Let T be a tree rooted at e endowed with a nearest-neighbor transition probability that yields a recurrent random walk. We show that there exists a function K biharmonic off e whose Laplacian has potential theoretic importance and, in addition, has the following property: Any function f on T which is biharmonic outside a finite set has a representation, unique up to addition of a harmonic funct...

In this article, we prove energy quantization for approximate (intrinsic and extrinsic) biharmonic maps into spheres where the approximate map is in L logL. Moreover, we demonstrate that if the L logL norm of the approximate maps does not concentrate, the image of the bubbles are connected without necks.

We establish an optimal $$L^p$$ -regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions $$n\ge 5$$ : $$\begin{aligned} \Delta ^2 u=\Delta (D\cdot \nabla u)+\text {div}(E\cdot u) +(\Delta \Omega +G)\cdot u +f \qquad \ \mathrm{{in}}\ B^n, \end{aligned}$$ where $$\Omega \in W^{1,2}(B^n, so_m)$$ is and $$f\in L^p(B^n)$$ , $$D, ...

In this note we use the Nehari manifold and fibering maps to show existence of positive solutions for a nonlinear biharmonic equation in a bounded smooth domain in Rn, when n = 5, 6, 7. Mathematics Subject Classification: 35J35, 35J40

In this paper, we determine necessary and sufficient conditions for a non-Frenet Legendre curve to be $f$-harmonic, $f$-biharmonic, bi-$f$-harmonic, biminimal $f$-biminimal in three-dimensional normal almost paracontact metric manifold. Besides, obtain some nonexistence theorems.