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.