3.1. On a parametrized problem

Throughout the sequel we will take into account the results of Section 2 with $g=f$. Thus, for every $w\in C^1_0(\overline \Omega )$, the parametrized Dirichlet problem reads as follows

(P _w)\begin{equation} \left\{ \begin{array}{@{}ll@{}} -\Delta_p u= f(x,u, \nabla w) & {\rm in}\ \Omega \\ u=0 & {\rm on}\ \partial\Omega. \end{array}\right. \end{equation}

Recall that each solution $u$ of $(P_w)$ is in $C^1_0(\overline \Omega )$. Also, assumptions $(f_2)$ and $(f_3)$ imply that $f(x,0,\xi )=0$ for all $x,\xi$, thus the zero function is a solution of $(P)$ and $(P_w)$ for each $w\in C^1_0(\overline \Omega )$.

In [Reference Faraci, Motreanu and Puglisi13], we have proved the following

From the proof of [Reference Faraci, Motreanu and Puglisi13, Theorem 2.1] we deduce that

We conclude this subsection recalling the strong comparison principle for the $p$-Laplace operator ([Reference Arcoya and Ruiz2]). For $h_1, h_2 \in L^\infty (\Omega )$, we say that $h_1 \prec h_2$ if for any $\Omega _0 \subseteq \Omega$ compact subset, there exists $\varepsilon >0$ such that $h_1(x) + \varepsilon < h_2(x)$ for almost every $x \in \Omega _0$. In particular, if $h_1$ and $h_2$ are continuous functions such that $h_1(x)< h_2(x)$ for all $x\in \Omega$, then $h_1 \prec h_2$.

Proposition 3.1 [Reference Arcoya and Ruiz2, Proposition 2.6] For $\lambda \geq 0$ and $f, g \in L^\infty (\Omega )$, let $v, u$ be solutions of the problems$:$

\begin{align*} & \left\{ \begin{array}{@{}ll@{}} -\Delta_p v+ \lambda |v|^{p-2} v= f & {\rm in}\ \Omega \\ u=0 & {\rm on}\ \partial\Omega, \end{array}\right. \\ & \left\{ \begin{array}{@{}ll@{}} -\Delta_p u+ \lambda |u|^{p-2} u= g & {\rm in}\ \Omega \\ u=0 & {\rm on}\ \partial\Omega, \end{array}\right. \end{align*}

If $f \prec g$ and $u \in \mathrm {int}(C^1_0(\overline \Omega )_+)$, then $u-v\in \mathrm {int}(C^1_0(\overline \Omega )_+)$.

3.2. A pseudogradient vector field

Throughout the sequel, $u_0\in C^1_0(\overline \Omega )$ is a fixed function, $M>0$ is given by proposition 2.1 and we choose $m=m_M$ in assumption $(f_4)$. Thus, following the notation of Section 2 with $g=f$ and $\mu =m$ one has

\begin{align*} \|u\|_{m} & := \left(\int_{\Omega}(|\nabla u|^p + m |u|^p )\,{\rm d}x\right)^{1/p},\\ A_w(u)& := (-\Delta_p+ m h_p({\cdot}))^{{-}1}(f(x, u, \nabla w)+ m h_p(u)),\\ J_w(u) & = \frac1p \int_{\Omega}|\nabla u|^p\,{\rm d}x - \int_{\Omega} F(x, u, \nabla w)\,{\rm d}x, \end{align*}

where $F(x, t, \xi ) = \int _0^t f(x, s, \xi )\,{\rm d}s$.

Let us introduce the set $\Lambda ^w$ which will be crucial in our argument. Let $u_P^w \in \mathrm {int}(C^1_0(\overline \Omega )_+)$ and $u_N^w \in - \mathrm {int}(C^1_0(\overline \Omega )_+)$ the smallest positive solution and the biggest negative solution of $(P_w)$ (see theorem 3.1). Let us denote by $[u_N^w, u_P^w]$ the set of all $C^1_0$-functions $u$ such that $u_N^w \leq u \leq u_P^w$.

Consider then, the following set

\[ \Lambda^w=\{ u \in C^1_0(\overline\Omega) : \ u \in \text{int}_{C^1_0} [u_N^w, u_P^w]\}. \]

Proposition 3.2

\[ A_w(\Lambda^w) \subseteq \Lambda^w \quad \text{and} \quad A_w(\mathrm{int}(C^1_0(\overline\Omega)_+)) \subseteq \mathrm{int}(C^1_0(\overline\Omega)_+). \]

Proof. First, we show that $A_w(\Lambda ^w) \subseteq \Lambda ^w$. Let $u \in \Lambda ^w$ and $v=A_w(u)$:

\begin{align*} -\Delta_p v + m h_p(v) & = f(x, u, \nabla w) + m h_p(u)\\ (\text{by}\ (f_4)\ \text{and continuity of}\ f)\ & \prec f(x, u_P^w, \nabla w) + m h_p( u_P^w)\\ & ={-}\Delta_p u_P^w + m h_p( u_P^w). \end{align*}

By proposition 3.1, we conclude that $u_P^w-v \in \mathrm {int}(C^1_0(\overline \Omega )_+)$. Analogously one obtains that $v- u_N^w \in \mathrm {int}(C^1_0(\overline \Omega )_+)$. For the other inclusion, let $u \in \mathrm {int}(C^1_0(\overline \Omega )_+)$ and $v=A_w(u)$. Thus,

\[ -\Delta_p v + m h_p(v) = f(x, u, \nabla w) + m h_p(u) >0 \]

By the strong maximum principle [Reference Vázquez23], we conclude that $v \in \mathrm {int}(C^1_0(\overline \Omega )_+)$.

