### Nodal solutions of perturbed elliptic problem

DOI: http://dx.doi.org/10.12775/TMNA.2008.035

#### 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.

$$

\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.

#### Keywords

Nodal solutions; elliptic problem; perturbation from symmetry; essential values

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