Positive solutions for the Schrodinger-Poisson system with steep potential well

In this paper, we consider the following Schrodinger-Poisson system {-Delta u + lambda V(x)u + mu phi u = vertical bar u vertical bar(p-2)u in R-3, -Delta phi = u(2) in R-3, where lambda, mu > 0 are real parameters and 2 < p < 6. Suppose that V(x) represents a potential well with the bottom V-1(0), the system has been widely studied in the case 4 <= p < 6. In contrast, no existence result of solutions is available for the case 2 < p < 4 due to the presence of the nonlocal term phi u. With the aid of the truncation technique and the parameter-dependent compactness lemma, we first prove the existence of positive solutions for lambda large and mu small in the case 2 < p < 4. Then we obtain the nonexistence of nontrivial solutions for lambda large and mu large in the case 2 < p <= 3. Finally, we explore the decay rate of the positive solutions as vertical bar x vertical bar -> infinity as well as their asymptotic behavior as lambda -> infinity and mu -> 0.