Related papers: A Generalized CR equation with isolated singularit…

We show the linking-type result which allows us to study strongly indefinite problems with sign-changing nonlinearities. We apply the abstract theory to the singular Schr\"{o}dinger equation $$ -\Delta u + V(x)u + \frac{a}{r^2} u = f(u) -…

We study the generalized Lam\'e equation on an elliptic curve $E$ with multiple singularities. By restricting to the locus admitting solutions with quasi-periodic properties, we construct two curves: (i) The generalized Lam'e curve: with…

We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…

This paper focuses on the existence and multiplicity of normalized solutions for the coupled Schrodinger system with Sobolev critical coupling term. We present several existence and multiplicity results under some explicit conditions.…

We investigate generalizations of the Burgers and Burgers-Huxley equations. The investigations we offer focus attention mainly on presenting explict analytical solutions by means of relating these generalized equations to relativistic 1+1…

In this paper, we study a class of generalized extensible beam equations with a superlinear nonlinearity \begin{equation*} \left\{ \begin{array}{ll} \Delta ^{2}u-M\left( \Vert \nabla u\Vert _{L^{2}}^{2}\right) \Delta u+\lambda V(x) u=f(…

This article is devoted to the study of solutions of non-homogenous linear differential equations having entire coefficients. We get all non-trivial solutions of infinite order of equation $f^{(n)}+a_{n-1}(z)f^{(n-1)}+\ldots…

We consider the existence of \emph{normalized} solutions in $H^1(\R^N) \times H^1(\R^N)$ for systems of nonlinear Schr\"odinger equations which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz…

We study the local properties of positive solutions of the equation $-\Delta u+ae^{bu}=m|\nabla u|^q$ in a punctured domain $\Omega\setminus\{0\}$ of $\bf R^2$ where $m,a,b$ are positive parameters and $q>1$. We study particularly the…

The class of ordinary linear constant coefficient differential equations is naturally embedded into a wider class by associating differential equations to algebraic curves.

Normality arguments are applied to study the oscillation of solutions of $f''+Af=0$, where the coefficient $A$ is analytic in the unit disc $\mathbb{D}$ and $\sup_{z\in\mathbb{D}} (1-|z|^2)^2|A(z)| < \infty$. It is shown that such…

This paper deals with the existence and multiplicity of solutions for the generalized $(p, q)$-Laplacian equation \begin{align*} &-{\text{ div}}(A(x, u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x, u)|\nabla u|^p -{\text{ div}}(B(x, u)|\nabla…

Let $d\geq3$ and $g\geq1$ be integers. Using a geometric construction involving the symmetric product of a projective curve, we exhibit a $d$-dimensional complete local normal domain over $\mathbb{C}$ with an isolated singularity such that…

Preliminary group classification for a class of generalized inviscid Burger's equations in the general form $u_t+g(x, u)u_x = f(x, u)$ is given and additional equivalence transformations are found. Adduced results complete and essentially…

A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…

In this paper, we study the existence and regularity of solutions for a class of nonlinear singular elliptic equations involving unbounded coefficients and a singular right-hand side. Specifically, we are interested to problem whose…

Generalised Fermat equation (GFE) is the equation of the form $ax^p+by^q=cz^r$, where $a,b,c,p,q,r$ are positive integers. If $1/p+1/q+1/r<1$, GFE is known to have at most finitely many primitive integer solutions $(x,y,z)$. A large body of…

The Cauchy problem for the generalized Zakharov-Kuznetsov equation $$\partial_t u +\partial_x\Delta u=\partial_x u^{k+1}, \qquad \qquad u(0)=u_0$$ is considered in space dimensions $n=2$ and $n=3$ for integer exponents $k \ge 3$. For data…

We find a normalized solution $u=(u_1,\ldots,u_K)$ to the system of $K$ coupled nonlinear Schr\"odinger equations \begin{equation*} \left\{ \begin{array}{l} -\Delta u_i+ \lambda_i u_i = \sum_{j=1}^K\beta_{i,j}u_i|u_i|^{p/2-2}|u_j|^{p/2}…

The Riemann problem is studied in the case when the unknown function has nonisolated singularities, concentrated on the real axis. The problem is used for the factorization of functions, holomorphic outside of the unit circle and the real…