Let $B=B_w$ be as in lemma 2.2 with $D=\Lambda _w$. Thus, since $B_w$ is a convex combination of $A_w$, one has

(3.1)\begin{equation} B_w(\Lambda^w) \subseteq \Lambda^w \quad \text{and} \quad B_w(\mathrm{int}(C^1_0(\overline\Omega)_+)) \subseteq \mathrm{int}(C^1_0(\overline\Omega)_+). \end{equation}

For every $u \in C^1_0(\overline \Omega ) \setminus K_w$ (where $K_w$ is the set of all fixed points of $A_w$) consider the following Cauchy problem

(3.2)\begin{equation} \left\{\begin{array}{@{}ll@{}} \displaystyle{\dfrac{{\rm d}}{{\rm d}t} \varphi(t) ={-}\varphi(t) + B_w(\varphi(t))}\\ \varphi(0) = u. \end{array}\right. \end{equation}

Since $B_w$ is locally Lipschitz, the above problem admits a unique solution $\varphi ^t(u)$ in $C^1_0(\overline \Omega )$ called descending flow curve with maximal interval of existence $[0, \tau (u)[$. Notice that $\tau (u)$ can be either a positive number or $+\infty$.

By lemma 2.2 $(iii)$,

\begin{align*} \frac{{\rm d}}{{\rm d}t} J_w(\varphi^t(u)) & = \langle J_w^\prime(\varphi^t(u)), \frac{{\rm d}}{{\rm d}t}\varphi^t(u)\rangle\\ & ={-} \langle J_w^\prime(\varphi^t(u)), \varphi^t(u) - B_w(\varphi^t(u)\rangle\\ & <0 \end{align*}

and the inequality is strict since $u \notin K_w$ so that it is not a fixed point of $B_w$. Thus, $J_w(\varphi ^t(u))$ is strictly decreasing. Recall also that $J_w$ is coercive, hence bounded from below.

Moreover from (3.2), we have

\[ \int_0^t e^s \frac{{\rm d}}{{\rm d}s} \varphi^s(u)\,{\rm d}s ={-}\int_0^t e^s\varphi^s(u)\,{\rm d}s+ \int_0^t e^sB_w(\varphi^s(u)) \,{\rm d}s, \]

or

(3.3)\begin{equation} \varphi^t(u)= e^{{-}t} u+ e^{{-}t}\int_0^t e^sB_w(\varphi^s(u))\,{\rm d}s. \end{equation}

By (3.1), one has the following (see [Reference Liu and Sun19, proof of Lemma 3.2]).

For $D \subseteq C^1_0(\overline \Omega )$, let us denote by

\[ \mathcal{C}(D) = D\cup \{u \in C^1_0(\overline\Omega)\setminus K_w: \ \text{there exists}\ \bar t \geq 0\ \text{such that}\ \varphi^{\bar t}(u) \in D\}. \]

We recall that $D$ is called an invariant set of descending flow for $J_w$, if whenever $u\in D\setminus K_w$, the flow $\{\varphi ^t(u):t\in [0,\tau (u)[\}\subset D$. If $D=\mathcal {C}(D)$, then $D$ is said complete.

If $D$ is invariant then, by the definition above, $\mathcal {C}(D)$ is invariant. Moreover it is easy to see that $\mathcal {C}(D)$ is complete as $\mathcal {C}(\mathcal {C}(D))=\mathcal {C}(D)$. Then $\partial \mathcal {C}(D)$ is also an invariant set of descending flow [Reference Liu and Sun19, Lemma 2.3]. If $D$ is also open then $\mathcal {C}(D)$ is open [Reference Liu and Sun19, Lemma 2.4(i)].

