SEMILINEAR NONLOCAL ELLIPTIC EQUATIONS WITH CRITICAL AND SUPERCRITICAL EXPONENTS

We study the problem { (-Delta)(s) u = u(p) - u(q) in R-N,R- u is an element of H-s (R-N) boolean AND Lq+1 (R-N), u > 0 in R-N, where s is an element of (0; 1) is a fixed parameter, (-Delta)(s) is the fractional Laplacian in R-N, q > p >= N+2s/N-2s and N > 2s. For every s is an element of(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 is an element of (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.