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We consider a linearized partial data Calderón problem for biharmonic operators extending the analogous result for harmonic operators (Dos Santos Ferreira et al., 2009). We construct special solutions and utilize Segal-Bargmann transform to recover lower order perturbations. [ABSTRACT FROM AUTHOR]

Given three measurable functions V r ≥ 0 , K r > 0 and Q r ≥ 0 , r > 0 , we consider the bilaplacian equation Δ 2 u + V (| x |) u = K (| x |) f (u) + Q (| x |) in R N and we find radial solutions thanks to compact embeddings of radial spaces of Sobolev functions into sum of weighted Lebesgue spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem to the density-dependent liquid crystal system in R 3 . We establish the local existence and uniqueness of strong solutions to this system. In order to overcome the difficulties caused by the high order coupling terms, a biharmonic regularization of the system, as an auxiliary system, is introduced, and we make full use of the intrinsic cancellation properties between the high order coupling terms. [ABSTRACT FROM AUTHOR]

We consider a two-obstacle problem for the parabolic biharmonic equation in a bounded domain. We prove long time existence of solutions via an implicit time discretization scheme, and we investigate the regularity properties of solutions. [ABSTRACT FROM AUTHOR]

This note collects results related to parabolic Muckenhoupt A p weights for a doubly nonlinear parabolic partial differential equation. A general approach is proposed, which extends the theory beyond the quadratic growth case. In particular, the natural parabolic geometry for the equation and the unavoidable time lag are incorporated in the definitions and results. The limiting Muckenhoupt conditions of A ∞ type are also discussed and several open questions are posed. [ABSTRACT FROM AUTHOR]

Abstract: We consider some qualitative properties of entire radial solutions of a semilinear biharmonic equation with negative exponent: where and . In this paper we study the asymptotic rate of any regular positive radial entire solution of (0.1) at which the limit is attained. [Copyright &y& Elsevier]

Abstract: Using the estimates of the principal eigenvalue of the -biharmonic operator on the ball from below and from above, we provide its asymptotic analysis as . [Copyright &y& Elsevier]

We establish the boundedness properties of the Neumann extension operator in half-space in the setting of Morrey–Lorentz spaces. As a by-product we derive estimates on the restriction operator in block spaces. Direct application to solvability of a fourth order nonlinear equation related to higher order boundary conformally invariant problem is considered. By employing a nonvariational approach, we obtain a unique solution under suitable smallness conditions on the boundary data and potentials prescribed in Morrey–Lorentz spaces which allow for singular functions. Moreover, these solutions are shown to be C ∞ in the interior, C l o c 2 , μ ( R + n + 1 ¯) , μ ∈ (0 , 1) in some special case and satisfy interesting qualitative properties including positivity. The results are extended to a larger class of problems involving the polyharmonic operator. In particular, the higher order nonlocal equation (− Δ) 2 m − 1 2 v = K (x) | v | σ − 1 v + H (x) v + g in R n , m ∈ N for 2 ≤ 2 m < n + 1 is solvable in a suitable Morrey–Lorentz space and admits positive solutions whenever σ > n / (n − 2 m + 1). We also prove that no positive solution in the latter framework exists if σ ≤ n / (n − 2 m + 1). [ABSTRACT FROM AUTHOR]

In this paper, we consider the asymptotic behavior of positive solutions of the biharmonic equation Δ 2 u = u p in B 2 ∖ { 0 } with an isolated singularity, where the punctured ball B 2 ∖ { 0 } ⊂ R n with n > 5 and p > (n + 4) / (n − 4). We classify isolated singularities of positive solutions and describe the asymptotic behavior of positive singular solutions without the sign assumption for − Δ u. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we first determine conditions on the analytic polynomials and such that the family of univalent sense-preserving biharmonic polynomials of the form is dense in the family of univalent sense-preserving biharmonic mappings defined on the unit disk of the form . Next, we consider the univalency of harmonic function whenever is univalent in . Finally, we give a partial answer to the problem raised by Muhanna about the radius of univalence of a family of sense-preserving biharmonic mappings with the form , where is univalent in . [Copyright &y& Elsevier]

Abstract: By using critical point theory, we establish the existence of infinitely many weak solutions for a class of elliptic Navier boundary value problems depending on two parameters and involving the -biharmonic operator. [Copyright &y& Elsevier]

