Finer analysis of the Nehari set associated to a class of Kirchhoff-type equations

Abstract

We extend and improve the results of Chen-Ou (Comput. Math. Appl. 77(10):2859–2866, 2019), which concern a Kirchhoff-type equation depending on two real parameters $\lambda ,\mu$. Our technique, which relies upon a refined analysis of the Nehari set associated to the problem, permit us prove existence and multiplicity of solutions by minimizing the associated energy functional over components of the Nehari set. We also analyze the threshold of the method and prove existence of solutions even in the case where the Nehari set is not a manifold.

Appendix

Lemma 7

There holds:

(i) :

Let $a\lambda _1<\lambda \le \lambda '< \lambda ^*$ and $b\mu _1\le \mu \le \mu '\le \mu ^*(\lambda ')$ then ${I}^+_{\lambda ,\mu }>{I}^+_{\lambda ',\mu '}$, where $(\lambda ,\mu )\ne (\lambda ',\mu ')$;

(ii) :

Take $u\in H^1_0(\varOmega )\setminus {0}$. Let $B \subset {\mathbb {R}}^2$ be an open such that $s_{\lambda ,\mu }(u)$ are well defined for all $(\lambda ,\mu ) \in B$. Then the function $(\lambda ,\mu )\mapsto {\varPhi }_{\lambda ,\mu |{\mathcal {N}}^+_{\lambda ,\mu }}(u)$ is continuous. Moreover, if $\lambda \le \lambda '$, $\mu \le \mu '$ and $u\in {\mathcal {N}}^+_{\lambda ,\mu }\cap {\mathcal {N}}^+_{\lambda ',\mu '}$ then $\varPhi _{\lambda ,\mu }(u)>\varPhi _{\lambda ',\mu '}(u)$, where $(\lambda ,\mu )\ne (\lambda ',\mu ')$

Proof

• (i) By Proposition 8 there exists an $u\in H_0^1(\varOmega )$ such that ${\hat{J}}^+_{\lambda ,\mu }=\varPhi _{\lambda ,\mu }(u)$, since that $(\lambda ,\mu )\ne (\lambda ',\mu ')$, we can suppose without loss of generality that $\mu <\mu '$ which implies that $H_{\lambda '}(u)\le H_{\lambda }(u)<0$ and $G_{\mu '}(u)<G_\mu (u)$ hence

  $$\begin{aligned} \varPhi _{\lambda ,\mu }(u)=\frac{1}{2}H_\lambda (u)+\frac{1}{4}G_\mu (u)>\frac{1}{2}H_{\lambda '}(u)+\frac{1}{4}G_{\mu '}(u). \end{aligned}$$

  (28)

  We claim that $G_{\mu '}(u)>0$. Indeed, since $H_{\lambda '}(u)<0$ it follows that $u\in {\mathcal {M}}_{\lambda '}$ and by definition of the $\mu ^*(\lambda ')$ and by Corollary 1 we conclude that $\mu ^*(\lambda ')< b\frac{\Vert u\Vert ^4}{\Vert u\Vert ^4_4}$ what implies that $G_{\mu ^*(\lambda ')}(u)>0$, once that $\mu '\le \mu ^*(\lambda ')$ we conclude $G_{\mu '}(u)>0$. Therefore $H_{\lambda '}(u)<0<G_{\mu '}(u)$ and from Proposition 1 there exists a $s_{\lambda ',\mu '}(u)>0$ such that $s_{\lambda ',\mu '}(u)u\in {\mathcal {N}}_{\lambda '.\mu '}^+$ it follows by (28)

  $$\begin{aligned} {I}^+_{\lambda ',\mu '}\le \varPhi ^+_{\lambda ',\mu '}(u)<\varPhi _{\lambda ',\mu '}(u)<\varPhi _{\lambda ,\mu }(u)={I}^+_{\lambda ,\mu }. \end{aligned}$$

• (ii) It is immediately of $\varPhi _{\lambda ,\mu }(u)=\frac{1}{4}H_\lambda (u)$. $\square$

Remark 3

It is worthwhile to point out that Lemma 7 also holds for ${I}^-_{\lambda ,\mu }$ and $\varPhi _{\lambda ,\mu |{\mathcal {N}}^-_{\lambda ,\mu }}$.

Now consider the mountain pass critical values

$$\begin{aligned} c_{\lambda ,\mu }:=\inf _{\varphi \in \varGamma _\lambda }\max _{t\in [0,1]}\varPhi _{\lambda ,\mu }(\gamma (t)), \end{aligned}$$

(29)

where $\varGamma _\lambda =\{\gamma \in C([0,1],H_0^1(\varOmega )): \gamma (0)=0,\ \varPhi _{\lambda ,\mu }(\gamma (1))<0\}$.

Proposition 7

Let $(\lambda ,\mu )\in (-\infty ,a\lambda _1)\times (b\mu _1,+\infty )$. Then

$$\begin{aligned} c_{\lambda ,\mu }={\hat{J}}^-_{\lambda ,\mu }. \end{aligned}$$

Proof

Indeed, by one hand, we know from [5], via mountain pass theorem, that there exists $u\in H^1_0(\varOmega )$ a positive solution for (1) with $\varPhi _{\lambda ,\mu }(u)=c_{\lambda ,\mu }$. On the other hand we have by Theorem 2 that there exists ${\bar{u}}\in H^1_0(\varOmega )$ such that ${I}^-_{\lambda ,\mu }=\varPhi _{\lambda ,\mu }({\bar{u}})$. From Propositions 2 and 4, we have that $u\in {\mathcal {N}}_{\lambda ,\mu }^-$ and hence

$$\begin{aligned} {I}^-_{\lambda ,\mu }\le c_{\lambda ,\mu }. \end{aligned}$$

We claim that ${I}^-_{\lambda ,\mu }=c_{\lambda ,\mu }$. Indeed, once that ${I}^-_{\lambda ,\mu }=\varPhi _{\lambda ,\mu }({\bar{u}})$ then we have $G_\mu ({\bar{u}})<0<H_\lambda ({\bar{u}})$ hence there exists $t_0>0$ sufficiently large such that $\varPhi _{\lambda ,\mu }(t_0{\bar{u}})<0$. However by definition of $c_{\lambda ,\mu }$ and ${I}_{\lambda ,\mu }$ we obtain

$$\begin{aligned} c_{\lambda ,\mu }\le \varPhi _{\lambda ,\mu }(s_{\lambda ,\mu }(t_0{\bar{u}})t_0{\bar{u}})=\varPhi _{\lambda ,\mu }({\bar{u}})={I}^-_{\lambda ,\mu }, \end{aligned}$$

therefore ${I}^-_{\lambda ,\mu }=c_{\lambda ,\mu }$. $\square$