We study the a priori estimates, existence/nonexistence of radial sign changing solution, and the Palais-Smale characterisation of the problem −We will also prove the existence of sign changing solution to the Hardy-Sobolev-Mazya equation and the critical Grushin problem.

We study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to second-order interpolation inequalities with weights. More precisely, we deal with the following family of equationswhere N ≥ 5, q > 2, β = N −q(N − 4)/2 and λ ∈ R is smaller than the Rellich constant.

We study the regularity, Palais-Smale characterization and existence/nonexistence of solutions of the Hardy-Sobolev-Maz'yashow different behaviors of PS sequences depending on t = 0 or t > 0.

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.

We study the existence/nonexistence of positive solution to the problem of the type:where Ω is a smooth bounded domain in R N , N ≥ 5, a, b, f are nonnegaive functions satisfying certain hypothesis which we will specify later. µ, λ are positive constants. Under some suitable conditions on functions a, b, f and the constant µ, we show that there exists λ * > 0 such that when 0 < λ < λ * , (P λ ) admits a solution in W 2,2 (Ω) ∩ W 1,2 0 (Ω) and for λ > λ * , it does not have any solution in W 2,2 (Ω) ∩ W 1,2 0 (Ω). Moreover as λ ↑ λ * , minimal positive solution of (P λ ) converges in W 2,2 (Ω) ∩ W 1,2 0 (Ω) to a solution of (P λ * ). We also prove that there existsλ * < ∞ such that λ * ≤λ * and for λ >λ * , the above problem (P λ ) does not have any solution even in the distributional sense/very weak sense and there is complete blow-up. Under an additional integrability condition on b, we establish the uniqueness of positive solution of (P λ * ) in W 2,2 (Ω)∩ W 1,2 0 (Ω).Keywords: Semilinear biharmonic equations, singular potential, navier boundary condition, existence/nonexistence results, blow-up phenomenon, stability results, uniqueness of extremal solution 1 1 2010 Mathematics Subject Classification: 35B09, 35B25, 35B35, 35G30, 35J91.