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We consider the problem ǫ 2 ∆u − V (y)u + u p = 0, u > 0in Ω, ∂u ∂ν = 0 on ∂Ω,where Ω is a bounded domain in R 2 with smooth boundary, the exponent p > 1, ǫ > 0 is a small parameter, V is a uniformly positive, smooth potential onΩ, and ν denotes the outward normal of ∂Ω. Let Γ be a curve intersecting orthogonally with ∂Ω at exactly two points and dividing Ω into two parts. Moreover, Γ satisfies stationary and non-degeneracy conditions with respect to the functional Γ V σ , where σ = p+1 p−1 − 1 2 . We prove the existence of a solution u ǫ concentrating along the whole of Γ, exponentially small in ǫ at any positive distance from it, provided that ǫ is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by A. Ambrosetti, A. [4]).

We consider the nonlinear problem of inhomogeneous Allen-Cahn equation 2 ∆u + V (y) (1 − u 2) u = 0 in Ω, ∂u ∂ν = 0 on ∂Ω,

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This paper deals with the following non-linear equation with a fractional Laplacian operator and almost critical exponents: \[ (-\Delta)^{s} u=K(|y'|,y'')u^{({N+2s})/(N-2s)\pm\epsilon},\quad u > 0,\quad u\in D^{1,s}(\mathbb{R}^{N}), \] where N ⩾ 4, 0 < s < 1, (y′, y″) ∈ ℝ2 × ℝN−2, ε > 0 is a small parameter and K(y) is non-negative and bounded. Under some suitable assumptions of the potential function K(r, y″), we will use the finite-dimensional reduction method and some local Pohozaev identities to prove that the above problem has a large number of bubble solutions. The concentration points of the bubble solutions include a saddle point of K(y). Moreover, the functional energies of these solutions are in the order $\epsilon ^{-(({N-2s-2})/({(N-2s)^2})}$ .