[Paper Review] Existence of solutions for a class of nonlinear Choquard equations with critical growth

Yong Ao|arXiv (Cornell University)|Aug 25, 2016

Nonlinear Partial Differential Equations13 references3 citations

TL;DR

This paper establishes the existence of nontrivial solutions for a class of nonlinear Choquard equations with critical growth in $[\mathbb{R}^N$ for $N \geq 4$, using variational methods and Nehari manifold techniques. The key result proves the existence of a minimizer for a constrained minimization problem involving a nonlocal term and a subcritical term, under conditions $p = \frac{N+\alpha}{N-2}$ and $2 < q < 2^*$, resolving the existence question in the critical growth regime.

ABSTRACT

In this paper, we consider a class of nonlinear Choquard equations with critical growth, and we show the existence of solutions of the equations. Besides, we consider the corresponding minimizing problem and prove the existence of a minimizer.

Motivation & Objective

• To establish the existence of nontrivial solutions for a class of nonlinear Choquard equations with critical growth in $\mathbb{R}^N$ for $N \geq 4$.
• To analyze the structure of the energy functional associated with the equation, particularly the interplay between the nonlocal term $(I_\alpha * |u|^p)|u|^{p-2}u$ and the subcritical term $|u|^{q-2}u$.
• To prove the existence of a minimizer for a constrained minimization problem involving the Nehari manifold and radial symmetry, under critical growth conditions.
• To overcome the difficulty of lacking scaling invariance due to the coexistence of nonlocal and subcritical terms, which prevents standard scaling arguments from yielding least energy solutions.

Proposed method

• The problem is studied via variational methods by seeking critical points of the energy functional $I(u)$ defined on $H^1(\mathbb{R}^N)$.
• The Nehari manifold is used to characterize the energy level of potential solutions, ensuring the functional is bounded from below on this manifold.
• A constrained minimization problem is formulated to find a minimizer of the Dirichlet energy $T(u) = \int |\nabla u|^2 dx$ under the constraint $H(u) = 1$, where $H(u)$ combines the nonlocal, subcritical, and $L^2$-terms.
• Radial symmetry is exploited by considering the Schwarz spherical rearrangement of functions to construct a radial minimizing sequence.
• The Brezis-Lieb lemma and Hardy-Littlewood-Sobolev inequality are applied to handle convergence and norm estimates in the limit process.
• A contradiction argument based on scaling properties of the functional under $u_\sigma(x) = u(x/\sigma)$ is used to rule out vanishing or dichotomy in the weak limit, proving the existence of a nontrivial minimizer.

Experimental results

Research questions

• RQ1 Does the nonlinear Choquard equation with critical growth $p = \frac{N+\alpha}{N-2}$ and subcritical $q$-growth admit a nontrivial solution in $\mathbb{R}^N$ for $N \geq 4$?
• RQ2 Can a minimizer be found for the constrained minimization problem involving the nonlocal term and the subcritical term when scaling arguments fail due to mixed growth types?
• RQ3 What happens to the concentration-compactness structure when the nonlocal term and the subcritical term coexist and break the scaling invariance of the problem?
• RQ4 Is the Nehari manifold approach sufficient to guarantee the existence of a ground state solution in the critical growth regime under these conditions?
• RQ5 Can the minimizer be shown to be radial and nontrivial, even when the standard scaling method fails to produce the least energy solution?

Key findings

• For $N \geq 5$, $2 < q < 2^*$, or $N = 4$, $3 < q < 4$, the equation $-\Delta u + u = (I_\alpha * |u|^p)|u|^{p-2}u + |u|^{q-2}u$ in $\mathbb{R}^N$ admits a nontrivial solution.
• The constrained minimization problem $A = \inf \left\{ \frac{1}{2}\int |\nabla u|^2 dx : H(u) = 1 \right\}$ has a minimizer in $H^1_{\text{rad}}(\mathbb{R}^N)$, where $H(u)$ combines the nonlocal, subcritical, and $L^2$-terms.
• The minimizer $u_0$ is radial and nontrivial, and its existence is proven by contradiction, ruling out vanishing and dichotomy via scaling arguments.
• The failure of scaling invariance due to the coexistence of nonlocal and subcritical terms prevents the use of standard scaling to derive the least energy solution, but the minimizer still exists via compactness and variational analysis.
• The proof relies on the radial symmetry of the minimizing sequence, the Brezis-Lieb lemma for norm convergence, and the Hardy-Littlewood-Sobolev inequality to control the nonlocal term.
• The contradiction in the case $\lambda_0 = 0$ arises from a lower bound on $\|v_n\|_{2^*}$ derived from the functional constraint, leading to a contradiction with the Sobolev constant $S$.