On Dancer-type solutions for the Lane--Emden equation via semivirial-vanishing geometry

Abstract: Aubin--Talenti bubbles describe the decaying positive solutions of the zero-frequency critical Lane--Emden equation in Euclidean space. By appealing to bifurcation methods, Dancer constructed in his seminar paper \cite{DancerSolution} positive-frequency solutions to the Lane--Emden equation which decay in the noncompact directions and are periodic in one direction. Alternatively, we give in this paper an energy-based variational construction of such Dancer-type solutions via the semivirial-vanishing geometry developed in author's recent work for studying focusing NLS on waveguide manifolds. The main new ingredient is a strict sub-bubbling estimate below the Euclidean Sobolev threshold. Unlike the usual Brezis--Nirenberg mechanism, no lower-order focusing perturbation is available in our model. Instead, the energy drop is produced by the bounded periodic direction: truncating a Euclidean bubble to one period removes a leading-order part of the gradient tail, while the nonlinear tail is of lower order. This restores compactness of minimizing sequences and yields normalized ground states for every prescribed mass, thereby answering an open question from \cite{Luo_energy_crit}.

• The paper develops a semivirial-vanishing variational method that constructs positive normalized ground states for every prescribed mass on the waveguide manifold R^d × T.
• A strict energy bound m_c < S^{D/2}/D is established using truncated Aubin–Talenti bubbles, where the periodic direction creates an energy drop through distinct gradient and nonlinear tail rates.
• The results produce Dancer-type, partially periodic solutions for all sufficiently small masses and close the endpoint compactness gap, while uniqueness, stability, and nontrivial periodic dependence at larger masses remain open.

Setting and problem

The paper studies positive solutions of the energy-critical Lane–Emden equation with a positive frequency term,

$-\Delta_{x,y}u+\beta u=u^{2^*-1},\qquad \beta>0,$

on the waveguide manifold $\mathbb X=\mathbb R^d_x\times\mathbb T_y$, where $D=d+1$ and $2^*=2D/(D-2)$. Such solutions decay in the noncompact directions and are periodic in the compact one; following Dancer's bifurcation construction for semilinear elliptic equations on $\mathbb R^n$, they are referred to as Dancer-type solutions. The purpose of the paper is to give an energy-based variational construction of these solutions, thereby obtaining quantitative information not visible from bifurcation theory.

The variational framework is the semivirial-vanishing geometry developed previously by the author for focusing NLS on waveguides. For prescribed mass $c>0$, one minimizes the energy

$$E(u)=\frac12\|\nabla_{x,y}u\|_2^2-\frac{D-2}{2D}\|u\|_{2^*}^{2^*}$$

over the constraint set $V(c)=\{u: M(u)=c,\ Q(u)=0\}$, where $M(u)=\|u\|_2^2$ is the mass and $Q(u)=\|\nabla_xu\|_2^2-\frac dD\|u\|_{2^*}^{2^*}$ is the semivirial functional, which involves only the $\mathbb X=\mathbb R^d_x\times\mathbb T_y$0-gradient because the virial identity in the dispersive directions sees only those coordinates. The value $\mathbb X=\mathbb R^d_x\times\mathbb T_y$1 is well posed: the scaling $\mathbb X=\mathbb R^d_x\times\mathbb T_y$2 preserves mass, satisfies $\mathbb X=\mathbb R^d_x\times\mathbb T_y$3, and a projection lemma guarantees that any nonzero function can be rescaled onto $\mathbb X=\mathbb R^d_x\times\mathbb T_y$4 while decreasing the auxiliary functional $\mathbb X=\mathbb R^d_x\times\mathbb T_y$5.

Prior work established two facts. First, there exists $\mathbb X=\mathbb R^d_x\times\mathbb T_y$6 such that any optimizer of $\mathbb X=\mathbb R^d_x\times\mathbb T_y$7 with $\mathbb X=\mathbb R^d_x\times\mathbb T_y$8 must depend nontrivially on the periodic variable — hence any small-mass optimizer would be Dancer-type. Second, the non-strict bound $\mathbb X=\mathbb R^d_x\times\mathbb T_y$9 holds, where $D=d+1$0 is the sharp Sobolev constant on $D=d+1$1. The existence of optimizers, however, was left open. This paper closes that gap.

Main results

Two theorems are proved.

Strict sub-bubbling estimate: for every $D=d+1$2,

$D=d+1$3

Existence of normalized ground states: for every $D=d+1$4, the variational problem defining $D=d+1$5 admits an optimizer, which may be chosen strictly positive and solves the Lane–Emden equation for some $D=d+1$6.

Combined with the $D=d+1$7-dependence theorem from prior work, the existence result yields Dancer-type partially periodic ground states at every sufficiently small mass, answering the open question left in the author's earlier energy-critical work. Together with the intercritical theory also developed by the author, this gives a unified variational treatment of Dancer-type solutions from the mass-supercritical regime up to the energy-critical endpoint.

