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


Author: Alexander Nabutovsky
Journal: Trans. Amer. Math. Soc. 332 (1992), 135-166
MSC: Primary 35J65; Secondary 34B15, 35B20, 58C25
DOI: https://doi.org/10.1090/S0002-9947-1992-1066447-8
MathSciNet review: 1066447
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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\Vert u\right\Vert _{{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\Vert h\right\Vert _{{C^0}(\Omega )} \leq \varepsilon ?$ We present here an explicit upper bound in terms of $ \varepsilon $ , $ S$ and

$\displaystyle \mathop {\max }\limits_{\vert u\vert \leq S,x \in \bar \Omega } \left\Vert D_u^i f(u,x)\right\Vert \quad (i \in \{ 0,1,2\} ).$

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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DOI: https://doi.org/10.1090/S0002-9947-1992-1066447-8
Article copyright: © Copyright 1992 American Mathematical Society

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