TY - JOUR
AU - Li, Yi
AU - Liu, Zhaoli
AU - Zhao, Cunshan
PY - 2008/09/01
Y2 - 2022/01/22
TI - Nodal solutions of perturbed elliptic problem
JF - Topological Methods in Nonlinear Analysis
JA - TMNA
VL - 32
IS - 1
SE -
DO -
UR - https://apcz.umk.pl/TMNA/article/view/TMNA.2008.035
SP - 49 - 68
AB - 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 nodalsolutions. Except $C^{1}$ continuity no further condition isneeded for $g$. We also prove a similar result for a continuoussublinear 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.
ER -