In this paper, the main tool for obtaining the existence and localization of positive solutions to problem (1.1) is given by the norm-version of Krasnosel’skiǐ’s theorem in a cone [11].

In the case of systems, we use the vector version of Theorem 2.1 given in [12], which allows us to localize individually the components of a solution under different behaviors of the nonlinear terms of the system.

Looking for Carathéodory solutions of equations we are led to use Sobolev spaces.

The Sobolev space $H_0^1(0,1)$ given by

$$H_0^1(0,1)=\{u:[0,1]\to ℝ:u\text{is absolutely continuous,}{u}^{\prime 2}(0,1),u(0)=u(1)=0\}$$

is endowed with scalar product and energy norm

$${(u,v)}_{H_0^1}={\int }_0^{1}uv,|u{|}_{H_0^1}={({\int }_0^1{u}^{\prime }{}_{}{}^2)}^{\frac{1}{2}}.$$

Its dual space is denoted by $H^{-1}(0,1)$ and for any $h\in H^{-1}(0,1)$ and $\nu \in H_0^1(0,1),$ by $(h,\nu )$ we mean the value of the linear functional $h$ on $\nu$. In case that $h\in L^2(0,1),$ one has $(h,v)={(h,v)}_{L^2}={\int }_0^{1}hv.$

Also, $H^2(0,1)$ is the space

$$H^2(0,1)=\{u\in C^1[0,1]:u^{\prime }\text{is absolutely continuous,}{u}^{\prime \prime }\in L^2(0,1)\}.$$

Note that for any $h\in H^{-1}(0,1),$ there is a unique $u\in H_0^1(0,1)$ (weak solution) such that $-u^{\prime \prime }=h$ in the sense of distributions, i.e.

$${(u,\nu )}_{H_0^1}=(h,\nu )\text{for all}\nu \in H_0^1(0,1)$$

and one has

$${|u|}_{H_0^1}={|h|}_{H^{-1}}.$$

(2.1)

In particular, if $h\in L^2(0,1),$ then $u$ is a Carathéodory solution and can be expressed using the Green function as

$$u(x)={\int }_0^{1}G(x,s)h(s)𝑑s,$$

where $G(x,s)=s(1-x)$ for $0\le s\le x\le 1,$ $G(x,s)=x(1-s)$ for $0\le x\le s\le 1.$

The following continuous embeddings take place

$$H_0^1(0,1)\subset L^2(0,1)\subset H^{-1}(0,1)$$

and the Poincaré inequalities hold

$${|u|}_{L^2}\le \frac{1}{\sqrt{{\lambda }_1}}{|u|}_{H_0^1}(u\in H_0^1(0,1)),$$

$${|h|}_{H^{-1}}\le \frac{1}{\sqrt{{\lambda }_1}}{|h|}_{L^2}(h\in L^2(0,1)),$$

(2.2)

where ${\lambda }_1={\pi }^2$ is the first eigenvalue of the Dirichlet problem for the operator $Lu:=-u^{\prime \prime }$ in $(0,1)$ (see Remark 3.3 and Lemma 9.2 [13]).

Details about Sobolev spaces in one dimersion can be found in [13] and [14].

We seek positive solutions of $(\text{<a href="\#S1.E1" title="In 1 Introduction ‣ Energy-based localization of positive solutions for stationary Kirchhoff type equations and systems" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">1.1</span></a>})$ which are symmetric with respect to the middle of the interval $[0,1]$, that is $u(1-x)=u(x)$ for every $x\in [0,\frac{1}{2}]$. Note that the symmetry on $[0,1]$ of any $C^1$ function implies that $u^{\prime }\left(\frac{1}{2}\right)=0,$ which is useful in the proof of the next theorem. In this respect we consider the Hilbert space

$$X={\widehat{H}}_0^1(0,1):=\{u\in H_0^1(0,1):u(1-x)=u(x)\text{for all}x\in [0,\frac{1}{2}]\},$$

endowed with the scalar product and norm  ${(.,.)}_{H_0^1},|.{|}_{H_0^1}.$

In order to apply Theorem 2.1 we need a weak Harnack type inequality for the differential operator $Lu:=-\eta ({|u|}_{H_0^1})u^{\prime \prime }$ subject to the boundary conditions $u(0)=u(1)=0$.

