Positive ground states for nonlinear Schrödinger–Kirchhoff equations with periodic potential or potential well in R 3 $\mathbf{R}^{3}$

Abstract

Abstract This work is devoted to the nonlinear Schrödinger–Kirchhoff-type equation − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) , in R 3 , $$ - \biggl( a+b \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2} \,\text{d}x \biggr) \Delta u+V(x)u=f(x,u), \quad \text{in } \mathbb{R}^{3}, $$ where a > 0 $a>0$ , b ≥ 0 $b\geq 0$ , the nonlinearity f ( x , ⋅ ) $f(x,\cdot )$ 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.

Keywords

• Schrödinger–Kirchhoff equations
• Ground states
• Potential well
• Variational methods