Nonstandard solutions for a perturbed nonlinear Schrodinger system with small coupling coefficients

作者：An, Xiaoming; Wang, Chunhua^*

来源：Mathematische Nachrichten, 2021, 294(6): 1052-1084.

DOI：10.1002/mana.202000121

• concentrating synchronously
• penalized technique
• Schrodingersystents
• shall coupling coeflicients
• variational methods

In this paper, we consider the following weakly coupled nonlinear Schrodinger system @@@ {-epsilon(2)Delta u(1) + V-1(x)u(1) = vertical bar u(1)vertical bar 2(p-2)u(1) + beta vertical bar u(1)vertical bar(p-2)vertical bar u(2)vertical bar(p)u(1), x is an element of R-N, @@@ -epsilon(2)Delta u(2) + V-2(x)u(2) = vertical bar u(2)vertical bar(2p-2)u(2) + beta vertical bar u(2)vertical bar(p-2)vertical bar u(1)vertical bar(p)u(2), x is an element of R-N, @@@ where epsilon > 0, beta is an element of R is a coupling consta.nt, 2p is an element of (2, 2*) with 2* = 2N/N-2 if N >= 3 and +infinity if N = 1, 2, V-1 and V-2 belong to C (R-N, [0, infinity)). @@@ When p >= 2 and beta > 0 is suitably small, we show that the problem has a family of nonstandard solutions {w(epsilon), = (u(epsilon)(1),u(epsilon)(2)) : 0 < epsilon < epsilon(0)} concentrating synchronously at the common local minimum of V-1 and V-2. All decay rates of V-i(i = 1, 2) are admissible and we can allow that beta > 0 is close to 0 in this paper. Moreover, the location of concentration points is given by local I'ohozaevidentities. Our proofs arc based on variational methods and the penalized technique.