Infinitely many solutions for Hamiltonian system with critical growth

Abstract

Abstract In this article, we consider the following elliptic system of Hamiltonian-type on a bounded domain: − Δ u = K 1 ( ∣ y ∣ ) ∣ v ∣ p − 1 v , in B 1 ( 0 ) , − Δ v = K 2 ( ∣ y ∣ ) ∣ u ∣ q − 1 u , in B 1 ( 0 ) , u = v = 0 on ∂ B 1 ( 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 K 1 ( r ) {K}_{1}\left(r) and K 2 ( r ) {K}_{2}\left(r) are positive bounded functions defined in [ 0 , 1 ] \left[0,1] , B 1 ( 0 ) {B}_{1}\left(0) is the unit ball in R N {{\mathbb{R}}}^{N} , and ( p , q ) \left(p,q) is a pair of positive numbers lying on the critical hyperbola 1 p + 1 + 1 q + 1 = N − 2 N . \frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}. Under some suitable further assumptions on the functions K 1 ( r ) {K}_{1}\left(r) and K 2 ( 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.