By lemma 3.2, $\Lambda ^w$ and $\mathrm {int}(C^1_0(\overline \Omega )_+)$ are invariant sets of descending flow for $J_w$, so $\mathcal {C}(\Lambda ^w)$ and $\mathcal {C}(\mathrm {int}(C^1_0(\overline \Omega )_+))$ are invariant, as well as $\partial \mathcal {C}(\Lambda ^w)$ and $\partial \mathcal {C}(\mathrm {int}(C^1_0(\overline \Omega )_+))$. Moreover, $\mathcal {C}(\Lambda ^w)$ is open in $C^1_0(\overline \Omega )$ and $\mathcal {C}(\Lambda ^w) \not = C^1_0(\overline \Omega )$ (indeed $u^w_P, u^w_N \in K_w\setminus \Lambda ^w$, so they cannot lie in $\mathcal {C}(\Lambda ^w)$). Also, $\mathcal {C}(\mathrm {int}(C^1_0(\overline \Omega )_+))$ is open in $C^1_0(\overline \Omega )$ and $\mathcal {C}(\mathrm {int}(C^1_0(\overline \Omega )_+)) \not = C^1_0(\overline \Omega )$ (indeed $0 \in K_w\setminus \mathrm {int}(C^1_0(\overline \Omega )_+)$). Since $C^1_0(\overline {\Omega })_+\subseteq \overline {\mathcal {C}(\mathrm {int}(C^1_0(\overline \Omega )_+))}$ and $\partial \mathcal {C}(\Lambda ^w) \cap C^1_0(\overline {\Omega })_+ \not =\emptyset$, we have $\partial \mathcal {C}(\Lambda ^w) \cap \overline {\mathcal {C}(\mathrm {int}(C^1_0(\overline \Omega )_+))} \not = \emptyset$. Because $\partial \mathcal {C}(\Lambda ^w) \cap (-C^1_0(\overline {\Omega })_+) \not = \emptyset$, we obtain that $\partial \mathcal {C}(\Lambda ^w) \cap \partial \mathcal {C}(\mathrm {int}(C^1_0(\overline \Omega )_+)) \not = \emptyset$. Let us fix

\[ u^* \in \partial \mathcal{C}(\Lambda^w) \cap \partial \mathcal{C}(\mathrm{int}(C^1_0(\overline\Omega)_+)). \]

Proof. Let $0< t_1< t_2<\tau (u^*)$. Then,

\begin{align*} \|\varphi^{t_2}(u^*)-\varphi^{t_1}(u^*)\|_m& \leq \int_{t_1}^{t_2} \| \frac{{\rm d}}{{\rm d}t}\varphi^t(u^*)\|_m\,{\rm d}t \\ & =\int_{t_1}^{t_2}\|\varphi^t(u^*)-B_w(\varphi^t(u^*))\|_m\\ \text{(by lemma 2.2}, (ii)) & \leq 2\int_{t_1}^{t_2}\|\varphi^t(u^*)-A_w(\varphi^t(u^*))\|_m. \end{align*}

Assume that $p\geq 2$.

Applying Hölder inequality, lemma 2.2 $\it {(iii)}$, the monotonicity of the flow $t\to J_w(\varphi ^{t}(u^*))$ and the boundedness from below of $J_w$, we obtain

\begin{align*} \int_{t_1}^{t_2}\|\varphi^t(u^*)-A_w(\varphi^t(u^*))\|_m& \leq \left(\int_{t_1}^{t_2}\|\varphi^t(u^*)-A_w(\varphi^t(u^*))\|^p_m\right)^{1/p} (t_2-t_1)^{1/{p'}} \\ & \leq c_1 \left(\int_{t_1}^{t_2}\frac{{\rm d}}{{\rm d}t} J_w(\varphi^t(u^*))\,{\rm d}t\right)^{1/p} \ (t_2-t_1)^{1/{p'}} \\ & =c_1(J_w(\varphi^{t_2}(u^*))-J_w(\varphi^{t_1}(u^*)))^{1/p} \ (t_2-t_1)^{1/{p'}} \\ & \leq c_2(t_2-t_1)^{1/{p'}}. \end{align*}

Putting together the above outcomes we get that

\[ \|\varphi^{t_2}(u^*)-\varphi^{t_1}(u^*)\|_m \leq c (t_2-t_1)^{1/{p'}}. \]

Assume now $1< p< 2$. By coercivity of $J_w$, we deduce that the set $\{u: J_w(u)\leq J_w(\varphi ^0(u^*))=J_w(u^*)\}\subset \overline B(0,b)$ for some $b>0$ where $\overline B(0,b)$ denotes the closed ball in $W^{1,p}_0(\Omega )$ centred at zero of radius $b$. By the monotonicity of the flow, $J_w(\varphi ^t(u^*)\leq J_w(u^*)$, thus there exists a constant $b_1>0$, such that $\|\varphi ^t(u^*)\|_m, \|A_w(\varphi ^t(u^*))\|_m\leq b_1$ for each $t$.

Then, by Hölder inequality, lemma 2.2 $(iii)$

\begin{align*} \int_{t_1}^{t_2}\|\varphi^t(u^*)-A_w(\varphi^t(u^*))\|_m& \leq \left(\int_{t_1}^{t_2}\|\varphi^t(u^*)-A_w(\varphi^t(u^*))\|^2_m (\|\varphi^t(u^*)\|_m\right.\nonumber\\ & \quad + \left.\|A_w(\varphi^t(u^*))\|_m)^{p-2}\,{\rm d}t\right)^{1/2} \\ & \quad \cdot\left(\int_{t_1}^{t_2}(\|\varphi^t(u^*)\|_m + \|A_w(\varphi^t(u^*))\|_m)^{2-p}\,{\rm d}t\right)^{1/2}\\ & \leq c_1 \left(\int_{t_1}^{t_2}\frac{{\rm d}}{{\rm d}t} J_w(\varphi^t(u^*))\right)^{1/2} (t_2-t_1)^{1/2} \\ & =c_1(J_w(\varphi^{t_2}(u^*))-J_w(\varphi^{t_1}(u^*)))^{1/2} (t_2-t_1)^{1/2} \\ & \leq c_2(t_2-t_1)^{1/2}. \end{align*}

