We study the problemwhere s ∈ (0, 1) is a fixed parameter, (−∆) s is the fractional Laplacian in R N , q > p ≥ N+2s N−2s and N > 2s. For every s ∈ (0, 1), we establish regularity results of solutions of above equation (whenever solution exists) and we show that every solution is a classical solution. Next, we derive certain decay estimate of solutions and the gradient of solutions at infinity for all s ∈ (0, 1). Using those decay estimates, we prove Pohozaev type identity in R N and we show that the above problem does not have any solution when p = N+2s N−2s . We also discuss radial symmetry and decreasing property of the solution and prove that when p > N+2s N−2s , the above problem admits a solution . Moreover, if we consider the above equation in a bounded domain with Dirichlet boundary condition, we prove that it admits a solution for every p ≥ N+2s N−2s and every solution is a classical solution.2010 Mathematics Subject Classification. Primary 35B08, 35B40, 35B44.

In this article we study the existence of sign changing solution of the following p-fractional problem with concave-critical nonlinearities:where s ∈ (0, 1) and p ≥ 2 are fixed parameters, 0 < q < p − 1, µ ∈ R + and p * s = Np N−ps . Ω is an open, bounded domain in R N with smooth boundary with N > ps .2010 Mathematics Subject Classification. 47G20, 35J20, 35J60, 35J62.

We study the nonlocal scalar field equation with a vanishing parameterwhere s ∈ (0, 1), N > 2s, q > p > 2 are fixed parameters and ǫ > 0 is a vanishing parameter. For ǫ > 0 small, we prove the existence of a ground state solution and show that any positive solution of (Pǫ) is a classical solution and radially symmetric and symmetric decreasing. We also obtain the decay rate of solution at infinity. Next, we study the asymptotic behavior of ground state solutions when p is subcritical, supercritical or critical Sobolev exponent 2 * = 2N N−2s . For p < 2 * , the solution asymptotically coincides with unique positive ground state solution of (−∆) s u + u = u p . On the other hand, for p = 2 * the asymptotic behaviour of the solutions is given by the unique positive solution of the nonlocal critical Emden-Fowler type equation. For p > 2 * , the solution asymptotically coincides with a ground-state solution of (−∆) s u = u p − u q . Furthermore, using these asymptotic profile of solutions, we prove the local uniqueness of solution in the case p ≤ 2 * .

In this paper we show the uniqueness of the critical point for semi-stable solutions of the problemwhere ⊂ R 2 is a smooth bounded domain whose boundary has nonnegative curvature and f (0) ≥ 0. It extends a result by Cabré-Chanillo to the case where the curvature of ∂ vanishes.

Abstract. We study the fractional Laplacian problemwhereand ε > 0 is a parameter. Here Ω ⊆ R N is a bounded star-shaped domain with smooth boundary and N > 2s. We establish existence of a variational positive solution uε and characterise the asymptotic behaviour of uε as ε → 0. When p = N+2s N−2s, we describe how the solution uε blows up at a interior point of Ω. Furthermore, we prove the local uniqueness of solution of the above problem when Ω is a convex symmetric domain of R N with N > 4s and p = N+2s N−2s.