The strict sub-bubbling estimate

A standard route to strict inequality below the Euclidean bubble level is the Brezis–Nirenberg mechanism, in which a lower-order focusing perturbation lowers the energy of bubble test functions. That mechanism is unavailable here: the equation contains only the critical power, with no lower-order term whose sign could be exploited. The key observation of the paper is that the bounded periodic direction itself supplies the missing energy drop.

The test functions are truncated Aubin–Talenti bubbles. Writing $D=d+1$8 for the Sobolev optimizer on $D=d+1$9, one places a rescaled copy of $2^*=2D/(D-2)$0 into a single period of $2^*=2D/(D-2)$1; in rescaled coordinates this amounts to restricting $2^*=2D/(D-2)$2 to the slab $2^*=2D/(D-2)$3 and discarding the tails $2^*=2D/(D-2)$4, with an additional $2^*=2D/(D-2)$5-cutoff included only to guarantee uniform $2^*=2D/(D-2)$6-admissibility in low dimensions.

The quantitative core is a tail-asymptotic comparison. If $2^*=2D/(D-2)$7 denote the gradient tails of $2^*=2D/(D-2)$8 outside the slab and $2^*=2D/(D-2)$9 its nonlinear tail, then

$\mathbb R^n$0

That is, truncation removes gradient energy at order $\mathbb R^n$1 but nonlinear energy only at order $\mathbb R^n$2, so the net effect after projection back onto $\mathbb R^n$3 is a strict energy deficit. Concretely, the truncated profile satisfies $\mathbb R^n$4 and

$\mathbb R^n$5

for some $\mathbb R^n$6 and all large $\mathbb R^n$7. The projection parameter $\mathbb R^n$8 solving $\mathbb R^n$9 satisfies $c>0$0, and a mean-value argument shows the projection changes the energy only by $c>0$1, preserving the strict inequality.

The masses $c>0$2 tend to zero as $c>0$3 — via distinct estimates in the cases $c>0$4, $c>0$5 (where a logarithmic factor appears), and $c>0$6 (where the $c>0$7-cutoff is essential). Monotonicity of $c>0$8 then transfers the strict bound from arbitrarily small masses to every $c>0$9. This completes the proof of the strict sub-bubbling estimate.

Compactness and existence

With the strict bound in hand, compactness follows the standard concentration–compactness scheme, using the static critical profile decomposition on $$E(u)=\frac12\|\nabla_{x,y}u\|_2^2-\frac{D-2}{2D}\|u\|_{2^*}^{2^*}$$0 (the time-zero version of the Hani–Pausader decomposition), which separates a minimizing sequence into scale-one profiles, Euclidean concentrating bubbles, and a remainder small in $$E(u)=\frac12\|\nabla_{x,y}u\|_2^2-\frac{D-2}{2D}\|u\|_{2^*}^{2^*}$$1.

The non-vanishing lemma proceeds by contradiction: if all $$E(u)=\frac12\|\nabla_{x,y}u\|_2^2-\frac{D-2}{2D}\|u\|_{2^*}^{2^*}$$2-translations of a minimizing sequence converge weakly to zero, no scale-one profiles can occur (a nonzero scale-one profile would produce a nonzero weak limit after translation), so all profiles are Euclidean. The Pythagorean expansions of the semivirial identity force at least one Euclidean profile $$E(u)=\frac12\|\nabla_{x,y}u\|_2^2-\frac{D-2}{2D}\|u\|_{2^*}^{2^*}$$3 to satisfy $$E(u)=\frac12\|\nabla_{x,y}u\|_2^2-\frac{D-2}{2D}\|u\|_{2^*}^{2^*}$$4, where $$E(u)=\frac12\|\nabla_{x,y}u\|_2^2-\frac{D-2}{2D}\|u\|_{2^*}^{2^*}$$5 and $$E(u)=\frac12\|\nabla_{x,y}u\|_2^2-\frac{D-2}{2D}\|u\|_{2^*}^{2^*}$$6. An anisotropic Sobolev inequality, optimized over the ratio of $$E(u)=\frac12\|\nabla_{x,y}u\|_2^2-\frac{D-2}{2D}\|u\|_{2^*}^{2^*}$$7- and $$E(u)=\frac12\|\nabla_{x,y}u\|_2^2-\frac{D-2}{2D}\|u\|_{2^*}^{2^*}$$8-scalings, then gives

$$E(u)=\frac12\|\nabla_{x,y}u\|_2^2-\frac{D-2}{2D}\|u\|_{2^*}^{2^*}$$9

contradicting the strict bound. Hence some translated subsequence has a nonzero weak limit $V(c)=\{u: M(u)=c,\ Q(u)=0\}$0.

