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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…

In this paper, the following critical biharmonic elliptic problem \begin{eqnarray*} \begin{cases} \Delta^2u= \lambda u+\mu u\ln u^2+|u|^{2^{**}-2}u, &x\in\Omega,\\ u=\dfrac{\partial u}{\partial \nu}=0, &x\in\partial\Omega \end{cases}…

We prove the multiplicity and concentration of normalized solutions of critical biharmonic equations with combined nonlinearities in $\mathbb{R}^{N}$ \begin{equation*} \Delta^{2}u+V(\varepsilon x)u=\lambda u+\mu |u|^{q-2}u+|u|^{2^{**}-2}u…

We give a new bound on the exponent for the nonexistence of stable solutions to the biharmonic problem $$\Delta^2u=u^p,\quad u>0 in \mathbb{R}^n $$ where $p>1, n \geq 20$.

The biharmonic equation, as well as its nonlinear and inhomogeneous generalizations, plays an important role in engineering and physics. In particular the focusing biharmonic nonlinear Schr\"{o}dinger equation, and its standing wave…

In this note we devise and analyse well-posed variational formulations and operator theoretical methods for boundary value problems associated to the biharmonic operator. Of particular interest are Neumann type and over- and underdetermined…

This paper considers a pair of coupled nonlinear Helmholtz equations \begin{align*} -\Delta u - \mu u = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2} \right)|u|^{\frac{p}{2} - 2}u, \end{align*} \begin{align*} -\Delta v - \nu v = a(x)…

\ In this paper, the following biharmonic elliptic problem \begin{eqnarray*} \begin{cases} \Delta^2u-\lambda\frac{|u|^{q-2}u}{|x|^s}=|u|^{2^{**}-2}u+ f(x,u), &x\in\Omega,\\ u=\dfrac{\partial u}{\partial n}=0, &x\in\partial\Omega \end{cases}…

We investigate the focusing inhomogeneous nonlinear biharmonic Schr\"odinger equation \[ i\partial_t u + \Delta^2 u - |x|^{-b}|u|^p u = 0 \quad \text{on } \mathbb{R} \times \mathbb{R}^N, \] in the energy-critical regime, $p = \frac{8 -…

We are concerned on the fourth-order elliptic equation \begin{equation}\tag{$P_\lambda$} \left\{ \begin{array}[c]{ll} \Delta^2 u- \Delta u + V(x)u -\lambda \Delta[\rho(u^2)]\rho'(u^2)u= f(u)\, \, \mbox{in} \, \, \mathbb{R}^N, & u\in…

This article investigates nonlocal, fully nonlinear generalizations of the classical biharmonic operator $(-\Delta)^2$. These fractional $p$-biharmonic operators appear naturally in the variational characterization of the optimal fractional…

In this paper, we introduce a new higher-order Laplacian operator in the framework of Orlicz-Sobolev spaces, the biharmonic g-Laplacian $$\Delta_g^2 u:=\Delta \left(\dfrac{g(|\Delta u|)}{|\Delta u|} \Delta u\right),$$ where $g=G'$, with $G$…

The biharmonic supercritical equation $\Delta^2u=|u|^{p-1}u$, where $n>4$ and $p>(n+4)/(n-4)$, is studied in the whole space $\mathbb{R}^n$ as well as in a modified form with $\lambda(1+u)^p$ as right-hand-side with an additional eigenvalue…

This paper is concerned with the existence of solutions to the problem $$-\left(a+ b\int_{\mathbb{R}^{N}}|\nabla u|^{2} dx \right)\Delta u +V(x)u+\lambda u = |u|^{p-2}u,\ \ x \in \mathbb{R}^{N},\ \ \lambda \in \mathbb{R}^{+} $$ where $a,…

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 consider the Schr\"odinger-Poisson system \begin{eqnarray}\left\{\begin{array} [c]{ll} -\Delta u+V(x) u+|u|^{p-2}u=\lambda \phi u, & \mbox{in}\mathbb{R}^{3},\\ -\Delta\phi= u^{2}, & \mbox{in}\mathbb{R}^{3}. \end{array} \right.\nonumber…

We study the bifurcation of solutions of semilinear elliptic boundary value problems of the form \begin{align*} \begin{aligned} -\Delta u &= f_\lambda(|x|,u,|\nabla u|) &&\text{in }\Omega, u &= 0 &&\text{on }\partial\Omega, \end{aligned}…

Using dual method we establish the existence of nodal ground state solution for the following class of problems $$ \left\{ \begin{array}{l} \Delta^2 u = f(u), \quad \mbox{in} \quad \Omega, \\ u =Bu=0,\quad\mbox{on} \quad \partial \Omega…

In this paper, we consider the weighted fourth order equation $$\Delta(|x|^{-\alpha}\Delta u)+\lambda \text{div}(|x|^{-\alpha-2}\nabla u)+\mu|x|^{-\alpha-4}u=|x|^\beta u^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\},$$ where $n\geq…

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…