2023/06/01 by Guo, Qing, Liu, Junyuan, Peng, Shuangjie
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2306.00811
We consider the following supercritical problem for the Lane-Emden system: \begincases -Δu1=|u2|p-1u2 amp;in D,
-Δu2=|u1|q-1u1 amp;in D,
u1=u2=0 amp;on ∂ D, \endcases where D is a bounded smooth domain in ℝN, N≥4. What we mean by supercritical is that the exponent pair (p,q)∈(1,∞)×(1,∞) satisfies \frac1p+1+\frac1q+1<\fracN-2N. We prove that for some suitable domains D⊂ℝN, there exist positive solutions with layers concentrating along one or several k-dimensional sub-manifolds of ∂ D as \frac1p+1+\frac1q+1 → (n-2)/(n), (n-2)/(n)lt;\frac1p+1+\frac1q+1lt;\fracN-2N, where n:=N-k with 1≤ k≤ N-3. By transforming the original problem \eqrefeq00 into a lower n-dimensional weighted system, we carry out the reduction framework and apply the blow-up analysis. The properties of the ground states related to the limit problem play a crucial role in this process. The corresponding exponent pair (p0,q0), which represents the limit pair of (p,q), lies on the critical hyperbola \frac np0+1+\frac nq0+1=n-2. It is widely recognized that the range of the smaller exponent, say p0, has a profound impact on the solutions, with p0=\frac nn-2 being a threshold. It is worth emphasizing that this paper tackles the problem by considering two different ranges of p0, which is contained in p0>\frac nn-2 and p0<\frac nn-2 respectively. The coupling mechanisms associated with these ranges are completely distinct, necessitating different treatment approaches. This represents the main challenge overcome and the novel element of this study..