[Paper Review] Entire radial and nonradial solutions for systems with critical growth

Francesca Gladiali, Massimo Grossi+1 more|arXiv (Cornell University)|Dec 12, 2016

Nonlinear Partial Differential Equations10 references4 citations

TL;DR

This paper establishes the existence of both radial and nonradial entire solutions for a class of nonlinear elliptic systems with critical growth in $[\mathbb{R}^N$ ($N \geq 3$), using global bifurcation theory and symmetry-breaking arguments. The key contribution is proving the existence of nonradial solutions for noncooperative systems via spectral analysis of the linearized operator and invariant subspace decomposition, particularly for systems on the critical hyperbola.

ABSTRACT

In this paper we establish existence of radial and nonradial solutions to the system $$ \begin{array}{ll} -Δu_1 = F_1(u_1,u_2) & ext{in }\mathbb{R}^N, ewline -Δu_2 = F_2(u_1,u_2) & ext{in }\mathbb{R}^N, ewline u_1\geq 0,\ u_2\geq 0 & ext{in }\mathbb{R}^N, ewline u_1,u_2\in D^{1,2}(\mathbb{R}^N), \end{array} $$ where $F_1,F_2$ are nonlinearities with critical behavior.

Motivation & Objective

• To establish the existence of entire radial and nonradial solutions for a class of nonlinear elliptic systems with critical growth in $[\mathbb{R}^N$.
• To address the lack of compactness and variational structure in critical systems by employing bifurcation theory and spectral analysis.
• To investigate symmetry breaking in noncooperative systems where the maximum principle does not apply, leading to nonradial solutions.
• To extend previous results on radial solutions to include nonradial solutions using invariant subspace techniques and spherical harmonics.
• To analyze the structure of the linearized operator around trivial solutions and identify conditions under which nontrivial solutions emerge.

Proposed method

• Use global bifurcation theory to construct solutions from the trivial solution family $U_{\delta,y}$, parameterized by $\delta > 0$ and $y \in \mathbb{R}^N$.
• Apply spectral analysis to the linearized system at the trivial solution to detect bifurcation points where eigenvalues cross zero.
• Decompose the solution space using spherical harmonics and exploit symmetry properties under rotations and reflections to isolate invariant subspaces.
• Define invariant subspaces $X_{\mathcal{S}_m}$ and $\mathcal{Z}_m$ to restrict the problem to functions with specific symmetry and periodicity properties in angular variables.
• Use the expansion of spherical harmonics in terms of Gegenbauer polynomials to characterize the angular dependence of solutions.
• Prove that the kernel of the linearized operator is one-dimensional in the invariant subspace $\mathcal{Z}_m$, enabling application of bifurcation theorems.

Experimental results

Research questions

• RQ1 Under what conditions does a noncooperative system with critical growth in $\mathbb{R}^N$ admit nonradial entire solutions?
• RQ2 How does the structure of the nonlinearity, particularly on the critical hyperbola, influence the existence and symmetry of solutions?
• RQ3 Can bifurcation theory be applied to non-variational systems with critical growth to construct nontrivial solutions?
• RQ4 What role do symmetry-breaking mechanisms play in generating nonradial solutions when radial solutions exist?
• RQ5 What is the dimension of the kernel of the linearized operator in symmetric subspaces, and how does this affect bifurcation behavior?

Key findings

• The system admits nontrivial radial solutions when the eigenvalues of the coupling matrix $A$ reach specific values, as shown via bifurcation theory.
• Nonradial solutions exist for noncooperative systems where $1 - \alpha < 0$, indicating repulsive interactions between components.
• For $m \geq 2$, the kernel of the linearized operator in the invariant subspace $\mathcal{Z}_m$ is one-dimensional, with $\gamma(m) = 1$, enabling bifurcation.
• The nonradial solutions are constructed as perturbations of the trivial solution family $U_{\delta,y}$, with angular dependence given by $\operatorname{Im}((x_1 + i x_2)^m)$.
• The solution $(0, W_{m,m}(r) Y_m)$ belongs to $\mathcal{Z}_m$, confirming the existence of nonradial solutions with prescribed symmetry.
• The analysis confirms that the linearized operator has no nontrivial solutions in $X_{\mathcal{S}_m}$, ensuring the bifurcation occurs only in the $\mathcal{Z}_m$-invariant sector.