1 Introduction

Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^{N}\) (\(N\ge 2)\) having a smooth boundary \(\partial \Omega .\) Given \(1<p<N\), we consider the Neumann quasilinear elliptic problem with general gradient dependence

$$\begin{aligned} \left( \textrm{P}\right) \qquad \left\{ \begin{array}{ll} -\Delta _{p}u+\frac{|\nabla u|^{p}}{u+\delta }=f(x,u,\nabla u) &{}\quad \text {in} \;\Omega , \\ |\nabla u|^{p-2}\frac{\partial u}{\partial \eta }=0 &{}\quad \text {on}\;\partial \Omega , \end{array} \right. \end{aligned}$$

where \(f:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^{N}\rightarrow {\mathbb {R}}\) is a Carathéodory function, \(\delta >0\) is a small parameter, \(\eta \) is the unit outer normal to \(\partial \Omega \) while \(\Delta _{p}\) denotes the p-Laplace operator, namely \(\Delta _{p}:=\textrm{div}(|\nabla u|^{p-2}\nabla u)\), \(\forall \,u\in W^{1,p}(\Omega ).\)

We say that \(u\in W^{1,p}(\Omega )\) is a (weak) solution of \(\left( \textrm{P }\right) \) provided \(u+\delta >0\) a.e. in \(\Omega ,\) \(\frac{|\nabla u|^{p}}{ u+\delta }\in L^{1}(\Omega )\) and

$$\begin{aligned} \int _{\Omega }|\nabla u|^{p-2}\nabla u\nabla \varphi { } \textrm{d}x+\int _{\Omega }\frac{|\nabla u|^{p}}{u+\delta }\varphi { } \textrm{d}x=\int _{\Omega }f(x,u,\nabla u)\varphi { }\textrm{d}x, \end{aligned}$$
(1)

for all \(\varphi \in W^{1,p}(\Omega )\cap L^{\infty }(\Omega )\). The requirement of \(\varphi \) to be bounded is necessary since \(\frac{|\nabla u|^{p}}{u+\delta }\) is only in \(L^{1}(\Omega )\).

Problem \(\left( \textrm{P}\right) \) brings together a lower-order term with natural growth with respect to the gradient \(\frac{|\nabla u|^{p}}{u+\delta },\) called absorption, as well as a reaction–convection term \(f(x,u,\nabla u)\). Both depend on the solution and its gradient. The absorption describes a natural polynomial growth in the \(\nabla u\)-variable while the convection outlines a p-sublinear one (cf. Sect. 3). Note that the absorption term dominates the diffusion operator by its growth at infinity.

Absorption and/or reaction–convection terms appear in various nonlinear processes that occur in engineering and natural systems. In biology, they arise in heat transfer of gas and liquid flow in plants and animals while in geology, they are involved in thermoconvective motion of magmas and during volcanic eruptions. They also appear in chemical processes such as in catalytic and noncatalytic reactions, in exothermic and endothermic reacting, as well as in global climate energy balance models [17, 20]. Moreover, convective–absorption problem \(\left( \textrm{P}\right) \) can be associated with different class of nonlinear equations including nonlinear Fokker–Planck equations and multidimensional formulation of generalized viscous Burgers’ equations. These equations are involved in diverse physical phenomenon such as plasma physics, astrophysics, physics of polymer fluids and particle beams, nonlinear hydrodynamics and neurophysics [19]. We also mention that problems like \(\left( \textrm{P}\right) \) arise in stochastic control theory and have been first studied in [26]. The study of Dirichlet problems involving absorption term has raised considerable interest in recent years and has been the subject of substantial number of papers that it is impossible to quote all of them. A significant part are carried on semilinear problems with quadratic growth (i.e., \(p=2\)). Among them, we quote [5, 13, 14, 24, 35] and the references therein. For quasilinear Dirichlet problems we refer, for instance, to [2, 15, 18, 36, 39, 40]. Surprisingly enough, so far we were not able to find previous results dealing with Neumann boundary conditions. This case is considered only when the absorption term is canceled, see [21, 38]. We also mention [3, 4, 8, 9, 16, 22, 37] where convective problem \(\left( \textrm{P}\right) \) (without absorption) subjected to Dirichlet boundary conditions is examined.