Thus Theorem 3.1 is proved.    

Notice that such kind of estimations from below in terms of the energy norm have been given for the first time in [15] (see also [ppv]).

To give the operator form of the Dirichlet problem (1.1) we need to associate the solution operator.

(b) Uniqueness: Assume that $u_1,$ $u_2$ are two solutions of (3.5). Then

$$\eta ({|u_1|}_{H_0^1}){|u_1|}_{H_0^1}^2=(h,u_1),$$

$$\eta ({|u_2|}_{H_0^1}){(u_1,u_2)}_{H_0^1}=(h,u_1).$$

It follows that

$$\eta ({|u_1|}_{H_0^1}){|u_1|}_{H_0^1}^2=\eta ({|u_2|}_{H_0^1}){(u_1,u_2)}_{H_0^1}\le \eta ({|u_2|}_{H_0^1}){|u_1|}_{H_0^1}{|u_2|}_{H_0^1}.$$

Simplifying gives

$$\eta ({|u_1|}_{H_0^1}){|u_1|}_{H_0^1}\le \eta ({|u_2|}_{H_0^1}){|u_2|}_{H_0^1}.$$

The function $\eta (x)x$ being strictly increasing on ${ℝ}_{+},$ the last inequality gives

$${|u_1|}_{H_0^1}\le {|u_2|}_{H_0^1}.$$

By symmetry, the converse inequality also holds. Thus

$${|u_1|}_{H_0^1}={|u_2|}_{H_0^1}$$

and correspondingly

$$\eta ({|u_1|}_{H_0^1})=\eta ({|u_2|}_{H_0^1}).$$

Now the uniqueness of solution of the Dirichlet problem for the operator  $-u^{\prime \prime }$ yields $u_1=u_2.$

(c) Continuity: Let $h_k\to h$ in $H^{-1}(0,1)$ and let $u_k:=S(h_k).$ Using (3.7) we have that the sequence $(u_k)$ is bounded. Hence, passing if necessary to a subsequence, we may assume that the sequence of real numbers $({|u_k|}_{H_0^1})$ is convergent. We now prove that the sequence $(u_k)$ is Cauchy. From

$$-u_k^{\prime \prime }=\frac{1}{\eta ({|u_k|}_{H_0^1})}{h}_k,$$

we have

$$-{(u_k-u_p)}^{\prime \prime }=\frac{1}{\eta ({|u_k|}_{H_0^1})}{h}_k-\frac{1}{\eta ({|u_p|}_{H_0^1})}{h}_p$$

in the weak sense. Consequently

$${\left|u_k-u_p\right|}_{H_0^1}^2=(\frac{1}{\eta ({|u_k|}_{H_0^1})}{h}_k-\frac{1}{\eta ({|u_p|}_{H_0^1})}{h}_p,u_k-u_p)=$$

$$=\frac{1}{\eta ({|u_k|}_{H_0^1})}(h_k-h_p,u_k-u_p)+\left(\frac{1}{\eta ({|u_k|}_{H_0^1})}-\frac{1}{\eta ({|u_p|}_{H_0^1})}\right)(h_p,u_k-u_p).$$

Since $\eta$ is Lipschitz continuous on ${ℝ}_{+},$ there is an $L>0$  such that

$$\left|\eta ({|u_p|}_{H_0^1})-\eta ({|u_k|}_{H_0^1})\right|\le L\left|{|u_p|}_{H_0^1}-{|u_k|}_{H_0^1}\right|.$$

