Existence of non-radial entire solutions for the Hénon equation beyond even exponents

Qinfeng Jiang; Jingang Xiong

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arXiv · 2606.31670

Existence of non-radial entire solutions for the Hénon equation beyond even exponents

Qinfeng Jiang·Jingang Xiong

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Abstract

This paper is concerned with the existence of non-radial positive classical solutions for the critical Hénon equation \[ -Δu=|x|^αu^{\frac{N+2+2α}{N-2}} \qquad \text{in }\mathbb R^N, \] where $α>0$ and $N\ge3$, satisfying the Newtonian-type decay condition at infinity. Gladiali, Grossi and Neves (2013) proved existence for the discrete sequence $α_k=2(k-1)$, $k\in\mathbb N$, and conjectured that non-radial solutions may exist only at these special values. We disprove this conjecture by establishing existence for a continuum of exponents near each $α_k$: for every even $k>\frac{N-2}{2}$, non-radial solutions persist for parameters $α$ close to, and different from, $α_k$. We recast the problem as a semilinear elliptic equation with Sobolev-supercritical exponent on the cylinder via the Emden--Fowler change of variables. Our argument is formulated directly on the cylindrical domain, thereby streamlining the characterization of the kernel of the linearized operator via Pöschl--Teller spectral theory, avoiding the ball-exhaustion technique employed in the original work, and allowing us to compute the bifurcation slope and verify the non-verticality condition.

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Qinfeng Jiang, Jingang Xiong. 2026-06-30. Existence of non-radial entire solutions for the Hénon equation beyond even exponents. https://arxiv.org/abs/2606.31670

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