Related papers: On a Fourth Order Lichnerowicz Type Equation Invol…

The squared Laplace operator acting on symmetric rank-two tensor fields is studied on a (flat) Riemannian manifold with smooth boundary. Symmetry of this fourth-order elliptic operator is obtained provided that such tensor fields and their…

We study the copolynomials of $n$ variables, i.e. $K$-linear mappings from the ring of polynomials $K[x_1,...,x_n]$ into the commutative ring $K$. We prove an existence and uniqueness theorem for a linear differential equation of infinite…

We consider a critical problem in a bounded domain involving the $p$-Grushin operator $\Delta_\alpha^p$. After a truncation argument, we obtain infinitely many solutions to our problem via Krasnoselskii's genus, extending a previous result…

We develop an operator approach to the integration of linear differential equations based on intertwining relations between differential operators. Conditions for the existence of intertwining operators are obtained, and it is shown that,…

The paper considers a boundary value problem for the high-order Lavrent'ev-Bitsadze equation. Necessary and sufficient conditions for the uniqueness of the solution are found. When substantiating the existence, the problem of "small…

The Riccati equation method is used to establish Kamenev type conditions for the existence of oscillatory solutions to third order linear ordinary differential equations. Three oscillatory theorems are proved, which generalize the Lazer's…

In a recent paper, Soner, Touzi and Zhang [20] have introduced a notion of second order backward stochastic differential equations (2BSDEs for short), which are naturally linked to a class of fully non-linear PDEs. They proved existence and…

In this article, we study the fractional Br\'{e}zis-Nirenberg type problem on whole domain $\mathbb{R}^N$ associated with the fractional $p$-Laplace operator. To be precise, we want to study the following problem: \begin{equation*}…

We study the following Brezis-Nirenberg problem of Kirchhoff type $$ \left\{\aligned &-(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u = \lambda|u|^{q-2}u + \delta |u|^{2}u, &\quad \text{in}\ \Omega, \\ &u=0,& \text{on}\ \partial\Omega,…

In this paper, we establish a fundamental inequality for fourth order partial differential operator $\cal P=\alpha\partial_s+\beta\partial_{ss}+\Delta^2$ ($\alpha, \beta\in\mathbb{R}$) with an abstract exponential-type weight function. Such…

In the paper, a newly developed three-point fourth-order compact operator is utilized to construct an efficient compact finite difference scheme for the Benjamin-Bona-Mahony-Burgers' (BBMB) equation. Detailed derivation is carried out based…

Several spectra of analytically Riesz operators will be characterized. These results will led to prove Weyl and Browder type theorems for the aforementioned class of operators.

Infinite-dimensional Newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations (PDEs). In computer-assisted proofs of PDEs, the original problem is transformed into the…

We study the resonant prescribed T-curvature problem on a compact 4-dimensional Riemannian manifold with boundary. We derive sharp energy and gradient estimates of the associated Euler-Lagrange functional to characterize the critical points…

Necessary and sufficient conditions are given for the existence of solutions to the discrete Lp Minkowski problem for the critical case where 0 < p < 1.

We study an eigenvalue problem involving a degenerate and singular elliptic operator on the whole space $\mathbb{R}^N$. We prove the existence of an unbounded and increasing sequence of eigenvalues. Our study generalizes to the case of…

In this paper we are concerned with existence results for a coupled system of quadratic functional differential equations. This system is reduced to a fixed point problem for a block operator matrix with nonlinear inputs. To prove the…

An algebraic Riccati equation for linear operators is studied, which arises in systems theory. For the case that all involved operators are unbounded, the existence of infinitely many selfadjoint solutions is shown. To this end, invariant…

In this paper we consider a class of critical concave convex Ambrosetti-Prodi type problems for the fractional $p$-Laplacian operator. By applying the Linking Theorem and the Mountain Pass Theorem as well, the interaction of the…

In this paper, we deal with the logarithmic weighted fourth order elliptic equation in the unit disk of $B\subset\R^{4}$: $$\displaystyle(P_{\lambda})~~\Delta(w(x) \Delta u) = \lambda\ f(x,u) \quad\mbox{ in }\quad B, \quad u=\frac{\partial…