Number of solutions with a norm bounded by a given constant of a semilinear elliptic PDE with a generic right-hand side

Nabutovsky, Alexander

doi:10.1090/S0002-9947-1992-1066447-8

Number of solutions with a norm bounded by a given constant of a semilinear elliptic PDE with a generic right-hand side
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by Alexander Nabutovsky PDF

Trans. Amer. Math. Soc. 332 (1992), 135-166 Request permission

Abstract:

We consider a semilinear boundary value problem $- \Delta u + f(u,x) = 0$ in $\Omega \subset {\mathbb {R}^N}$ and $u = 0$ on $\partial \Omega$. We assume that $f$ is a ${C^\infty }$-smooth function and $\Omega$ is a bounded domain with a smooth boundary. For any ${C^\alpha }$-smooth perturbation $h(x)$ of the right-hand side of the equation we consider the function ${N_h}(S)$ defined as the number of ${C^{2 + \alpha }}$-smooth solutions $u$ such that $\left \| u\right \| _{{C^0}(\Omega )} \leq S$ of the perturbed problem. How "small" ${N_h}(S)$ can be made by a perturbation $h(x)$ such that $\left \| h\right \| _{{C^0}(\Omega )} \leq \varepsilon ?$ We present here an explicit upper bound in terms of $\varepsilon$ , $S$ and \[ \max \limits _{|u| \leq S,x \in \bar \Omega } \left \| D_u^i f(u,x)\right \| \quad (i \in \{ 0,1,2\} ).\] If $S$ is fixed then $h$ can be chosen by such a way that the upper bound persists under small in ${C^0}$-topology perturbations of $h$ . We present an explicit lower bound for the radius of the ball of such admissible perturbations.

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