Thus, in both cases ($p\geq 2$ and $1< p<2$), if $\tau (u^*)<\infty$, there exists $u_w\in W^{1,p}_0(\Omega )$ such that

\[ \lim_{t\to \tau(u^*)}\|\varphi^{t}(u^*)-u_w\|_m=0. \]

Since the interval $[0,\tau (u^*)[$ is maximal it has to be $u_w\in K_w$.

If $\tau (u^*)=\infty$, the boundedness from below of $J_w$ allows us to fix an increasing sequence of positive numbers $(t_n)_n$, $t_n\to \infty$ such that

\[ \lim_n \frac{{\rm d}}{{\rm d}t} J_w(\varphi^t(u^*))|_{t=t_n}=0. \]

If $p\geq 2$, one has

\begin{align*} \frac{{\rm d}}{{\rm d}t} J_w(\varphi^t(u^*))|_{t=t_n}& ={-}\langle J_w'(\varphi^{t_n}(u^*)),\varphi^{t_n}(u^*)-B_w(\varphi^{t_n}(u^*)) \rangle\\ (\text{by lemma 2.2}, \ (iii))& \leq{-}c_1\|\varphi^{t_n}(u^*)-A_w(\varphi^{t_n}(u^*))\|_m^p, \end{align*}

which says that

\[ \lim_n \|\varphi^{t_n}(u^*)-A_w(\varphi^{t_n}(u^*))\|_m=0. \]

Now, let us observe that $(\varphi ^{t_n}(u^*))_n$ is bounded in $W^{1,p}_0(\Omega )$. Indeed, by the monotonicity of the flow, $J_w(\varphi ^{t_n}(u^*)\leq J_w(u^*)$ for every $n\in \mathbb {N}$, that means that $\varphi ^{t_n}(u^*)\in J_w^{-1}(]-\infty, J_w(u^*)])$ for every $n\in \mathbb {N}$, and the latter is a bounded set because of the coercivity of $J_w$. Moreover, since $A_w$ is a compact operator, it follows (eventually passing to a subsequence) that there exists $u_w\in W^{1,p}_0(\Omega )$ such that

\[ \lim_n \|\varphi^{t_n}(u^*)-u_w\|_m=\lim_n \|A_w(\varphi^{t_n}(u^*))-u_w\|_m=0. \]

In particular, $u_w\in K_w$. Also, for some $b_2>0$, one has $\|\varphi ^{t_n}(u^*)\|_m,\|A_w(\varphi ^{t_n}(u^*))\|_m \leq b_2$.

If $1< p<2$,

\begin{align*} \frac{{\rm d}}{{\rm d}t} J_w(\varphi^t(u^*))|_{t=t_n}& ={-}\langle J_w'(\varphi^{t_n}(u^*)),\varphi^{t_n}(u^*)-B_w(\varphi^{t_n} (u^*)) \rangle\\ (\text{by lemma 2.2}, (iii))& \leq{-}c_1\|\varphi^{t_n}(u^*)-A_w(\varphi^{t_n}(u^*))\|_m^2 (\|\varphi^{t_n}(u^*)\|_m\\ & \quad +\|A_w(\varphi^{t_n}(u^*))\|_m)^{p-2}\\ & \leq{-}c_2\|\varphi^{t_n}(u^*)-A_w(\varphi^{t_n}(u^*))\|_m^2 \end{align*}

which says that

\[ \lim_n \|\varphi^{t_n}(u^*)-A_(\varphi^{t_n}(u^*))\|_m=0 \]

and we conclude as above.

In the next lemma we refine the previous result.

Proof. As in the proof of lemma 3.3, we first observe that the set $\{\varphi ^t(u^*) \ : \ t\in [0,\tau (u^*)[\}$ is bounded in $W^{1,p}_0(\Omega )$. Recalling (3.3),

\[ \varphi^t(u^*)= e^{{-}t} u^*+ e^{{-}t}\int_0^t e^sB_w(\varphi^s(u^*))\,{\rm d}s. \]

Since $B_w: C^{1}_0(\overline {\Omega }) \to C^{1}_0(\overline {\Omega })$ is a compact operator, the set

\[ \left\{e^{{-}t}\int_0^t e^sB_w(\varphi^s(u^*))\,{\rm d}s \ : \ t\in [0,\tau(u^*)[\right\} \]

is bounded in $C^{1,\alpha }(\overline {\Omega })$, thus relatively compact in $C^1_0(\overline \Omega )$. This clearly implies that $\{\varphi ^t(u^*) \ : \ t\in [0,\tau (u^*)[\}$ is relatively compact in $C^1_0(\overline \Omega )$. The thesis follows from lemma 3.3.

