In this paper, we study the following generalized Kadomtsev-Petviashvili equation u t + u x x x + ( h ( u ) ) x = D x − 1 Δ y u , {u}_{t}+{u}_{xxx}+{\left(h\left(u))}_{x}={D}_{x}^{-1}{\Delta }_{y}u, where ( t , x , y ) ∈ R + × R × R N − 1 \left(t,x,y)\in {{\mathbb{R}}}^{+}\times {\mathbb{R}}\times {{\mathbb{R}}}^{N-1} , N ≥ 2 N\ge 2 , D x − 1 f ( x , y ) = ∫ − ∞ x f ( s , y ) d s {D}_{x}^{-1}f\left(x,y)={\int }_{-\infty }^{x}f\left(s,y){\rm{d}}s , f t = ∂ f ∂ t {f}_{t}=\frac{\partial f}{\partial t} , f x = ∂ f ∂ x {f}_{x}=\frac{\partial f}{\partial x} and Δ y = ∑ i = 1 N − 1 ∂ 2 ∂ y i 2 {\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 R N {{\mathbb{R}}}^{N} .

中文翻译：

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一类广义Kadomtsev-Petviashvili方程的多重解和基态解

在本文中，我们研究以下广义Kadomtsev-Petviashvili方程ut + uxxx +（h（u））x = D x − 1Δyu，{u} _ {t} + {u} _ {xxx} + {\ left（h \ left（u））} _ {x} = {D} _ {x} ^ {-1} {\ Delta} _ {y} u，其中（t，x，y）∈R +×R ×RN − 1 \ left（t，x，y）\ in {{\ mathbb {R}}} ^ {+} \ times {\ mathbb {R}} \ times {{\ mathbb {R}}} ^ { N-1}，N≥2 N \ ge 2，D x − 1 f（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}和Δy = ∑ i = 1 N − 1 ∂2∂yi 2 {\ Delta} _ {y} = {\ sum} _ {i = 1} ^ {N-1} \ frac {{\ partial} ^ {2}} {{\ partial} _ {{ y} _ {i}} ^ {2}}。在没有Ambrosetti-Rabinowitz条件的有界域中，在某些假设下，我们得到了无穷多个非平凡解的存在。而且，