Multiple Boundary Concentrating Solutions to Dirichlet Problem of Hénon Equation

Abstract

Let Ω be the unit ball centered at the origin in R N (N ≥ 4), 2 * = 2N N −2 , τ > 0, ε > 0. We study the following problemBy a constructive argument, we prove that for any k = 1, 2, · · ·, if ε is small enough, then the above problem has positive a solution uε concentrating at k distinct points which tending to the boundary of Ω as ε goes to 0 + .

Asymptotic behavior on the Hénon equation with supercritical exponent

“…After that, there are many results on problem (1.1) which are mainly on constructing the concentration solutions when the parameter p → 2 * or α → +∞. We can refer to [6][7][8][9][10] and the references therein. In [11,12], solutions with partially radial symmetry were verified to exist.…”

Multiple solutions for a Hénon-like equation on the annulus

“…Indeed, multiplying Eq. (22) by a smooth function ψ with supp ψ Ω and integrate, we obtain Ω ∇u α ∇ψ dx = Ω S p/2 α,p Ψ α u p−1 α ψ dx → 0, α → +∞, since, by (13), S p/2 α,p Ψ α → 0 uniformly on supp ψ and u α is uniformly bounded in L q for 1 q < 2 * . Hence Ω ∇u · ∇ϕ dx = 0 for all ϕ ∈ C ∞ 0 (Ω).…”

“…This kind of result was improved in [13], where multibump solutions for the Hénon equation with almost critical Sobolev exponent p are found, by means of a finite-dimensional reduction. These solution are not radial, though they are invariant under the action of suitable subgroups of O(N), and they concentrate at boundary points of the unit ball of R N as p → 2 * .…”

Multi-peak solutions for the Hénon equation with slightly subcritical growth

“…We would like to thank the referee for pointing out to us the very recent paper [17], where results very similar to ours are proved in great generality.…”