We are ready to prove the main result of this subsection.

Proof. Clearly $u_w$ is a solution of $(P_w)$ since it belongs to $K_w$. Since the initial point $u^*$ belongs to $\partial \mathcal {C}(\Lambda ^w)\cap \partial \mathcal {C}(\mathrm {int}(C^1_0(\overline \Omega )_+))$, and both sets $\partial \mathcal {C}(\Lambda ^w)$, $\partial \mathcal {C}(\mathrm {int}(C^1_0(\overline \Omega )_+))$ are invariant sets of descending flow, then $(\varphi ^{t_n}(u^*))_n \subseteq \partial \mathcal {C}(\Lambda ^w) \cap \partial \mathcal {C}(\mathrm {int}(C^1_0(\overline \Omega )_+))$. Moreover, $\partial \mathcal {C}(\Lambda ^w) \cap \partial \mathcal {C}(\mathrm {int}(C^1_0(\overline \Omega )_+))$ is a closed set, so that $u_w \in \partial \mathcal {C}(\Lambda ^w) \cap \partial \mathcal {C}(\mathrm {int}(C^1_0(\overline \Omega )_+))$. Being $u_w \in \partial \mathcal {C}(\Lambda ^w)$, we obtain that $u_w \not \in \text {int}_{C^1_0} [u_N^w, u_P^w]$, in particular $u_w \not = 0$. Actually, by remark 3.1, $u_w\not \in \text {int}_{C^1_0} [-\varepsilon \varphi _1, \varepsilon \varphi _1]$. On the other hand, since $u_w \in \partial \mathcal {C}(\mathrm {int}(C^1_0(\overline \Omega )_+))$, we also have $u_w \not \in \mathrm {int}(C^1_0(\overline \Omega )_+) \cup (-\mathrm {int}(C^1_0(\overline \Omega )_+))$. This ensures that $u_w$ can not have constant sign. Indeed, if $u_w\geq 0$, it would be a non negative, non trivial solution of

\[ \left\{ \begin{array}{@{}ll@{}} -\Delta_p u+mh_p(u)= f(x,u, \nabla w) +mh_p(u) & \text{in}\ \Omega \\ u=0 & {\rm on}\ \partial\Omega. \end{array}\right. \]

and by assumption $(f_4)$ and the strong maximum principle by Vazquez [Reference Vázquez23], we would deduce $u_w\in \mathrm {int}(C^1_0(\overline \Omega )_+)$. Thus, $u_w$ is a nodal solution of $(P_w)$.

3.3. Existence of a nodal solution for $(P)$

In this subsection through an iteration procedure we prove our first main result.

Proof of theorem 1.1. In theorem 3.2 choose $w=u_0$, where $u_0$ is the function we fixed at the beginning of Section 3.2. Thus, the existence of a nodal solution $u_1$ of $(P_{u_0})$ follows. Proceeding in such way, for each $n\in \mathbb {N}$ denote by $u_n$ the nodal solution of $(P_{u_{n-1}})$ given by theorem 3.2. Hence, we construct a sequence of functions $u_n\in C^1_0(\overline \Omega )$ such that

(3.4)\begin{gather} u_n\not\in \text{int}_{C^1_0} [-\varepsilon\varphi_1, \varepsilon \varphi_1] \end{gather}

(3.5)\begin{gather} u_n \not \in\mathrm{int}(C^1_0(\overline\Omega)_+) \cup (-\mathrm{int}(C^1_0(\overline\Omega)_+)) \end{gather}

By proposition 2.1, $\|u_n\|_{C^{1, \alpha }(\overline {\Omega })} \leq M$, and by the compactness of the embedding ${C^{1, \alpha }}(\overline {\Omega })\hookrightarrow C^1_0(\overline \Omega )$, $(u_n)_n$ is relatively compact in $C^1_0(\overline \Omega )$. Unless to pass to a subsequence, let

\[ \tilde u:=\lim_{n \rightarrow \infty} u_n \ \text{in}\ C^1_0(\overline \Omega). \]

Let us prove that $\tilde u$ is the solution of $(P)$ we are looking for. Since $u_n$ is a solution of $(P_{u_{n-1}})$, for every $\varphi \in W^{1,p}_0({\Omega })$ we have

\[ \int_{\Omega} |\nabla u_n|^{p-2}\nabla u_n \nabla \varphi\,{\rm d}x= \int_{\Omega }f(x, u_n, \nabla u_{n-1})\varphi\,{\rm d}x. \]

Since $u_n\to \tilde u$ in $C^1_0(\overline \Omega )$, we have that $f(x, u_n, \nabla u_{n-1})\rightarrow f(x, \tilde u, \nabla \tilde u)$ in $L^{p'}(\Omega )$ and passing to the limit in the above equality we get

\[ \int_{\Omega} |\nabla \tilde u|^{p-2}\nabla \tilde u \nabla \varphi\, {\rm d}x=\int_{\Omega }f(x, \tilde u, \nabla \tilde u)\varphi\,{\rm d}x, \]

which is our claim. By (3.4) and (3.5), it follows that $\tilde u\not \in \text {int}_{C^1_0} [-\varepsilon \varphi _1, \varepsilon \varphi _1]$ and $\tilde u \not \in \mathrm {int}(C^1_0(\overline \Omega )_+) \cup (-\mathrm {int}(C^1_0(\overline \Omega )_+))$. Thus, $\tilde u \not = 0$, and as in the proof of theorem 3.2, $\tilde u$ can not have constant sign.

