Many models in biology and ecology can be described by reaction-diffusion equations wit time delay. One of important solutions for these type of equations is the traveling wave solution that shows the phenomenon of wave propagation. The existence of traveling wave fronts has been proved for large class of equations, in particular, the monotone systems, such as the cooperative systems and some competition systems. However, the problem on the uniqueness of traveling wave (for a fixed wave speed)...

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is$$\left\{\begin{array}{cc}-div\left(a\left(x\right)\right(1+{\left|\nabla u\right|}^2{)}^{\frac{p-2}{2}}{\nabla u)+b\left(x\right)(1+\left|\nabla u\right|}^2{)}^{\frac{\lambda }{2}}=f\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$$where $\Omega$ is a bounded open subset of ${ℝ}^N$, $N\ge 2$, $2-1/N\<p\<N$, $a$ belongs to $L^{\infty }\left(\Omega \right)$, $a\left(x\right)\ge {\alpha }_0\>0$, $f$ is a function in $L^1\left(\Omega \right)$, $b$ is a function in $L^r\left(\Omega \right)$ and $0\le \lambda \<{\lambda }^{*}(N,p,r),$ for some $r$ and ${\lambda }^{*}(N,p,r)$.

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is
$$\left\{-\text{div}\left(a\left(x\right)\right(1+{\left|\nabla u\right|}^2{)}^{\frac{p-2}{2}}{\nabla u)+b\left(x\right)(1+\left|\nabla u\right|}^2{)}^{\frac{\lambda }{2}}=f\text{in}\Omega ,u=0\text{on}\partial \Omega ,\right.$$
where Ω is a bounded open subset of ${ℝ}^N$, N > 2, 2-1/N < p < N , a belongs to L∞(Ω), $a\left(x\right)\ge {\alpha }_0>0$, f is a function in L1(Ω), b is a function in $L^r\left(\Omega \right)$ and 0 ≤ λ < λ *(N,p,r), for some r and λ *(N,p,r).

We investigate the existence and uniqueness of solutions to the Dirichlet problem for a degenerate nonlinear elliptic equation $-{\sum }_{i,j=1}^n{D}_j\left(a_{ij}\left(x\right)D_{i}u\left(x\right)\right)+b\left(x\right)u\left(x\right)+div\left(\Phi \left(u\left(x\right)\right)\right)=g\left(x\right)-{\sum }_{j=1}^n{f}_j\left(x\right)$ on Ω in the setting of the space H₀(Ω).

We study a comparison principle and uniqueness of positive solutions for the homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations with lower order terms. A model example is given by $-\Delta u+\lambda \frac{{\left|\nabla u\right|}^2}{u^r}=f\left(x\right),\lambda ,r>0.$ The main feature of these equations consists in having a quadratic gradient term in which singularities are allowed. The arguments employed here also work to deal with equations having lack of ellipticity or some dependence on u in the right hand side. Furthermore, they...