Existence of single peak solutions for a nonlinear Schrodinger system with coupled quadratic nonlinearity

作者：Yang, Jing^*; Zhou, Ting

来源：Advances in Nonlinear Analysis, 2022, 11(1): 417-431.

DOI：10.1515/anona-2021-0201

• second-harmonic generation
• non-degenerate
• single peak solutions
• reduction method

We are concerned with the following Schrodinger system with coupled quadratic nonlinearity @@@ {-epsilon(2) Delta v + P(x)v = mu vw, x is an element of R-N, @@@ -epsilon(2) Delta w + Q(x)w = mu/2 v(2) + gamma w(2), x is an element of R-N, @@@ v > 0, w>0, v, w is an element of H-1(R-N), @@@ which arises from second-harmonic generation in quadratic media. Here epsilon > 0 is a small parameter, 2 <= N < 6, mu > 0 and mu > gamma, P(x), Q(x) are positive function potentials. By applying reduction method, we prove that if x(0) is a non-degenerate critical point of Delta(P + Q) on some closed N - 1 dimensional hypersurface, then the system above has a single peak solution (v(epsilon), w(epsilon)) concentrating at x(0) for epsilon small enough.

• 单位
  江苏科技大学