Abstract: In this paper we study the critical growth biharmonic problem with a parameter and establish uniform lower bounds for , which is the supremum of the set of , related to the existence of positive solutions of the biharmonic problem. [Copyright &y& Elsevier]

Abstract: By a variant version of Fountain Theorem due to Zou [W. Zou, Variant fountain theorems and their applications, Manuscripta Math. 104 (2001) 343–358], the existence of infinitely many solutions is obtained for a class of biharmonic equations where the nonlinearity involves a combination of superlinear and asymptotically linear terms. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence and multiplicity of nontrivial solutions for a class of biharmonic elliptic systems with Sobolev critical exponent in a bounded domain. By using the variational method and the Nehari manifold, we obtain the existence and multiplicity results of nontrivial solutions for the systems. [Copyright &y& Elsevier]

Abstract: We give strong theoretical and numerical evidence that solutions to some nonlinear fourth order ordinary differential equations blow up in finite time with infinitely many wild oscillations. We exhibit an explicit example where this phenomenon occurs. We discuss possible applications to biharmonic partial differential equations and to the suspension bridges model. In particular, we give a possible new explanation of the collapse of bridges. [Copyright &y& Elsevier]

Abstract: In this paper we study the asymptotic behavior of the ground state solutions of the Hénon type biharmonic equation in , in and on , where is the unit ball in , . We prove that the ground state solution concentrates on a boundary point and has a unique maximum point as , which deduce that the ground state solution is not radially symmetric. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the multiplicity of positive radially symmetric solutions of the Dirichlet boundary value problem for the following two-dimensional quasilinear biharmonic equation where is the unit ball in the plane. We apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of positive radially symmetric solutions. We have found that there exists a threshold , such that if , then the problem has no positive radially symmetric solution; while if , then the problem admits at least one positive radially symmetric solution. Especially, there exist at least two positive radially symmetric solutions for . [Copyright &y& Elsevier]

Abstract: In this paper, we establish the existence of at least three solutions to a Navier boundary problem involving the -biharmonic systems. The technical approach is mainly based on a three critical points theorem of B. Ricceri. [Copyright &y& Elsevier]

Abstract: The existence of at least three weak solutions is established for a class of Navier boundary value problem involving the -biharmonic. Our technical approach is based on the three critical points theorem obtained by B. Ricceri. [Copyright &y& Elsevier]

Abstract: In this paper, via a perturbation theory, we prove the existence of infinitely many sign-changing solutions for a class of biharmonic equations. [Copyright &y& Elsevier]

Abstract: We study the fourth order nonlinear critical problem in the unit ball of (), subject to the Steklov boundary conditions on . We provide the exact range of the parameter for which this problem admits a positive (radial) solution. We also show that the solution is unique in this range and in the class of radially symmetric functions. Finally, we study the behavior of the solution when tends to the extremals of this range. These results complement previous results in [E. Berchio, F. Gazzola, T. Weth, Critical growth biharmonic elliptic problems under Steklov-type boundary conditions, Adv. Differential Equations 12 (2007) 381–406]. [Copyright &y& Elsevier]

Abstract: We show that there exist at least three nontrivial solutions for a class of fourth elliptic equations under Navier boundary conditions by linking approaches. [Copyright &y& Elsevier]

Abstract: Contrary to the second-order case, biharmonic heat kernels are sign-changing. A deep knowledge of their behaviour may however allow us to prove positivity results for solutions of the Cauchy problem. We establish further properties of these kernels, we prove some Lorch–Szegö-type monotonicity results and we give some hints on how to obtain similar results for higher order polyharmonic parabolic problems. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the entire positive solutions for the inhomogeneous biharmonic equation where is the biharmonic operator, , and is a given nonnegative function. Based on the results on the biharmonic equation in [Q.Y. Dai, Entire positive solutions for inhomogeneous semilinear elliptic systems, Glasgow Math. J 47 (2005) 97–114], we obtain the optimal “decay coefficient” of the inhomogeneous term for existence and nonexistence. And also, we obtain that there exist at least two types of decay solutions at infinity with the assumptions on . [Copyright &y& Elsevier]

Abstract: We study the existence of solutions for the Cauchy problem of the non-isotropically perturbed nonlinear Schrödinger equation , where , are not simultaneously vanishing real constants, is a positive constant, and . By using Kato’s method, we establish some local existence results for initial data belonging to , where if either , , or , , , if , , and if , , . Global existence is also established under some additional conditions. [Copyright &y& Elsevier]

