Nodal radial solutions for a zero-mass $(p,q)$-Laplacian equation with Hardy-Hénon type weight

Abstract

We establish the existence of radial nodal solutions for a class of zero-mass $(p,q)$-Laplacian equations in $\mathbb{R}^N$ with a Hardy-Hénon type weight $|x|^α$. The absence of lower-order potential terms motivates the use of a homogeneous radial variational framework adapted to the mixed $p$- and $q$-growth structure. A key ingredient is a weighted Sobolev embedding theorem that establishes continuity and compactness in the appropriate range of exponents and identifies the sharp interval for continuous embedding. Building on these results, we construct sign-changing radial solutions through a variational method based on radial partitions and critical levels. The main result asserts that, for every $k\in\mathbb{N}$, the problem admits a radial weak solution with exactly $k$ nodal zeros. The proof combines minimization on Nehari manifolds over radial subdomains with an optimization procedure that determines the nodal radii and ensures the compatibility of the solution across nodal interfaces.