Semilinear Elliptic Equation with Biharmonic Operator and Multiple Critical Nonlinearities

Abstract

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.

Borderline Variational Problems Involving Fractional Laplacians and Critical Singularities

“…See [11] and references therein for the basics on the fractional Laplacian. Problems involving two non-linearities have been studied in the case of local operators such as the Laplacian −∆, the p-Laplacian −∆ p and the Biharmonic operator ∆ 2 (See [5], [17], [25] and [36]). Problem (1.1) above is the non-local counterpart of the one studied by Filippucci-Pucci-Robert in [17], who treated the case of the p-Laplacian in an equation involving both the Sobolev and the Hardy-Sobolev critical exponents.…”

“…The standard strategy to construct weak solutions of (1.1) is to find critical points of the corresponding functional on H α 2 (R n ). However, (1.1) is invariant under the following conformal one parameter transformation group, 5) which means that the convergence of Palais-Smale sequences is not a given. As it was argued in [17], there is an asymptotic competition between the energy carried by the two critical nonlinearities.…”

Entire Solutions for a Class of Fourth-Order Semilinear Elliptic Equations with Weights

“…In particular, in the case of the pure biharmonic operator, we quote, e.g., [1][2][3][4]6,[15][16][17]19,20]. Equations like (1.1) arise in a natural way from variational inequalities of the form…”

Improved Sobolev inequalities and critical problems

“…Nonlinear elliptic problems with partial Hardy term arise in astrophysics, it was considered in [1]. If p = 2, problem (1.1) has been studied in [2,4,5,6,7,12,16,17] etc for subcritical and critical cases by the variational method. In particular, it was shown in [17] that the functional related to problem (1.1) with the fractional Laplacian satisfies the (P S) c condition, where c is in certain interval.…”