Taiwanese Journal of Mathematics

Multiple Solutions for a Nonlinear Elliptic System Subject to Nonautonomous Perturbations

Danila Sandra Moschetto

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In this paper we consider the following Neumann problem $$ \left\{ \begin{array}{lll} -\Delta u = \alpha(x)(F_u(u,v)-u) + \lambda G_u(x,u,v) \ in \; \Omega \\ -\Delta v = \alpha(x) (F_v(u,v)-v) + \lambda G_v(x,u,v) \ in \; \Omega \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0 \ on \ \partial \Omega \end{array} \right. $$ In particular, by means of a multiplicity theorem obtained by Ricceri, we establish that if the set of all global minima of the function $\mathbb{R}^2 \ni y \longmapsto \frac{|y|^2}{2} - F(y)$ (where $F \in C^1(\mathbb{R}^2)$ and it satisfies the condition $F(0,0) = 0$) has at least $k \geq 2$ connected components, then the above Neumann problem admits at least $k+1$ weak solutions, $k$ of which are lying in a given set.

Article information

Taiwanese J. Math., Volume 15, Number 3 (2011), 1163-1169.

First available in Project Euclid: 18 July 2017

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Zentralblatt MATH identifier

Primary: 35J20: Variational methods for second-order elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations

Neumann problem multiplicity of solutions global minima connected components


Moschetto, Danila Sandra. Multiple Solutions for a Nonlinear Elliptic System Subject to Nonautonomous Perturbations. Taiwanese J. Math. 15 (2011), no. 3, 1163--1169. doi:10.11650/twjm/1500406292. https://projecteuclid.org/euclid.twjm/1500406292

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