High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition

Abstract

In this article, we study the following general Kirchhoff type equation: −M∫R3∣∇u∣2dxΔu+u=a(x)f(u)inR3,-M\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}| \nabla u{| }^{2}{\rm{d}}x\right)\Delta u+u=a\left(x)f\left(u)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3}, where infR+M>0{\inf }_{{{\mathbb{R}}}^{+}}M\gt 0 and ff is a superlinear subcritical term. By using the Pohozǎev manifold, we obtain the existence of high energy solutions of the aforementioned equation without the well-known Ambrosetti-Rabinowitz type condition.

Keywords

• kirchhoff type equation
• high energy solution
• pohozǎev manifold
• variational method
• 35r11
• 47g20
• 35s15
• 35j60