Fractional \(p \& q\) Laplacian problems in \(\mathbb{R}^N\) with critical growth (Q2197921)

Summary: We deal with the following nonlinear problem involving fractional \(p\&q\) Laplacians: \[ (-\Delta)_p^s u+(-\Delta)_q^s u+|u|^{p-2}u+|u|^{q-2}u=\lambda h(x) f(u)+|u|^{q_s^*-2}u \quad \text{in } \mathbb{R}^N, \] where \(s \in (0,1), 1 < p < q < \frac{N}{s}, q_s^*=\frac{Nq}{N-sq}, \lambda > 0\) is a parameter, \(h\) is a nontrivial bounded perturbation and \(f\) is a superlinear continuous function with subcritical growth. Using suitable variational arguments and concentration-compactness lemma, we prove the existence of a nontrivial non-negative solution for \(\lambda\) sufficiently large.