Semi-classical Analysis Around Local Maxima and Saddle Points for Degenerate Nonlinear Choquard Equations
Semi-classical Analysis Around Local Maxima and Saddle Points for Degenerate Nonlinear Choquard Equations
Abstract We study existence of semi-classical states for the nonlinear Choquard equation: $$\begin{aligned} -\varepsilon ^2\Delta v+ V(x)v = {1\over \varepsilon ^\alpha }(I_\alpha *F(v))f(v) \quad \text {in}\ {\mathbb {R}}^N, \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>ε</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>Δ</mml:mi> <mml:mi>v</mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> …