On the structure of solutions to a class of quasilinear elliptic Neumann problems. Part II

Chunshan Zhao; Yi Li
January 2010
Calculus of Variations & Partial Differential Equations;Jan2010, Vol. 37 Issue 1/2, p237
Academic Journal
We continue our work (Y. Li, C. Zhao in J Differ Equ 212:208�233, 2005) to study the structure of positive solutions to the equation e m? m u - u m-1 + f( u) = 0 with homogeneous Neumann boundary condition in a smooth bounded domain of $${\mathbb{R}^{N}\,\left(N\geq 2\right)}$$. First, we study subcritical case for 2 < m < N and show that after passing by a sequence positive solutions go to a constant in C1, a sense as e ? 8. Second, we study the critical case for 1 < m < N and prove that there is a uniform upper bound independent of $${\varepsilon \in \lbrack 1,\infty)}$$ for the least-energy solutions. Third, we show that in the critical case for 1 < m = 2 the least energy solutions must be a constant if e is sufficiently large and for 2 < m < N the least energy solutions go to a constant in C1, a sense as e ? 8.


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