Furthermore

$${\left|u_k-u_p\right|}_{H_0^1}^2\le \frac{1}{\eta (0)}{|h_k-h_p|}_{H^{-1}}{|u_k-u_p|}_{H_0^1}+\frac{L\left|{|u_p|}_{H_0^1}-{|u_k|}_{H_0^1}\right|}{\eta {(0)}^2}{|h_p|}_{H^{-1}}{\left|u_k-u_p\right|}_{H_0^1},$$

whence the simplification gives

$${|u_k-u_p|}_{H_0^1}\le \frac{1}{\eta (0)}{|h_k-h_p|}_{H^{-1}}+\frac{L}{{\eta }^2(0)}{|h_p|}_{H^{-1}}\left|{|u_p|}_{H_0^1}-{|u_k|}_{H_0^1}\right|.$$

Since ${|h_k|}_{H^{-1}}$ is bounded and $(h_k)$ and $({|u_k|}_{H_0^1})$ are convergent, one immediately obtain that the sequence $(u_k)$ is Cauchy. Hence there is $u$ with $u_k\to u$ and passing to the limit in

$$\eta ({|u_k|}_{H_0^1}){(u_k,\nu )}_{H_0^1}=(h_k,\nu ),v\in H_0^1(0,1)$$

we obtain that $u=S(h).$ Finally the uniqueness of the solution implies that the whole sequence $(u_k)$ converges to $S(h),$ that is $S(h_k)\to S(h).$

(d) Inequality (3.7) immediately follows from (3.6) by taking $v=u$ and using the obvious inequality $(h,u)\le {|h|}_{H^{-1}}{|u|}_{H_0^1}.$    

The next theorem expresses a monotonicity property of the solution operator.

We first note that $N$ is well-defined from $K$ to $K.$ Indeed, if $u\in K,$ then $f(.,u)\in L^2(0,1)$ and so $N(u)$ has the expression

$$N(u)(x)=\frac{1}{\eta ({|u|}_{H_0^1})}{\int }_0^{1}G(x,s)f(s,u(s))𝑑s.$$

Since the Green’s function $G$ is symmetric and so is $f$ accordingly with (3.9), the function $N\left(u\right)$ is also symmetric. Moreover, since $f(.,u)$ is nonnegative on $(0,1)$ and increasing on $(0,\frac{1}{2}),$ the function $N\left(u\right)$ satisfies the Harnack inequality

$$N(u)\ge \gamma {|N(u)|}_{H_0^1}.$$

Therefore $N(u)\in K$ and thus $N\left(K\right)\subset K.$

In addition, the operator $N$ is completely continuous. Indeed, if $ℳ\subset K$ is a bounded set, then it is easy to see that $N(ℳ)$ is bounded in $H^2(0,1)$ and thus it is compact in $H^1(0,1)$ as claimed.

Next we prove that

$${|u|}_{H_0^1}\ge {|N(u)|}_{H_0^1}\text{for all}u\in K\text{with}{|u|}_{H_0^1}=\alpha .$$

(3.14)

We suppose the contrary, i.e.,

(3.15)

Denote $\nu :=N(u)=Sf(\cdot ,u)$ and use (3.7) and (2.2), where $h=f(\cdot ,u),$ to deduce

$$\eta ({|\nu |}_{H_0^1}){|\nu |}_{H_0^1}\le {|f(\cdot ,u)|}_{H^{-1}}\le \frac{1}{\pi }{|f(\cdot ,u)|}_{L^2}.$$

(3.16)

Furthermore, since for every $x$ one has

$$u(x)={\int }_0^{x}1\cdot u^{\prime }(\tau )𝑑\tau \le {\left({\int }_0^x{1}^2𝑑\tau \right)}^{\frac{1}{2}}\cdot {\left({\int }_0^{x}u_{}^{\prime }{}_{}{}^2(\tau )𝑑\tau \right)}^{\frac{1}{2}}\le {|u|}_{H_0^1}=\alpha$$

and $f$ is increasing in the second variable, we have

$${|f(\cdot ,u)|}_{L^2}\le {|f(\cdot ,\alpha )|}_{L^2}.$$

(3.17)