The remainder of the argument is a Brezis–Lieb splitting plus Lagrange multiplier scheme. One first shows $V(c)=\{u: M(u)=c,\ Q(u)=0\}$1 (otherwise projecting the defect $V(c)=\{u: M(u)=c,\ Q(u)=0\}$2 onto $V(c)=\{u: M(u)=c,\ Q(u)=0\}$3 contradicts minimality since $V(c)=\{u: M(u)=c,\ Q(u)=0\}$4 for $V(c)=\{u: M(u)=c,\ Q(u)=0\}$5), then $V(c)=\{u: M(u)=c,\ Q(u)=0\}$6 (otherwise projecting $V(c)=\{u: M(u)=c,\ Q(u)=0\}$7 itself strictly lowers the energy), hence $V(c)=\{u: M(u)=c,\ Q(u)=0\}$8 and $V(c)=\{u: M(u)=c,\ Q(u)=0\}$9 is an optimizer for $M(u)=\|u\|_2^2$0 with $M(u)=\|u\|_2^2$1, satisfying $M(u)=\|u\|_2^2$2. Testing the Euler–Lagrange equation against $M(u)=\|u\|_2^2$3 and differentiating along the mass-changing scaling $M(u)=\|u\|_2^2$4 shows $M(u)=\|u\|_2^2$5. The case $M(u)=\|u\|_2^2$6 is excluded by lifting $M(u)=\|u\|_2^2$7 periodically to $M(u)=\|u\|_2^2$8: regularity and the strong maximum principle give a positive $M(u)=\|u\|_2^2$9 solution of the zero-frequency critical equation, which the Caffarelli–Gidas–Spruck classification forces to be an Aubin–Talenti bubble — impossible, since bubbles are not periodic. Finally, mass conservation ($Q(u)=\|\nabla_xu\|_2^2-\frac dD\|u\|_{2^*}^{2^*}$0) follows by contradiction: if $Q(u)=\|\nabla_xu\|_2^2-\frac dD\|u\|_{2^*}^{2^*}$1 were constant on $Q(u)=\|\nabla_xu\|_2^2-\frac dD\|u\|_{2^*}^{2^*}$2, then $Q(u)=\|\nabla_xu\|_2^2-\frac dD\|u\|_{2^*}^{2^*}$3 would be a local minimizer of $Q(u)=\|\nabla_xu\|_2^2-\frac dD\|u\|_{2^*}^{2^*}$4, forcing $Q(u)=\|\nabla_xu\|_2^2-\frac dD\|u\|_{2^*}^{2^*}$5 and hence $Q(u)=\|\nabla_xu\|_2^2-\frac dD\|u\|_{2^*}^{2^*}$6, contradicting $Q(u)=\|\nabla_xu\|_2^2-\frac dD\|u\|_{2^*}^{2^*}$7. Positivity follows from the strong maximum principle.

Limitations and open questions

The construction is purely variational and establishes existence of normalized ground states for every mass, but it does not address uniqueness or qualitative structure of the optimizers beyond positivity and $Q(u)=\|\nabla_xu\|_2^2-\frac dD\|u\|_{2^*}^{2^*}$8-dependence at small mass; whether Dancer-type ground states exist for all masses in the sense of nontrivial $Q(u)=\|\nabla_xu\|_2^2-\frac dD\|u\|_{2^*}^{2^*}$9-dependence above the threshold $\mathbb X=\mathbb R^d_x\times\mathbb T_y$00 remains open. The strict sub-bubbling mechanism relies on the specific tail-order separation $\mathbb X=\mathbb R^d_x\times\mathbb T_y$01 versus $\mathbb X=\mathbb R^d_x\times\mathbb T_y$02 for the Aubin–Talenti profile in one compact direction; the paper does not treat product geometries with multiple torus factors or different compact dimensions, nor does it address stability of the resulting ground states or their dynamical role for the associated focusing energy-critical NLS flow. The threshold constant $\mathbb X=\mathbb R^d_x\times\mathbb T_y$03 itself is not identified quantitatively.

Conclusion

The paper resolves the endpoint compactness gap in the semivirial-vanishing program on $\mathbb X=\mathbb R^d_x\times\mathbb T_y$04 by proving a strict sub-bubbling estimate below the Euclidean Sobolev threshold without any Brezis–Nirenberg-type lower-order perturbation. The energy drop is produced entirely by the bounded periodic direction, through the order separation between the gradient tail ($\mathbb X=\mathbb R^d_x\times\mathbb T_y$05) and the nonlinear tail ($\mathbb X=\mathbb R^d_x\times\mathbb T_y$06) of a slab-truncated Aubin–Talenti bubble. Concentration–compactness then yields, for every prescribed mass, a positive normalized ground state of the positive-frequency critical Lane–Emden equation, providing an energetic alternative to Dancer's bifurcation construction and completing a unified variational theory of Dancer-type solutions up to the energy-critical exponent.