Related papers: Elliptic Euler-Poisson-Darboux equation, critical …
Complete description of the singular sectors of the 1-layer Benney system (classical long wave equation) and dToda system is presented. Associated Euler-Poisson-Darboux equations E(1/2,1/2) and E(-1/2,-1/2) are the main tool in the…
We investigate the dependence on parameters for second order difference equations with two point boundary value conditions by using a variational method in case when the corresponding Euler action functional is coercive. Some applications…
In this paper we establish, using variational methods combined with the Moser-Trudinger inequality, existence and multiplicity of weak solutions for a class of critical fractional elliptic equations with exponential growth without a…
We consider a class of noncooperative Schr\"{o}dinger-Kirchhoff type system which involves a general variable exponent elliptic operator with critical growth. Under certain suitable conditions on the nonlinearities, we establish the…
We study inhomogeneous non-strictly hyperbolic systems of two equations, which are a formal generalization of the transformed one-dimensional Euler-Poisson equations. For such systems, a complete classification of the behavior of the…
We consider the Euler--Darboux equation with parameters modulo 1/2 and generalization to the space 3D analogue. Due to the fact that the Cauchy problem in its classical formulation is incorrect for such parameter values, the authors propose…
In the article a classification method for nonlinear integrable equations with three independent variables is discussed based on the notion of the integrable reductions. We call the equation integrable if it admits a large class of…
In this paper we investigate the behavior and the existence of positive and non-radially symmetric solutions to nonlinear exponential elliptic model problems defined on a solid torus $\bar{T}$ of $\mathbb{R}^3$, when data are invariant…
In this paper we provide a variational derivation of the Euler-Poincar\'e equations for systems subjected to external forces using an adaptation of the techniques introduced by Galley and others. Moreover, we study in detail the underlying…
This is the second part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using abstract theorems in the first part we obtain many new bifurcation results for quasi-linear…
We consider the problem of finding complete conformal metrics with prescribed curvature functions of the Einstein tensor and of more general modified Schouten tensors. To achieve this, we reveal an algebraic structure of a wide class of…
This paper concerns the existence of critical points for solutions to second order elliptic equations of the form $\nabla\cdot \sigma(x)\nabla u=0$ posed on a bounded domain $X$ with prescribed boundary conditions. In spatial dimension…
Local indices at isolated fixed points of a differentiable compact nonlinear map $T$ on Banach spaces will be discussed. These results are applied to establish the existence of nontrivial solutions. As an example, the existence of…
We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. For such a class of operators we establish a factorization into a product of first order operators…
We consider the fractional elliptic inequality with variable-exponent nonlinearity $$ (-\Delta)^{\frac{\alpha}{2}} u+\lambda\, \Delta u \geq |u|^{p(x)}, \quad x\in\mathbb{R}^N, $$ where $N\geq 1$, $\alpha\in (0,2)$, $\lambda\in\mathbb{R}$…
We investigate the critical threshold phenomena in a large class of one dimensional pressureless Euler--Poisson (EP) equations, with non-vanishing background states. First, we establish local-in-time well-posedness in proper regularity…
We study non-linear integrable partial differential equations naturally arising as bi-Hamiltonian Euler equations related to the looped cotangent Virasoro algebra. This infinite-dimensional Lie algebra (constructed in \cite{OR}) is a…
In this paper, we deal with a class of semilinear elliptic equation in a bounded domain $\Omega\subset\mathbb{R}^N$, $N\geq 3$, with $C\sp{1,1}$ boundary. Using a new fixed point result of the Krasnoselskii's type for the sum of two…
Study the following two-component elliptic system% \begin{equation*} \left\{\aligned&\Delta u-(\lambda a(x)+a_0)u+u^3+\beta v^2u=0\quad&\text{in }\bbr^4,\\% &\Delta v-(\lambda b(x)+b_0)v+v^3+\beta u^2v=0\quad&\text{in }\bbr^4,\\%…
We consider the Euler system set on a bounded convex planar domain, endowed with impermeability boundary conditions. This system is a model for the barotropic mode of the Primitive Equations on a rectangular domain. We show the existence of…