The isocritical regime for mixed local-nonlocal $(p,q)$ Laplacian: existence of ground state, and decay estimates

Abstract

We study the mixed local-nonlocal operator ${\mathcal{L}}_{p,q}:=-{\Delta }_p+(-\Delta {)}_q^s$ in the isocritical regime $p^{\ast }=q_s^{\ast }$, i.e. $1-N/p=s-N/q$, under which both operators become critical for the same nonlinearity. We consider \[ -Δ_p u + (-Δ)_q^s u = |u|^{p^*-2}u \qquad \text{in } \mathbb{R}^N, \] with $N\ge 2$, $1<p<N$, $0<s<1$, $1<sq<N$. In this regime the energy space reduces to ${\mathcal{D}}_0^{1,p}({\mathbb{R}}^N)$, and both best Sobolev constants enter the variational structure simultaneously. We prove: $(i)$ existence of a nonnegative radial ground state via Nehari manifold methods and a double-threshold concentration-compactness analysis; $(ii)$ a logarithmic energy estimate, weak comparison principle, and strong maximum principle for all admissible exponents; $(iii)$ a weak Harnack inequality; and $(iv)$ sharp two-sided decay $U(x)\asymp |x{|}^{-(N-p)/(p-1)}$ for positive radial solutions, matching the fundamental solution of the $p$-Laplacian.