Positive ground states for nonlinear Schrödinger–Kirchhoff equations with periodic potential or potential well in $\mathbf{R}^{3}$
Wei Chen, Zunwei Fu, Yue Wu
Abstract
Abstract This work is devoted to the nonlinear Schrödinger–Kirchhoff-type equation $$ - \biggl( a+b \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2} \,\text{d}x \biggr) \Delta u+V(x)u=f(x,u), \quad \text{in } \mathbb{R}^{3}, $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:msub> <mml:mo>∫</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:msub> <mml:mo>|</mml:mo> <mml:mi>∇</mml:mi> <mml:mi>u</mml:mi> <mml:msup> <mml:mo>|</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mspace/> <mml:mtext>d</mml:mtext> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <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:mi>f</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:mo>,</mml:mo> <mml:mspace/> <mml:mtext>in </mml:mtext> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>,</mml:mo> </mml:math> where $a>0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> , $b\geq 0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>b</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:math> , the nonlinearity $f(x,\cdot )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo>)</mml:mo> </mml:math> is 3-superlinear and the potential V is either periodic or exhibits a finite potential well. By the mountain pass theorem, Lions’ concentration-compactness principle, and the energy comparison argument, we obtain the existence of positive ground state for this problem without proving the Palais–Smale compactness condition.