### Critical point approaches to quasilinear second order differential equations depending on a parameter

#### Abstract

In this paper, we make application of some three-critical points

results to establish the existence of at least three solutions

for a boundary value problem for the quasilinear second order differential equation on a compact interval $[a,b]\subset\mathbb{R}$,

$$

\cases

-u''=(\lambda f(x,u)+g(x,u))h(x,u') &\text{\rm in } (a,b),\\

u(a)=u(b)=0,

\endcases

$$

under appropriate hypotheses. We exhibit the

existence of at least three (weak) solutions.

results to establish the existence of at least three solutions

for a boundary value problem for the quasilinear second order differential equation on a compact interval $[a,b]\subset\mathbb{R}$,

$$

\cases

-u''=(\lambda f(x,u)+g(x,u))h(x,u') &\text{\rm in } (a,b),\\

u(a)=u(b)=0,

\endcases

$$

under appropriate hypotheses. We exhibit the

existence of at least three (weak) solutions.

#### Keywords

Dirichlet problem; critical point; three solutions; variational methods

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