### Multiple positive solutions for quasilinear elliptic problems with sign-changing nonlinearities.

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In this paper we study the Sobolev trace embedding W(Ω) → L (∂Ω), where V is an indefinite weight. This embedding leads to a nonlinear eigenvalue problem where the eigenvalue appears at the (nonlinear) boundary condition. We prove that there exists a sequence of variational eigenvalues λ / +∞ and then show that the first eigenvalue is isolated, simple and monotone with respect to the weight. Then we prove a nonexistence result related to the first eigenvalue and we end this article...

We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ${u}_{t}=\Delta u$, ${v}_{t}=\Delta v$ in $\Omega \times (0,T)$; fully coupled by the boundary conditions $\frac{\partial u}{\partial \eta}={u}^{{p}_{11}}{v}^{{p}_{12}}$, $\frac{\partial v}{\partial \eta}={u}^{{p}_{21}}{v}^{{p}_{22}}$ on $\partial \Omega \times (0,T)$, where $\Omega $ is a bounded smooth domain in ${\mathbb{R}}^{d}$. We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation $(U,V)$. We prove that if $U$ blows up in finite time then $V$ can fail to blow up if and only if ${p}_{11}\>1$ and ${p}_{21}\<2({p}_{11}-1)$, which is the same condition as the one for non-simultaneous blow-up in the continuous problem. Moreover,...

We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations , in Ω x (0,); fully coupled by the boundary conditions $\frac{\partial u}{\partial \eta}={u}^{{p}_{11}}{v}^{{p}_{12}}$, $\frac{\partial v}{\partial \eta}={u}^{{p}_{21}}{v}^{{p}_{22}}$ on ∂Ω x (0,), where is a bounded smooth domain in ${\mathbb{R}}^{d}$. We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation . We prove that if blows up in finite time then can fail to blow up if and only if > 1 and < 2( - 1) , which is the same...

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