4.1. On a parametrized problem

Throughout the sequel we will take into account the results of Section 2 with $g=\lambda f$ and $\mu = \lambda m$ for some $m$ to be chosen later. For every $w\in C^1_0(\overline \Omega )$, let us consider the parametrized Dirichlet problem

(P _{λ, w})\begin{equation} \left\{ \begin{array}{@{}ll@{}} -\Delta_p u= \lambda f(x,u, \nabla w) & {\rm in}\ \Omega \\ u=0 & {\rm on}\ \partial\Omega. \end{array}\right. \end{equation}

From hypothesis $(\tilde f_1)$, it follows that for each $\lambda >0$ we can fix $\varepsilon =\varepsilon (\lambda )>0$ with $\varepsilon (1+\lambda _1^{\frac {1}{p'}})\lambda <\lambda _1$ and $k_0(\lambda ) \in \mathbb {R}$ such that

\[ \lambda|f(x,s,\xi)|\leq k_0(\lambda)+\varepsilon \lambda|s|^{p-1}+\varepsilon \lambda|\xi|^{p-1} \]

for all $x\in \Omega$, $s\in \mathbb {R}$, and $\xi \in \mathbb {R}^N$. This ensures that under the above conditions, the function $g=\lambda f$ fulfils assumption $(\mathcal {H})$ of Section 2 and proposition 2.1 applies, i.e. for every $\lambda >0$ and $u_0\in C^1_0(\overline \Omega )$, there exists $\alpha \in ]0,1[$ and a positive constant $M$ depending on $\lambda$ and $\|u_0\|$ such that if $u_n$ is a solution of $(P_{\lambda, u_{n-1}})$, $\|u_n\|_{C^{1,\alpha }(\overline \Omega )}\leq M$.

We will need the following abstract result.

Let $X$ be a Banach space, ${D}^\pm$ closed convex subsets of $X$, $A:X \longrightarrow X$ continuous and compact operator and $J:X\to \mathbb {R}$ a functional of class ${C}^1(X, \mathbb {R})$. Introduce the following conditions.

1. $(D_1)$ $\mathcal {O}=$int$({D}^+) \cap \text {int}({D}^-) \not =\emptyset$.

2. $(D_2)$ $A({D}^\pm )\subseteq \hbox {int}({D}^\pm )$.

3. $(J_3)$ For any $b \in \mathbb {R}$ there exists a constant $a=a(b)>0$ such that if $u \in \{u \in X: J(u) \leq b\}$ then

   \[ \|u\| + \|A(u)\| \leq a (1+ \|u-A(u)\|). \]

4. $(J_4)$ There exists a path $h:[0,1]\longrightarrow X$ such that $h(0) \in \text {int}({D}^+)\setminus {D}^-$ and $h(1) \in \text {int}({D}^-)\setminus {D}^+$ and

   \[ \max_{0\leq t\leq1} J(h(t)) < \alpha_0:=\inf_{{D}^+{\cap} {D}^-}J(u). \]

The next theorem is the abstract tool we will exploit to deduce the existence of a nodal solution for the parametrized problem $(P_{\lambda, w})$, which we state here in a convenient form for our purposes.

Following the notation of Section 2, $X=W^{1,p}_0(\Omega )$ endowed with the equivalent norm

\[ \|u\|_m^\lambda := \left(\int_{\Omega}(|\nabla u|^p + \lambda m |u|^p )\,{\rm d}x\right)^{1/p}, \]

where $\lambda$ and $m$ will be chosen later in a convenient way,

\[ A_w^\lambda(u):= (-\Delta_p+ \lambda m h_p({\cdot}))^{{-}1}(\lambda f(x, u, \nabla w)+ \lambda m h_p(u)), \]

and $B_w^\lambda$ as in lemma 2.2. Denote also by $J_w^\lambda : W^{1,p}_0(\Omega ) \longrightarrow {\mathbb {R}}$

\[ J_w^\lambda(u) = \frac1p \int_{\Omega}|\nabla u|^p\,{\rm d}x - \lambda\int_{\Omega} F(x, u, \nabla w)\,{\rm d}x, \]

\[ P=\{u\in W^{1,p}_0(\Omega): u\geq 0 \ \text{a.e. in} \ \Omega\} \]

and for $\varepsilon >0$

\[ D_\varepsilon^{{\pm}}=\{u \in W^{1,p}_0(\Omega): {\rm dist}_m(u,\pm P)\leq \varepsilon\}. \]

Proposition 4.1 For $\varepsilon >0$ small enough,

\[ A_w^\lambda(D_\varepsilon^{{\pm}}) \subseteq {\rm int} D_\varepsilon^{{\pm}}. \]

Proof. Let us prove $A_w^\lambda (D_\varepsilon ^+) \subseteq {\rm int} D_\varepsilon ^+$. Notice that by assumptions $(\tilde f_2)-(\tilde f_4)$, for any $M\geq \|w\|_{C^1}$ there exists $m_M>0$ such that

(4.1)\begin{equation} s f(x,s,\xi)+m_M s h_p(s)>0 \ \text{for each}\ x\in\Omega, s\neq0, |\xi|\leq M. \end{equation}

In the sequel put $m:=m_M$.

