Ground State Solutions for Quasilinear Schrödinger Equations with Critical Growth and Lower Power Subcritical Perturbation

Abstract

Abstract We study the following generalized quasilinear Schrödinger equation: - ( g 2 ⁢ ( u ) ⁢ ∇ ⁡ u ) + g ⁢ ( u ) ⁢ g ′ ⁢ ( u ) ⁢ | ∇ ⁡ u | 2 + V ⁢ ( x ) ⁢ u = h ⁢ ( u ) , x ∈ ℝ N , -(g^{2}(u)\nabla u)+g(u)g^{\prime}(u)|\nabla u|^{2}+V(x)u=h(u),\quad x\in% \mathbb{R}^{N}, where N ≥ 3 {N\geq 3} , g : ℝ → ℝ + {g\colon\mathbb{R}\rightarrow\mathbb{R}^{+}} is an even differentiable function such that g ′ ⁢ ( t ) ≥ 0 {g^{\prime}(t)\geq 0} for all t ≥ 0 {t\geq 0} , h ∈ C 1 ⁢ ( ℝ , ℝ ) {h\in C^{1}(\mathbb{R},\mathbb{R})} is a nonlinear function including critical growth and lower power subcritical perturbation, and the potential V ⁢ ( x ) : ℝ N → ℝ {V(x)\colon\mathbb{R}^{N}\rightarrow\mathbb{R}} is positive. Since the subcritical perturbation does not satisfy the (AR) condition, the standard variational method cannot be used directly. Combining the change of variables and the monotone method developed by Jeanjean in [L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on 𝐑 N {\mathbf{R}}^{N} , Proc. Roy. Soc. Edinburgh Sect. A 129 1999, 4, 787–809], we obtain the existence of positive ground state solutions for the given problem.