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 $\mathcal{L}_{p,q} := -Δ_p + (-Δ)_q^s$ in the isocritical regime $p^* = q_s^*$, 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 $N \geq 2$, $1 < p < N$, $0 < s < 1$, $1 < sq < N$. 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.