We exhibit new concentration phenomena for the equation −ε 2 u + u = u p in a smooth bounded domain ⊆ R 2 and with Neumann boundary conditions. The exponent p is greater than or equal to 2 and the parameter ε is converging to 0. For a suitable sequence ε n → 0 we prove the existence of positive solutions u n concentrating at the whole boundary of or at some component.

In the present paper, we establish the existence of Ground State Solutions for some class of Elliptic problems with Critical Growth in R N for N ≥ 2. Our results complete the study made in Berestycki and Lions (Arch Rat Mech Anal 82: 1983) and Berestycki, Gallouët and Kavian (C R Acad Sci Paris Ser I Math 297: [307][308][309][310] 1984), in the sense that, in those papers only the subcritical growth was considered.

No abstract

Let λ * > 0 denote the largest possible value of λ such that 8 < :has a solution, where B is the unit ball in R N and n is the exterior unit normal vector. We show that for λ = λ * this problem possesses a unique weak solution u * . We prove that u * is smooth if N ≤ 12 and singular when N ≥ 13, in which case u * (r) = −4 log r + log(8(N − 2)(N − 4)/λ * ) + o(1) as r → 0. We also consider the problem with general constant Dirichlet boundary conditions.1991 Mathematics Subject Classification. Primary 35J65, Secondary 35J40 .

No abstract