Now, from (3.16) and (3.17), we have

$$\eta ({|\nu |}_{H_0^1}){|\nu |}_{H_0^1}\le \frac{1}{\pi }{|f(\cdot ,\alpha )|}_{L^2}.$$

(3.18)

On the other hand, since from (3.15) ${|\nu |}_{H_0^1}>\alpha$ and $s\eta (s)$ is strictly increasing in ${ℝ}_{+},$ we have

$$\eta ({|\nu |}_{H_0^1}){|\nu |}_{H_0^1}>\alpha \eta (\alpha ).$$

(3.19)

Then (3.18) and (3.19) lead to

which contradicts our assumption (3.10). Thus (3.14) is proved.

Next we prove that

$${|u|}_{H_0^1}\le {|N(u)|}_{H_0^1}\text{for all}u\in K\text{with}{|u|}_{H_0^1}=\beta .$$

(3.20)

We suppose the contrary, i.e.,

$$\text{there exists}u\in K:{|u|}_{H_0^1}=\beta \text{and}{|u|}_{H_0^1}>{|N(u)|}_{H_0^1}.$$

(3.21)

Letting  $v:=N\left(u\right)$ gives

$$-\eta ({|\nu |}_{H_0^1}){\nu }^{\prime \prime }=f(\cdot ,u)$$

in the weak sense. Then also using (3.12) we obtain

$$-\eta ({|\nu |}_{H_0^1})({\nu }^{\prime \prime },\nu )=\eta ({|\nu |}_{H_0^1}){|\nu |}_{H_0^1}^2={(f(\cdot ,u),\nu )}_{L^2}={\int }_0^1\nu (x)f(x,u(x))𝑑x$$

$$\ge {\int }_{x_0}^{1-x_0}\nu (x)f(x,u(x))𝑑x\ge {\int }_{x_0}^{1-x_0}\nu (x)f(x,\gamma \beta )𝑑x\ge \gamma {|\nu |}_{H_0^1}{\int }_{x_0}^{1-x_0}f(x,\gamma \beta )𝑑x.$$

Consequently

$$\eta ({|\nu |}_{H_0^1}){|\nu |}_{H_0^1}\ge \gamma {\int }_{x_0}^{1-x_0}f(x,\gamma \beta )𝑑x.$$

(3.22)

On the other hand,   whence

$$\beta \eta (\beta )>\gamma {\int }_{x_0}^{1-x_0}f(x,\gamma \beta )𝑑x,$$

which contradicts (3.11). Thus (3.20) is proved. Now Krasnosel’skiǐ’s theorem applies and yields the result.    

In this section we extend the results from the equations to the case of systems. We look for solutions $u=(u_1,u_2,\mathrm{\dots },u_n)$ with $u_i\in K$ for all $i=1,2,\mathrm{\dots },n.$ Under the details given before, to each ${\eta }_i$ we associate a solution operator $S_i:H^{-1}(0,1)\to H_0^1(0,1)$. Then system $(\text{<a href="\#S1.E2" title="In 1 Introduction ‣ Energy-based localization of positive solutions for stationary Kirchhoff type equations and systems" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">1.2</span></a>})$ reads equivalently as

$$u_i=S_i{f}_i(\cdot ,u),i=1,2,\mathrm{\dots },n.$$

Let $N:K^n\to K^n,$ $N=(N_1,N_2,\mathrm{\dots },N_n)$ be defined by

$$N_i(u)=S_i{f}_i(\cdot ,u(\cdot ))(i=1,2,\mathrm{\dots },n).$$

If $u_i\in K$ for each $i,$ then $N_i(u)\in K.$ Thus the cone $K^n$ is invariant by $N.$ Moreover, the operator $N$ is completely continuous since, by standard arguments, the components $N_i$ are completely continuous.