No Arabic abstract
In this work we consider a system of k non-linear elliptic equations where the non-linear term is the sum of a quadratic form and a sub-critic term. We show that under suitable assumptions, e.g. when the non-linear term has a zero with non-zero coordinates, we can find a infinitely many solution of the eigenvalue problem with radial symmetry. Such problem arises when we search multiple standing-waves for a non-linear wave system.
The existence of positive weak solutions to a singular quasilinear elliptic system in the whole space is established via suitable a priori estimates and Schauders fixed point theorem.
We discuss counterexamples to the validity of the weak Maximum Principle for linear elliptic systems with zero and first order couplings and prove, through a suitable reduction to a nonlinear scalar equation, a quite general result showing that some algebraic condition on the structure of gradient couplings and a cooperativity condition on the matrix of zero order couplings guarantee the existence of invariant cones in the sense of Weinberger [21].
Let $E$ be an elliptic curve, with identity $O$, and let $C$ be a cyclic subgroup of odd order $N$, over an algebraically closed field $k$ with $operatorname{char} k mid N$. For $P in C$, let $s_P$ be a rational function with divisor $N cdot P - N cdot O$. We ask whether the $N$ functions $s_P$ are linearly independent. For generic $(E,C)$, we prove that the answer is yes. We bound the number of exceptional $(E,C)$ when $N$ is a prime by using the geometry of the universal generalized elliptic curve over $X_1(N)$. The problem can be recast in terms of sections of an arbitrary degree $N$ line bundle on $E$.
We consider positive solutions of $Delta u-mu u+Ku^{frac{n+2}{n-2}}=0$ on $B_1$ ($nge 5$) where $mu $ and $K>0$ are smooth functions on $B_1$. If $K$ is very sub-harmonic at each critical point of $K$ in $B_{2/3}$ and the maximum of $u$ in $bar B_{1/3}$ is comparable to its maximum over $bar B_1$, then all positive solutions are uniformly bounded on $bar B_{1/3}$. As an application, a priori estimate for solutions of equations defined on $mathbb S^n$ is derived.
We study the Fredholm properties of a general class of elliptic differential operators on $R^n$. These results are expressed in terms of a class of weighted function spaces, which can be locally modeled on a wide variety of standard function spaces, and a related spectral pencil problem on the sphere, which is defined in terms of the asymptotic behaviour of the coefficients of the original operator.