## Topological Methods in Nonlinear Analysis

### Nodal solutions of perturbed elliptic problem

#### Abstract

Multiple nodal solutions are obtained for the elliptic problem \begin{alignat}{2} -\Delta u&=f(x, u)+\varepsilon g(x, u)&\quad& \text{in } \Omega,\\ u&=0&\quad& \text{on } \partial \Omega , \end{alignat} where $\varepsilon$ is a parameter, $\Omega$ is a smooth bounded domain in ${{\mathbb R}}^{N}$, $f\in C(\overline{\Omega }\times {{\mathbb R}})$, and $g\in C(\overline{\Omega }\times {{\mathbb R}})$. For a superlinear $C^{1}$ function $f$ which is odd in $u$ and for any $C^{1}$ function $g$, we prove that for any $j\in {\mathbb N}$ there exists $\varepsilon _{j}> 0$ such that if $|\varepsilon |\leq \varepsilon _{j}$ then the above problem possesses at least $j$ distinct nodal solutions. Except $C^{1}$ continuity no further condition is needed for $g$. We also prove a similar result for a continuous sublinear function $f$ and for any continuous function $g$. Results obtained here refine earlier results of S. J. Li and Z. L. Liu in which the nodal property of the solutions was not considered.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 32, Number 1 (2008), 49-68.

Dates
First available in Project Euclid: 13 May 2016

https://projecteuclid.org/euclid.tmna/1463150462

Mathematical Reviews number (MathSciNet)
MR2466802

Zentralblatt MATH identifier
1173.35497

#### Citation

Li, Yi; Liu, Zhaoli; Zhao, Cunshan. Nodal solutions of perturbed elliptic problem. Topol. Methods Nonlinear Anal. 32 (2008), no. 1, 49--68. https://projecteuclid.org/euclid.tmna/1463150462

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