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In this paper, we consider the following quasilinear Schrödinger equation: −Δu+V(x)u+κ2Δ(u2)u=K(x)f(u),x∈RN,-\Delta u+V\left(x)u+\frac{\kappa }{2}\Delta \left({u}^{2})u=K\left(x)f\left(u),\hspace{1.0em}x\in {{\mathbb{R}}}^{N}, where N≥3N\ge 3, κ>0\kappa \gt 0, f∈C(R,R)f\in {\mathcal{C}}\left({\mathbb{R}},{\mathbb{R}}), V(x)V\left(x) and K(x)K\left(x) are positive continuous potentials. Under given conditions, by changing variables and truncation argument, the energy of ground state solutions of the Nehari type is achieved. We also prove the existence of ground state sign-changing solutions for the aforementioned equation. Our results are the generalization work of M. B. Yang, C. A. Santos, and J. Z. Zhou, Least action nodal solution for a quasilinear defocusing Schrödinger equation with supercritical nonlinearity, Commun. Contemp. Math. 21 (2019), no. 5, 1850026, https://doi.org/10.1142/S0219199718500268.

In this paper, we study the following generalized Kadomtsev-Petviashvili equation ut+uxxx+(h(u))x=Dx−1Δyu,{u}_{t}+{u}_{xxx}+{\left(h\left(u))}_{x}={D}_{x}^{-1}{\Delta }_{y}u, where (t,x,y)∈R+×R×RN−1\left(t,x,y)\in {{\mathbb{R}}}^{+}\times {\mathbb{R}}\times {{\mathbb{R}}}^{N-1}, N≥2N\ge 2, Dx−1f(x,y)=∫−∞xf(s,y)ds{D}_{x}^{-1}f\left(x,y)={\int }_{-\infty }^{x}f\left(s,y){\rm{d}}s, ft=∂f∂t{f}_{t}=\frac{\partial f}{\partial t}, fx=∂f∂x{f}_{x}=\frac{\partial f}{\partial x} and Δy=∑i=1N−1∂2∂yi2{\Delta }_{y}={\sum }_{i=1}^{N-1}\frac{{\partial }^{2}}{{\partial }_{{y}_{i}}^{2}}. We get the existence of infinitely many nontrivial solutions under certain assumptions in bounded domain without Ambrosetti-Rabinowitz condition. Moreover, by using the method developed by Jeanjean [13], we establish the existence of ground state solutions in RN{{\mathbb{R}}}^{N}.

In this paper, we study the following Schrödinger–Kirchhoff type equation: (a + b ∬ R 2 N | u (x) − u (y) | p | x − y | N + p s d x d y) (− Δ) p s u + λ V (x) | u | p − 2 u = h (x) | u | r − 2 u + g (x) | u | q − 2 u , x ∈ R N , where s ∈ (0 , 1) , 2 ≤ p < ∞ , N>ps, 1 < q < p < r < p s ∗ and a , λ > 0 , b ≥ 0 are three real parameters. By using the mountain pass theorem and a variant version of Ekeland's variational principle in Zhong [A generalization of Ekeland's variational principle and application to the study of relation between the weak P.S. condition and coercivity. Nonlinear Anal. 1997;29:1421–1431], we get some results. Firstly, we obtain a positive energy solution u b , λ + by a truncated functional and discuss their asymptotical behavior for λ → + ∞ when p0, we obtain a positive energy solution u λ + and discuss their asymptotical behavior for λ → + ∞ when 2 p ≤ r < p s ∗ , where p s ∗ = N p N − s p . Moreover, we obtain other asymptotical behavior of positive energy solution when p < r < p s ∗ . Finally, we get a negative energy solution and the non-existence of the non-trivial solutions. [ABSTRACT FROM AUTHOR]