Nodal solutions of perturbed elliptic problem

Abstract

Multiple nodal solutions are obtained for the elliptic problem
$$
\alignat 2
-\Delta u&=f(x,\ u)+\varepsilon g(x,\ u)&\quad& \text{in } \Omega,\\
u&=0&\quad& \text{on } \partial \Omega ,
\endalignat
$$
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.