Topological Methods in Nonlinear Analysis

Nodal solutions of perturbed elliptic problem

Yi Li, Zhaoli Liu, and Cunshan Zhao

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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

Permanent link to this document
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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