In this paper, a mathematical analysis of in-situ biorestoration is presented. Mathematical formulation of such process leads to a system of non-linear partial differential equations coupled with ordinary differential equations. First, we introduce a notion of weak solution then we prove the existence of at least one such a solution by a linearization technique used in Fabrie and Langlais (1992). Positivity and uniform bound for the substrates concentration is derived from the maximum principle...

We show that the critical nonlinear elliptic Neumann problem $\Delta u-\mu u+u^{7/3}=0$ in $\Omega$, $u>0$ in $\Omega$, $\frac{\partial u}{\partial \nu }=0$ on $\partial \Omega$, where $\Omega$ is a bounded and smooth domain in ${ℝ}^5$, has arbitrarily many solutions, provided that $\mu >0$ is small enough. More precisely, for any positive integer $K$, there exists ${\mu }_K>0$ such that for $0<\mu <{\mu }_K$, the above problem has a nontrivial solution which blows up at $K$ interior points in $\Omega$, as $\mu \to 0$. The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional...

Let $W$ be a non-negative function of class ${\mathrm{C}}^3$ from ${ℝ}^2$ to $ℝ$, which vanishes exactly at two points $𝐚$ and $𝐛$. Let $S^1(𝐚,𝐛)$ be the set of functions of a real variable which tend to $𝐚$ at $-\infty$ and to $𝐛$ at $+\infty$ and whose one dimensional energy$$E_1\left(v\right)={\int }_{ℝ}\left[W\left(v\right)+|v^{\text{'}}{|}^2/2\right]\mathrm{d}x$$is finite. Assume that there exist two isolated minimizers $z_{+}$ and $z_{-}$ of the energy $E_1$ over $S^1(𝐚,𝐛)$. Under a mild coercivity condition on the potential $W$ and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at $z_{+}$ and $z_{-}$, it is possible to prove...

Let W be a non-negative function of class C3 from ${}^2$ to $$, which vanishes exactly at two points a and b. Let S1(a, b) be the set of functions of a real variable which tend to a at -∞ and to b at +∞ and whose one dimensional energy $$E_1\left(v\right)={\int }_{\left[W\left(v\right)+{\left|v^{\text{'}}\right|}^2/2\right]}x$$ is finite. Assume that there exist two isolated minimizers z+ and z- of the energy E1 over S1(a, b). Under a mild coercivity condition on the potential W and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at z+ and...

We consider the semilinear Lane–Emden problem $$ where $p>1$ and $\Omega$ is a smooth bounded domain of ${ℝ}^2$. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ${(}_p)$, as $p\to +\infty$. Among other results we show, under some symmetry assumptions on $\Omega$, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p\to +\infty$, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville...

We show that nonnegative solutions of $$\begin{array}{cc}& u_t-u_{xx}+f\left(u\right)=0,x\in ℝ,t>0,\hfill \\ & u=\alpha \overline{u},x\in ℝ,t=0,supp\overline{u}\text{compact}\hfill \end{array}$$ either converge to zero, blow up in $L^2$-norm, or converge to the ground state when $t\to \infty$, where the latter case is a threshold phenomenon when $\alpha >0$ varies. The proof is based on the fact that any bounded trajectory converges to a stationary solution. The function $f$ is typically nonlinear but has a sublinear growth at infinity. We also show that for superlinear $f$ it can happen that solutions converge to zero for any $\alpha >0$, provided $supp\overline{u}$ is sufficiently small.

2000 Mathematics Subject Classification: 35Q02, 35Q05, 35Q10, 35B40.We consider the stationary one dimensional Schrödinger-Poisson system on a bounded interval with a background potential describing a quantum well. Using a partition function which forces the particles to remain in the quantum well, the limit h®0 in the nonlinear system leads to a uniquely solved nonlinear problem with concentrated particle density. It allows to conclude about the convergence of the solution.