“Bubble-Tower” phenomena in a semilinear elliptic equation with mixed Sobolev growth

In this work we consider the following problem { Δ u + u p + u q = 0 in R N u > 0 in R N lim | x | → ∞ u ( x ) → 0 with N / ( N − 2 ) < p < p ∗ = ( N + 2 ) / ( N − 2 ) < q , N ≥ 3 . 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 critical, we prove the existence of a radial solution which resembles a superposition of flat bubbles centered at the origin.