Related papers: On the numerical approximation of $p$-Biharmonic a…

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear…

Nonlocal models and their associated theories have been extensively investigated in recent years. Among these, nonlocal versions of the classical Laplace operator have attracted the most attention, while higher-order nonlocal operators have…

We study the following semilinear biharmonic equation $$ \left\{\begin{array}{lllllll} \Delta^{2}u=\frac{\lambda}{1-u}, &\quad \mbox{in}\quad \B, u=\frac{\partial u}{\partial n}=0, &\quad \mbox{on}\quad \partial\B, \end{array} \right.…

In this paper, we are concerned with the following equation involving higher-order fractional Lapalacian \begin{equation*} \left\{\begin{aligned} &(-\Delta)^{p+{\frac{\alpha}{2}}}u(x)=u_+^\gamma~~ \mbox{ in }\mathbb{R}^n,\\…

The biharmonic equation with Dirichlet and Neumann boundary conditions discretized using the mixed finite element method and piecewise linear (with the possible exception of boundary triangles) finite elements on triangular elements has…

In this article, we deal with the following involving $p$-biharmonic critical Choquard-Kirchhoff equation $$ \left(a+b\left(\int_{\mathbb R^N}|\Delta u|^p dx\right)^{\theta-1}\right) \Delta_{p}^{2}u = \alpha…

The aim of this work is to derive new explicit solutions to the $\infty$-Laplace equation, the fundamental PDE arising in Calculus of Variations in the space $L^\infty$. These solutions obey certain symmetry conditions and are derived in…

In this paper we initiate the study of $2$nd order variational problems in $L^\infty$, seeking to minimise the $L^\infty$ norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the…

We are interested in entire solutions for the semilinear biharmonic equation $\Delta^{2}u=f(u)$ in $\R^N$, where $f(u)=e^{u}$ or $-u^{-p}\ (p>0)$. For the exponential case, we prove that any classical entire solution verifies $-\Delta u>0$…

The paper is concerned with a free boundary problem generated by the biharmonic operator and an obstacle. The main goal is to deduce a fully guaranteed upper bound of the difference between the exact minimizer u and any function…

This work is devoted to the study of the existence of at least one (non-zero) solution to a problem involving the discrete $p$-Laplacian. As a special case, we derive an existence theorem for a second-order discrete problem, depending on a…

Following ideas of Caffarelli and Silvestre in~\cite{CS}, and using recent progress in hyperbolic fillings, we define fractional $p$-Laplacians $(-\Delta_p)^\theta$ with $0<\theta<1$ on any compact, doubling metric measure space…

We investigate here the degenerate bi-harmonic equation: $$\Delta_{m}^2 u=f(x,u)\; \;\;\mbox{in} \O,\quad u = \Delta u = 0\quad \mbox{on }\; \p\Omega,$$ with $m\ge 2,$ and also the degenerate tri-harmonic equation: $$ -\Delta_{m}^3…

In this work, we exploit the capability of virtual element methods in accommodating approximation spaces featuring high-order continuity to numerically approximate differential problems of the form $\Delta^p u =f$, $p\ge1$. More…

In this paper, we prove the existence of a weak solution for the Dirichlet boundary value problem related to the $p(x)-$Laplacian $$ -\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)+u\in -[\underline{g}(x,u),\overline{g}(x,u)], $$ by using the…

Using variational methods, we establish existence of multi-bump solutions for the following class of problems $$ \left\{ \begin{array}{l} \Delta^2 u +(\lambda V(x)+1)u = f(u), \quad \mbox{in} \quad \mathbb{R}^{N}, u \in…

We consider weak non-negative solutions to the critical $p$-Laplace equation in $\mathbb{R}^N$, $-\Delta_p u =u^{p^*-1}$ in the singular case $1<p<2$. We prove that if the nonlinearity is locally Lipschitz continuous, namely $p^*\geqslant2$…

Let $ \mathbb{R}^{n} $ denote Euclidean $ n $ space and given $k$ a positive integer let $ \Lambda_k \subset \mathbb{R}^{n} $, $ 1 \leq k < n - 1, n \geq 3, $ be a $k$-dimensional plane with $ 0 \in \Lambda_k.$ If $n-k < p <\infty$, we…

For $n \geq 5$, we consider positive solutions $u$ of the biharmonic equation \[ \Delta^2 u = u^\frac{n+4}{n-4} \qquad \text{on}\ \mathbb R^n \setminus \{0\} \] with a non-removable singularity at the origin. We show that…

Assume that $f$ is Dunkl polyharmonic in $\mathbb{R}^n$ (i.e. $(\Delta_h)^p f=0$ for some integer $p$, where $\Delta_h$ is the Dunkl Laplacian associated to a root system $R$ and to a multiplicity function $\kappa$, defined on $R$ and…