{2\alpha}(-\Delta)^{\alpha}\phi = u^{2}. \end{cases}$$5 an $$\begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u + V(x)u + \phi u = u \log u^{2} + |u|^{p-2} u,\ \varepsilon^{2\alpha}(-\Delta)^{\alpha}\phi = u^{2}. \end{cases}$$6-function satisfying the $$\begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u + V(x)u + \phi u = u \log u^{2} + |u|^{p-2} u,\ \varepsilon^{2\alpha}(-\Delta)^{\alpha}\phi = u^{2}. \end{cases}$$7-condition. This leads to the Orlicz–Sobolev space setting

$$\begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u + V(x)u + \phi u = u \log u^{2} + |u|^{p-2} u,\ \varepsilon^{2\alpha}(-\Delta)^{\alpha}\phi = u^{2}. \end{cases}$$8

This space admits the required compactness and regularity properties, enabling the definition of a well-posed $$\begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u + V(x)u + \phi u = u \log u^{2} + |u|^{p-2} u,\ \varepsilon^{2\alpha}(-\Delta)^{\alpha}\phi = u^{2}. \end{cases}$$9 variational energy functional.

The energy functional on $0 < \varepsilon \ll 1$0 is given by

$0 < \varepsilon \ll 1$1

where $0 < \varepsilon \ll 1$2 is the unique solution to $0 < \varepsilon \ll 1$3.

A variational framework is established by restricting to the associated Nehari manifold: $0 < \varepsilon \ll 1$4 This set decomposes the functional $0 < \varepsilon \ll 1$5 into a natural constraint, supporting the search for nontrivial critical points that correspond to weak solutions of the system.

Overcoming Lack of Compactness and Nonlocality

The analysis must address the loss of compactness due to the unboundedness of $0 < \varepsilon \ll 1$6 and the non-compactness induced by nonlocal terms. The authors adapt concentration–compactness principles, develop precise splitting lemmas for the key nonlinear and nonlocal terms, and employ the profile decomposition method for sequences in Orlicz–Sobolev spaces, exploiting the precise structure of $0 < \varepsilon \ll 1$7 and $0 < \varepsilon \ll 1$8.

A key quantitative threshold is established for the autonomous problem associated with constant $0 < \varepsilon \ll 1$9 and $\alpha \in (\frac{3}{4}, 1)$0, enabling the authors to identify the critical level for compactness via analysis of the limiting energies $\alpha \in (\frac{3}{4}, 1)$1. The range $\alpha \in (\frac{3}{4}, 1)$2 is essential for appropriate decay, Sobolev embedding, and the quartic nonlocal term's well-definedness.

Main Existence, Multiplicity, and Concentration Theorem

The central result demonstrates that for every fixed neighborhood $\alpha \in (\frac{3}{4}, 1)$3 around $\alpha \in (\frac{3}{4}, 1)$4 (i.e., the global minimum set of the potential), and for all sufficiently small $\alpha \in (\frac{3}{4}, 1)$5, system (1.1) admits at least $\alpha \in (\frac{3}{4}, 1)$6 distinct positive solutions. Moreover, these solutions concentrate, as $\alpha \in (\frac{3}{4}, 1)$7, near $\alpha \in (\frac{3}{4}, 1)$8, in the sense that their maximum points approach $\alpha \in (\frac{3}{4}, 1)$9 and all solutions are strictly positive for $4 < p < 2_\alpha^* = \frac{6}{3-2\alpha}$0 small enough: $4 < p < 2_\alpha^* = \frac{6}{3-2\alpha}$1 where $4 < p < 2_\alpha^* = \frac{6}{3-2\alpha}$2 denotes the location of the maximum of each solution.

The proof utilizes the Lusternik–Schnirelmann (LS) category theory on the Nehari manifold, which relates the topology of the critical point set $4 < p < 2_\alpha^* = \frac{6}{3-2\alpha}$3 with the number of distinct positive solutions. The authors construct a barycenter map and test functions concentrated near $4 < p < 2_\alpha^* = \frac{6}{3-2\alpha}$4, proving that the LS category bounds from below the number of positive energy solutions.

Contradictory to prior literature on the logarithmic fractional Schrödinger–Poisson system, wherein only existence or penalization-based multiple solution results were available, the paper asserts that the multiplicity phenomenon is governed directly by the topology of $4 < p < 2_\alpha^* = \frac{6}{3-2\alpha}$5 under the generic potential assumption $4 < p < 2_\alpha^* = \frac{6}{3-2\alpha}$6, even in the presence of both logarithmic and nonlocal nonlinearities. The analysis avoids penalization and works directly on an unmodified functional.

Analytical Tools and Regularity

The paper develops fine elliptic and regularity estimates for the fractional Laplacian, applications of the Hardy–Littlewood–Sobolev inequality for the nonlocal term, and uniform $4 < p < 2_\alpha^* = \frac{6}{3-2\alpha}$7 and decay estimates for concentrating solutions. The strong maximum principle for fractional Laplacians is exploited to establish positivity of critical points, and translation compactness arguments ensure the limiting profiles are ground states of the autonomous problem.

Implications and Outlook

The results provide a comprehensive nonlinear analysis for systems combining fractional operators, Poisson coupling, and logarithmic nonlinearities. The techniques amalgamate Orlicz space theory, critical point methods on the Nehari manifold, and topological category arguments, demonstrating robust control over multiscale interacting nonlocalities.

Practically, these results extend variational and multiplicity techniques to complex quantum and nonlinear optics models characterized by nonlocality and nonstandard nonlinear growth. The methodology is adaptable to higher order fractionalities, more general potentials, and possibly critical exponent cases.

Theoretically, this approach establishes that the concentration and multiplicity behavior in nonlocal Schrödinger–Poisson systems with critical logarithmic growth is, under suitable conditions, dictated by the underlying topology of the potential minima, echoing the principle for local and standard nonlinearities but in a much more intricate analytical setting.

Conclusion

This work rigorously establishes the existence and topologically governed multiplicity of positive semiclassical states for the logarithmic fractional Schrödinger–Poisson system with a general potential. The combination of subtle variational analysis in Orlicz–Sobolev spaces, precise control of nonlocal and critical terms, and LS category theory advances the understanding of solution structures in a prominent class of nonlocal PDEs, confirming that topological methods remain effective in highly nonstandard, nonlocal, and critical growth contexts.