arXiv subjects

Tobias Weth

Publications and source records attributed to Tobias Weth.

75 records · Page 5Linked to original sources

arXiv2010

Existence and nonexistence of entire solutions for non-cooperative cubic elliptic systems

In this paper we deal with the cubic Schrödinger system $ -Δu_i = \sum_{j=1}^n β_{ij}u_j^2 u_i$, $u_1,\dots,u_n \geq 0$ in $\mathbb{R}^N (N\leq 3)$, where $β=(β_{i,j})_{ij}$ is a symmetric matrix with real coefficients and $β_{ii}\geq 0$ for every $i=1,\ldots,n$. We analyse the existence and nonexistence of nontrivial solutions in connection with the properties of the matrix $β$, and provide a complete characterization in dimensions $N=1,2$. Extensions to more general power-type nonlinearities are given.

math.AP↗

arXiv2009

Existence of solutions to nonlinear, subcritical higher-order elliptic Dirichlet problems

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $Ω\subset\R^N$ with Dirichlet boundary conditions on $\partialΩ$. The operator $L$ is a uniformly elliptic linear operator of order $2m$ whose principle part is of the form $\big(-\sum_{i,j=1}^N a_{ij}(x) \frac{\partial^2}{\partial x_i\partial x_j}\big)^m$. We assume that $f$ is superlinear at the origin and satisfies $\lim \limits_{s\to\infty}\frac{f(x,s)}{s^q}=h(x)$, $\lim \limits_{s\to-\infty}\frac{f(x,s)}{|s|^q}=k(x)$, where $h,k\in C(\overlineΩ)$ are positive functions and $q>1$ is subcritical. By combining degree theory with new and recently established a priori estimates, we prove the existence of a nontrivial solution.

math.AP↗

arXiv2007

A priori bounds and a Liouville theorem on a half-space for higher order elliptic Dirichlet problems

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $Ω$ in $R^N$ with Dirichlet boundary conditions. The operator $L$ is a uniformly elliptic operator of order $2m$. We assume that for $s\to \pm\infty$ the nonlinearity $f(x,s)$ behaves like $|s|^q$ multiplied by a continuous and positive function of $x$. Here the exponent $q$ is subcritical, i.e., $q>1$ if $N<=2m$, $1 2m$. We prove a priori bounds, i.e, we show that the $L^\infty$-norm of every solution $u$ is bounded by a constant independent of $u$. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if $u$ is a classical, bounded, non-negative solution of $(-Δ)^m u = u^q$ in a half-space with Dirichlet boundary conditions and if $q>1$ is subcritical then $u$ vanishes identically.

math.AP↗