Abstract: We consider two-dimensional mixed problems in an exterior domain for a semilinear strongly damped wave equation with a power-type nonlinearity . If the initial data have a small weighted energy, we shall derive a global existence and energy decay results in the case when the power of the nonlinear term satisfies . [Copyright &y& Elsevier]

Abstract: We consider a spatially distributed hybrid system consisting of a convection/reaction system in which the reaction switches discontinuously in time between modes, independently at each spatial point on reaching “switching thresholds”. The model involves a novel formulation for evolution of the free boundary between the modal regions. [Copyright &y& Elsevier]

Abstract: This paper deals with the blow-up phenomenon of classical solutions to the Cauchy problem for quasilinear hyperbolic systems with general small initial data which possess certain decaying properties. We prove that it is the envelope of the same family of characteristics which yields the blowup of classical solutions to the Cauchy problem. Moreover, the blow-up rate is also discussed. [Copyright &y& Elsevier]

Abstract: As the main result of the paper we prove that if is a strong global solution of the homogeneous Navier–Stokes equations in a smooth bounded domain endowed with homogeneous Dirichlet boundary conditions then for every there exist , and such that [Copyright &y& Elsevier]

Abstract: In this work, we show that the eigenvalues of the problem are generically simple in the set of regular regions of , . In fact, we prove that there exists a residual set of regions diffeomorphic to a given such that all the eigenvalues for a supported plate equation are simple. [Copyright &y& Elsevier]

In this paper, we deal with the nonlinear fourth-order boundary value problem with general Robin-type boundary conditionswhere λ∈R, p,q>1, M,N are 2×4 matrices, rank M=rank N=2, andUnder some restrictions on M and N, we show that every positive eigenvalue λ is simple. On the other hand, we give a counterexample showing that without these restrictions it need not be true. We also discuss the question of positivity of all eigenvalues. [Copyright &y& Elsevier]

In this paper, we study the biharmonic operator on a manifold with conical singularities and consider the existence of non-trivial weak solution for the corresponding semilinear degenerate elliptic equation with critical cone Sobolev exponents by the cone Sobolev inequality and Poincaré inequality. Furthermore, we show an interesting integral identity, which can be used to discuss the non-existence results for some biharmonic equations under Navier boundary conditions. [ABSTRACT FROM AUTHOR]

We consider critical points u : Ω → N of the bi-energy ∫ Ω | Δ u | 2 d x , where Ω ⊂ R m is a bounded smooth domain of dimension m ≥ 5 and N ⊂ R L a compact submanifold without boundary. More precisely, we consider variationally biharmonic maps u ∈ W 2 , 2 (Ω , N) , which are defined as critical points of the bi-energy that satisfy a certain stationarity condition up to the boundary. For weakly convergent sequences of variationally biharmonic maps, we demonstrate that the only obstruction that can prevent the strong compactness up to the boundary is the presence of certain non-constant biharmonic 4-spheres or 4-halfspheres in the target manifold. As an application, we deduce full boundary regularity of variationally biharmonic maps provided such spheres do not exist. [ABSTRACT FROM AUTHOR]

In this paper by variational methods, we study the multiplicity of solutions to the following fourth-order elliptic equations of Kirchhoff type with critical nonlinearity in R N : Δ 2 u − M ∫ R N | ∇ u | 2 d x Δ u + V (x) u = k (x) | u | q − 2 u + λ | u | 2 ∗ ∗ − 2 u , x ∈ R N , where Δ 2 u = Δ (Δ u) is the biharmonic operator, M : R + → R + is a continuous function, λ > 0 is a parameter, V : R N → R + is a potential function, and k (x) is a nonnegative continuous real valued function satisfying some conditions. The compactness condition is proved by the Lions' second Concentration-compactness principle and Concentration-compactness principle at infinity. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the multiplicity existence of solutions for the following p -biharmonic equation: Δ p 2 u = λ | u | p − 2 u + | u | q − 2 u , in Ω , u = Δ u = 0 , on ∂ Ω , where Ω is a bounded domain in R N , Δ p 2 u = Δ ( | Δ u | p − 2 Δ u) , p < q ≤ p ∗ = N p N − 2 p , λ ∈ R is a parameter. When p < q < p ∗ , we prove that the above problem possesses infinitely many solutions. While when q = p ∗ , a multiplicity existence result is obtained. [ABSTRACT FROM AUTHOR]