# Three solutions for pertubed Dirichilet problem

Giuseppe Rao, Giuseppe Cordaro

Risultato della ricerca: Article

### Abstract

\begin{abstract} In this paper we prove the existence of at least three distinct solutions to the following perturbed Dirichlet problem \begin{displaymath} \left\{ \begin{array}{ll} -\Delta u= f(x,u)+\lambda g(x,u) & \mbox{in\ } \Omega\\ u=0 & \mbox{on\ } \partial \Omega, \end{array}\right. \end{displaymath} where $\Omega\subset\mathbb{R}^N$ is an open bounded set with smooth boundary $\partial \Omega$ and $\lambda\in \mathbb{R}$. Under very mild conditions on $g$ and some assumptions on the behaviour of the potential of $f$ at $0$ and $+\infty$, our result assures the existence of at least three distinct solutions to the above problem for $\lambda$ small enough. Moreover such solutions belong to a ball of the space $W_0^{1,2}(\Omega)$ centered in the origin and with radius not dependent on $\lambda$. \end{abstract}
Lingua originale English 3879-3883 NONLINEAR ANALYSIS 68 Published - 2008

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Three Solutions
Distinct
Partial
Bounded Set
Open set
Dirichlet Problem
Ball
Subset
Dependent

### All Science Journal Classification (ASJC) codes

• Mathematics(all)
• Applied Mathematics
• Analysis

### Cita questo

Three solutions for pertubed Dirichilet problem. / Rao, Giuseppe; Cordaro, Giuseppe.

In: NONLINEAR ANALYSIS, Vol. 68, 2008, pag. 3879-3883.

Risultato della ricerca: Article

Rao, G & Cordaro, G 2008, 'Three solutions for pertubed Dirichilet problem', NONLINEAR ANALYSIS, vol. 68, pagg. 3879-3883.
Rao, Giuseppe ; Cordaro, Giuseppe. / Three solutions for pertubed Dirichilet problem. In: NONLINEAR ANALYSIS. 2008 ; Vol. 68. pagg. 3879-3883.
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AB - \begin{abstract} In this paper we prove the existence of at least three distinct solutions to the following perturbed Dirichlet problem \begin{displaymath} \left\{ \begin{array}{ll} -\Delta u= f(x,u)+\lambda g(x,u) & \mbox{in\ } \Omega\\ u=0 & \mbox{on\ } \partial \Omega, \end{array}\right. \end{displaymath} where $\Omega\subset\mathbb{R}^N$ is an open bounded set with smooth boundary $\partial \Omega$ and $\lambda\in \mathbb{R}$. Under very mild conditions on $g$ and some assumptions on the behaviour of the potential of $f$ at $0$ and $+\infty$, our result assures the existence of at least three distinct solutions to the above problem for $\lambda$ small enough. Moreover such solutions belong to a ball of the space $W_0^{1,2}(\Omega)$ centered in the origin and with radius not dependent on $\lambda$. \end{abstract}

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