Quasilinear Schrödinger equations with superlinear terms describing the Heisenberg ferromagnetic spin chain
Quasilinear Schrödinger equations with superlinear terms describing the Heisenberg ferromagnetic spin chain
Abstract In this paper, we consider a model problem arising from a classical planar Heisenberg ferromagnetic spin chain: $$ -\Delta u+V(x)u-\frac{u}{\sqrt{1-u^{2}}}\Delta \sqrt{1-u^{2}}=c \vert u \vert ^{p-2}u,\quad x\in \mathbb{R}^{N}, $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>−</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mi>u</mml:mi> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mi>u</mml:mi> <mml:msqrt> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> …