By $(\tilde f_2)$ and $(\tilde f_3)$, for each $\varepsilon >0$ and $q>p$ there exists another constant $c_M>0$ such that

(4.2)\begin{equation} |f(x,s,\xi)+m h_p(s)|\leq (\varepsilon +m)|s|^{p-1}+c_M|s|^{q-1},\end{equation}

for each $x\in \Omega, s\neq 0, |\xi |\leq M$.

Let $u \in D_\varepsilon ^+$ and $v=A_w^\lambda (u)$. Thus,

\[ -\Delta_p v + \lambda m h_p(v) = \lambda f(x, u, \nabla w) + \lambda m h_p(u) \]

and so testing the above equation with $-v^-$, where $v^-=\max \{-v, \ 0\}$, we deduce

\begin{align*} \|v^-\|_m^p& =\lambda \int_\Omega (f(x,u,\nabla w)+m h_p(u))(- v^-)\\ \text{[by (4.1), (4.2)]} \ & \leq \lambda(\varepsilon+m)\int_\Omega (u^{-})^{p-1}v^-{+}\lambda c_M\int_\Omega (u^{-})^{q-1} v^-\\ & \leq \lambda(\varepsilon+m)\|u^-\|_p^{p-1}\|v^-\|_p+\lambda c_M \|u^-\|_q^{q-1}\|v^-\|_q\\ & \leq \frac{\lambda(\varepsilon+m)}{(\lambda_1 + \lambda m)^{1/p}}\|u^-\|_p^{p-1}\|v^-\|_m+ \frac{\lambda c_M}{(\lambda_1 + \lambda m)^{1/p}} \|u^-\|_q^{q-1}\|v^-\|_m. \end{align*}

Hence,

\[ d_m(v, P)^{p-1} \leq \|v^-\|_m^{p-1} \leq \frac{\lambda(\varepsilon+m)}{(\lambda_1 + \lambda m)^{1/p}}\|u^-\|_p^{p-1}+ \frac{\lambda c_M}{(\lambda_1 + \lambda m)^{1/p}} \|u^-\|_q^{q-1}. \]

Since $\|u^-\|_p \leq \|u-w\|_p$ for all $w \in P$, we get

\begin{align*} d_m(v, P)^{p-1} & \leq \frac{\lambda(\varepsilon+m)}{(\lambda_1 + \lambda m)^{1/p}} \|u-w\|_p^{p-1}+ \frac{\lambda c_M}{(\lambda_1 + \lambda m)^{1/p}} \|u-w\|_p^{q-1} \\ & \leq \frac{\lambda(\varepsilon+m)}{(\lambda_1 + \lambda m)} \|u-w\|_m^{p-1}+ \frac{\lambda c_M}{(\lambda_1 + \lambda m)^{\frac{q}{p}}} \|u-w\|_m^{q-1}. \end{align*}

Thus,

\[ d_m(v, P)^{p-1} \leq \frac{\lambda(\varepsilon+m)}{(\lambda_1 + \lambda m)} d_m(u,P)^{p-1}+ \frac{\lambda c_M}{(\lambda_1 + \lambda m)^{\frac{q}{p}}} d_m(u,P)^{q-1}. \]

Thus, since $q>p$, there exists $\varepsilon _0>0$ such that

(4.3)\begin{equation} d_m(v, P) < d_m(u, P) \quad\ \text{if} \ 0< d_m(u, P) \leq \varepsilon_0, \end{equation}

and the proof is concluded.

Proof. Fix $w \in C^1_0(\overline \Omega )$. Let us show that all the hypotheses of theorem 4.1 are verified with $A=A_w^\lambda$, $J=J_w^\lambda$, $D^+={D}_\varepsilon ^+$, $D^-={D}_\varepsilon ^-$ and $\lambda$ big enough.

