Concentrating solutions for singularly perturbed fractional (N/s)-Laplacian equations with nonlocal reaction

Abstract

Abstract This paper is concerned with the following fractional (N/s)-Laplacian Choquard equation: ε N ⁢ ( - Δ ) N / s s ⁢ u + V ⁢ ( x ) ⁢ | u | N s - 2 ⁢ u = ε μ ⁢ ( 1 | x | N - μ ∗ F ⁢ ( u ) ) ⁢ f ⁢ ( u ) , x ∈ ℝ N , \varepsilon^{N}(-\Delta)_{N/s}^{s}u+V(x)\lvert u\rvert^{\frac{N}{s}-2}u=% \varepsilon^{\mu}\Bigl{(}\frac{1}{\lvert x\rvert^{N-\mu}}\ast F(u)\Bigr{)}f(u)% ,\quad x\in{\mathbb{R}}^{N}, where ( - Δ ) N / s s {(-\Delta)_{N/s}^{s}} denotes the (N/s)-Laplacian operator, 0 < μ < N {0<\mu<N} , and V and f are continuous real functions satisfying some mild assumptions. Applying the weak growth conditions on the exponential critical nonlinearity f and without using the strictly monotone condition, we use some refined analysis and develop the arguments in the existing results to establish the existence of the ground state solution of the fractional (N/s)-Laplacian Choquard equation. Moreover, we also study the concentration phenomenon of the ground state solutions. As far as we know, our results seem to be new concerning the fractional (N/s)-Laplacian equation with the Choquard reaction.