Infinitely many solutions for Hamiltonian system with critical growth

Abstract

In this article, we consider the following elliptic system of Hamiltonian-type on a bounded domain:−Δu=K1(∣y∣)∣v∣p−1v,inB1(0),−Δv=K2(∣y∣)∣u∣q−1u,inB1(0),u=v=0on∂B1(0),\left\{\begin{array}{ll}-\Delta u={K}_{1}\left(| y| ){| v| }^{p-1}v,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ -\Delta v={K}_{2}\left(| y| ){| u| }^{q-1}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ u=v=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}_{1}\left(0),\end{array}\right. where K1(r){K}_{1}\left(r) and K2(r){K}_{2}\left(r) are positive bounded functions defined in [0,1]\left[0,1], B1(0){B}_{1}\left(0) is the unit ball in RN{{\mathbb{R}}}^{N}, and (p,q)\left(p,q) is a pair of positive numbers lying on the critical hyperbola 1p+1+1q+1=N−2N.\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}. Under some suitable further assumptions on the functions K1(r){K}_{1}\left(r) and K2(r){K}_{2}\left(r), we prove the existence of infinitely many nonradial positive solutions whose energy can be made arbitrarily large. Our proof is based on the reduction method. The most ingredients of the article are using the Green representation and estimating the Green function and its regular part very carefully. For this purpose, some more extra ideas and techniques are needed. We believe that our method and techniques can be applied to other related problems.

Keywords

• hamiltonian system
• critical hyperbola
• infinitely many solutions
• reduction method
• 35a01
• 35b33
• 35j57