## Tohoku Mathematical Journal 2016 December SECOND SERIES VOL. 68, NO. 4

 Tohoku Math. J. 68 (2016), 591-605

Title CONSTRUCTION OF SIGN-CHANGING SOLUTIONS FOR A SUBCRITICAL PROBLEM ON THE FOUR DIMENSIONAL HALF SPHERE

Author Rabeh Ghoudi and Kamal Ould Bouh

(Received September 22, 2014, revised March 5, 2015)
Abstract. This paper is devoted to studying the nonlinear problem with subcritical exponent $(S_\varepsilon) : -\Delta_g u+2u = K|u|^{2-\varepsilon}u$, in $S^4_+$, ${\partial u}/{\partial\nu} =0$, on $\partial S^4_+,$ where $g$ is the standard metric of $S^4_+$ and $K$ is a $C^3$ positive Morse function on $\overline{S_+^4}$. We construct some sign-changing solutions which blow up at two different critical points of $K$ in interior. Furthermore, we construct sign-changing solutions of $(S_\varepsilon)$ having two bubbles and blowing up at the same critical point of $K$.

Mathematics Subject Classification. Primary 35J20; Secondary 35J60.

Key words and phrases. Critical points, Variational problem, Scalar curvature, Bubble-tower solutions.