Existence of non-radial entire solutions for the H\'enon equation beyond even exponents

Abstract

This paper is concerned with the existence of non-radial positive classical solutions for the critical H\'enon equation \[ -\Delta u=|x|^\alpha u^{\frac{N+2+2\alpha}{N-2}} \qquad \text{in }\mathbb R^N, \] where \(\alpha>0\) and \(N\ge3\), satisfying the Newtonian-type decay condition at infinity. Gladiali, Grossi and Neves (2013) proved existence for the discrete sequence $\alpha_k=2(k-1)$, $k\in\mathbb N$, 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 \(\alpha_k\): for every even $k>\frac{N-2}{2}$, non-radial solutions persist for parameters \(\alpha\) close to, and different from, \(\alpha_k\). 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 P\"oschl--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.