``Bubble-Tower'' phenomena in a semilinear elliptic equation with mixed Sobolev growth

Juan Campos
Universite de Chile
email: jcampos@dim.uchile.cl

Abstract: We consider the following problem

$\displaystyle \Delta u + u^p+u^q = 0 \in \mathbb{R}^N u > 0 \in \mathbb{R}^N
\lim_{\left\vert x\right\vert \ {\rm to}\ \infty} u(x)\ {\rm to}\ 0$

with $ N/(N-2)< p < p^*=(N+2)/(N-2) < q , N> 2$. We prove that if $ p$ is fixed, and $ q$ is close enough to the critical exponent $ p^*$, then there exists a radial solution which behaves like a superposition of bubbles of different blow-up orders centered at the origin. Similarly when $ q$ is fixed and $ p$ is sufficiently close to the critic, we prove the existence of a radial solution which resembles a superposition of flat bubbles centered at the origin.