2019/03/22 by Tang, Xianhua, Chen, Sitong
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1903.10347
In the present paper, we consider the following singularly perturbed problem: \ -ε2\triangle u+V(x)u=ε-α(Iα*F(u))f(u), · amp; x∈ \RN; u∈ H1(\RN), . where ε>0 is a parameter, N≥ 3, α∈ (0, N), F(t)=∫0tf(s)ds and Iα: \RN→ \R is the Riesz potential. By introducing some new tricks, we prove that the above problem admits a semiclassical ground state solution (ε∈ (0,ε0)) and a ground state solution (ε=1) under the general "Berestycki-Lions assumptions" on the nonlinearity f which are almost necessary, as well as some weak assumptions on the potential V. When ε=1, our results generalize and improve the ones in [V. Moroz, J. Van Schaftingen, T. Am. Math. Soc. 367 (2015) 6557-6579] and [H. Berestycki, P.L. Lions, Arch. Rational Mech. Anal. 82 (1983) 313-345] and some other related literature. In particular, our approach is useful for many similar problems.