JournalsrmiVol. 29, No. 3pp. 997–1020

Infinitely many nonradial solutions for the Hénon equation with critical growth

  • Juncheng Wei

    University of British Columbia, Vancouver, Canada
  • Shusen Yan

    University of New England, Armidale, Australia
Infinitely many nonradial solutions for the  Hénon equation with critical growth cover

Abstract

We consider the following Hénon equation with critical growth:

(){Δu=yαuN+2N2,  u>0,yB1(0),u=0,on B1(0),(*) \begin{cases} - \Delta u = |y|^\alpha \, u^{\frac{N+2}{N-2}},\; u>0, & y\in B_1(0) , \\ u=0, &\text{on } \partial B_1(0), \end{cases}

where α>0\alpha>0 is a positive constant, B1(0)B_1(0) is the unit ball in RN\mathbb{R}^N, and N4N\ge 4. Ni [9] proved the existence of a radial solution and Serra [12] proved the existence of a nonradial solution for α\alpha large and N4N \geq 4. In this paper, we show the existence of a nonradial solution for any α>0\alpha>0 and N4N \geq 4. Furthermore, we prove that equation (*) has infinitely many nonradial solutions, whose energy can be made arbitrarily large.

Cite this article

Juncheng Wei, Shusen Yan, Infinitely many nonradial solutions for the Hénon equation with critical growth. Rev. Mat. Iberoam. 29 (2013), no. 3, pp. 997–1020

DOI 10.4171/RMI/747