Condition $(D_1)$ is trivial, $(D_2)$ follows by proposition 4.1 and $(J_1)$, $(J_2)$ by proposition 2.2. Moreover, the map $A_w^\lambda$ is compact (see [Reference Faraci, Motreanu and Puglisi13, Lemma 2.3]) and assumption $(J_3)$ is implied by the coercivity of $J_w^\lambda$. It remains to check the validity of condition $(J_4)$. By (4.3), we have that

\[ K \cap ({D}_\varepsilon^+{\cap} {D}_\varepsilon^-)=\{0\}. \]

Since $J_w^\lambda$ is decreasing over the flow and ${D}_\varepsilon ^+ \cap {D}_\varepsilon ^-$ is an invariant set, it follows that for every $u \in {D}_\varepsilon ^+ \cap {D}_\varepsilon ^-$

\[ J_w^\lambda(u) \geq J_w^\lambda(\varphi_t(u)) \geq J_w^\lambda(u^*) \]

where $u^*=\lim _{t \rightarrow \tau (u)} \varphi _t(u) \in K \cap ({D}_\varepsilon ^+ \cap {D}_\varepsilon ^-)=\{0\}$. Therefore

\[ J_w^\lambda(u) \geq J_w^\lambda(0)=0 \quad \text{for every}\ u \in {D}_\varepsilon^+{\cap} {D}_\varepsilon^-. \]

Let us show that there exists $\tilde \lambda$ such that for all $\lambda \geq \tilde \lambda$ $(J_4)$ holds; we follow the construction of [Reference Bartsch and Liu4, Lemma 3.2]. Let

\[ a:=\inf\{x_1: x=(x_1, \ldots, x_N) \in \Omega\} \ \text{and} \ b :=\sup\{x_1: x=(x_1, \ldots, x_N) \in \Omega\}. \]

we consider

\[ \Omega_t:=\{ x \in \Omega: \ (1-t)a + tb < x_1< b\} \ \text{for}\ t \in [0,1]. \]

Thus $\Omega _0=\Omega$ and $\Omega _1=\emptyset$. We define

\[ h^*(t) = s^+ \chi_{\Omega_t} + s^- \chi_{\Omega \setminus \Omega_t}. \]

Let

\[ \delta:=|\Omega| \inf_{(x, \xi) \in \Omega\times \mathbb{R}^N} F(x, s^{{\pm}}, \xi)>0 \]

(see $(\tilde {f}_5)$). We can approximate $h^*$ by a function $h \in C([0,1], W^{1,p}_0(\Omega )\cap C^1(\overline {\Omega )})$ such that

\[ \int_{\Omega} F(x, h(t), \nabla w)\,{\rm d}x \geq \delta/2>0. \]

We choose $\varepsilon >0$ such that $s^- < - \varepsilon <0<\varepsilon < s^+$, so that $h(0) \in \text {int}(\mathcal {D}_{\varepsilon }^+) \setminus \mathcal {D}_{\varepsilon }^-$ and $h(1) \in \text {int}(\mathcal {D}_{\varepsilon }^-) \setminus \mathcal {D}_{\varepsilon }^+$. Finally

\[ J_w^\lambda(h(t) \leq \frac1p \|\nabla h(t)\|_p^p - \frac12 \lambda \delta \leq C - \frac12 \lambda \delta. \]

Choose $\tilde \lambda =2C/\delta$. Notice that $\tilde \lambda$ does not depend on $w$. The existence of a nodal solution for $(P_{\lambda,w})$ follows at once by theorem 4.1.

4.2. Nodal solution of $(P_\lambda )$

In this subsection we prove theorem 1.2 using an iterative procedure.

Proof of theorem 1.2. Let us choose $\lambda > \tilde \lambda$, where $\tilde \lambda$ is as in theorem 4.2. Fix $u_0$ and let $M>0$, depending on $\lambda$ and $u_0$, be as in proposition 2.1. For each $n\in \mathbb {N}^+$ denote by $u_n$ the nodal solution in $C^1_0(\overline \Omega )$ of $(P_{\lambda, u_{n-1}})$ given by theorem 4.2. From its proof,

(4.4)\begin{equation} u_n\in \partial \mathcal{C}(\hbox{int}({D}^+_\varepsilon) \cap \text{int}({D}^+_\varepsilon))\setminus (\text{int}({D}^+_\varepsilon) \cup \hbox{int}({D}^-_\varepsilon)). \end{equation}

Let us stress that $\varepsilon$ depends on $M$ which is independent on $n$ (remark 4.2). Thus the set $\partial \mathcal {C}(\text {int}({D}^+_\varepsilon ) \cap \text {int}({D}^+_\varepsilon ))\setminus (\text {int}({D}^+_\varepsilon ) \cup \text {int}({D}^-_\varepsilon ))$ is a closed invariant set and does not depend on $n$.

By proposition 2.1, $\|u_n\|_{C^{1, \alpha }} \leq M$, and by the compactness of the embedding ${C^{1, \alpha }}(\overline {\Omega })\hookrightarrow C^1_0(\overline \Omega )$, $(u_n)_n$ is relatively compact in $C^1_0(\overline \Omega )$. Unless to pass to a subsequence, let

\[ \tilde u:=\lim_{n \rightarrow \infty} u_n \ \text{in}\ C^1_0(\overline \Omega). \]

As in the proof of theorem 1.1, $\tilde u$ is a solution of $(P_\lambda )$. By the closedness of the set $\partial \mathcal {C}(\text {int}({D}^+_\varepsilon ) \cap \text {int}({D}^+_\varepsilon ))\setminus (\text {int}({D}^+_\varepsilon ) \cup \text {int}({D}^-_\varepsilon ))$, it follows from (4.4) that $\tilde u$ is a sign changing function as we claimed.