Problem \(\left( \textrm{P}\right) \) exhibits interesting features resulting from the interaction between absorption and reaction–convection terms. Their involvement in \(\left( \textrm{P}\right) \) gives rise to nontrivial difficulties such as the loss of variational structure thereby making it impossible applying variational methods. Obviously, the mere fact of their presence impacts substantially the structure of \(\left( \textrm{P}\right) \) as well as the nature of its solutions which, in some cases, leads to surprising situations, especially from a mathematical point of view. For instance, in [1, 36], it is shown that the absorption term regularizes solutions and it is sufficient to break down any resonant effect of the reaction term. In [7], it is established that a problem admits nontrivial weak solutions only under the effect of absorption. Otherwise, zero is the only solution for the problem.

In the present paper, we provide at least two nontrivial solutions for problem \((\textrm{P})\) with precise sign properties: one is nodal (i.e., sign changing) and the other is positive. According to our knowledge, this topic is a novelty. The study of the existence of nodal solutions has never been discussed for convection–absorption problems, just as the latter have never been handled under Neumann boundary conditions.

The multiplicity result is achieved in part through a location principle of nodal solutions which, in particular, helps distinguish between solutions of \((\textrm{P})\). However, this principle constitutes in itself a crucial part of our work since the multiplicity result depends on it. Indeed, under assumption \((\mathrm {H.}2)\) (cf. Sect. 3), if \(f(x,0,0)\ge 0\) a.e. in \(\Omega \), it is shown that every nodal solution of problem \((\textrm{P})\) should be bounded above by a positive solution. In particular, this provides the powerful fact that the existence of a nodal solution implies under the stated hypotheses that a positive solution must exists. In other words, nodal solutions generate positive solutions which is an unusual fact since generally, it is rather the opposite implication occurs (see [12, 32, 33]). This phenomenon happens also in the opposite unilateral sens: if \( f(x,0,0)\le 0\) a.e. in \(\Omega \), every nodal solution of problem \((\textrm{ P})\) should be bounded below by a negative solution. However, if \(f(x,0,0)=0\) a.e. in \(\Omega \), the both cases above are satisfied simultaneously and hence, every nodal solution to problem \((\textrm{P})\) is between two opposite constant-sign solutions.

The location principle of nodal solutions is stated in Theorem 7. The proof is chiefly based on Theorem 5, shown in Sect. 2 via monotone operator theory together with perturbation argument and adequate truncation. Theorem 5 is a version of sub-supersolutions result for quasilinear convective elliptic problems involving natural growth. It can be applied for large classes of Neumann elliptic problems since no sign condition on the nonlinearities is required and no specific structure is imposed. However, it is worth noting that due to the effect of the presence of absorption term, stretching out monotone operators theory’s scope to convection–absorption problems is not a straightforward task. This requires, on the one hand, truncation in order to stay inside the rectangle formed by sub-supersolution pair and, on the other hand, perturbation (regularization), by introducing a parameter \(\varepsilon >0\) in \((\textrm{P} )\), necessary to have a minimal control on the absorption term.

Another significant feature of our result lies in obtaining nodal solutions for problem \((\textrm{P})\). Taking advantage of Theorem 5, we construct a sign-changing sub-supersolution pair \(({\underline{u}},{\overline{u}} ) \) for problem \((\textrm{P})\) which inevitably leads to a nodal solution \( u_{0}\) for \((\textrm{P})\). The choice of suitable functions with an adjustment of adequate constants closely dependent on the small parameter \( \delta >0\) is crucial. By construction, the subsolution \({\underline{u}}\) is positive inside the domain \(\Omega \) while the supersolution \({\overline{u}}\) is negative near the boundary \(\partial \Omega \). Therefore, the solution \( u_{0}\) of \((\textrm{P}),\) being naturally imbued with these properties, is positive inside \(\Omega \) and negative once \(d(x)\rightarrow 0\). We emphasize that, without the implication of the absorption term, it would not have been possible to get nodal solutions for problem \((\textrm{P}),\) at least with the techniques developed in this work.

The rest of the paper is organized as follows. Section 2 contains the existence theorem involving sub-supersolutions. Section 3 focuses on a location principle of nodal solutions. Section 4 deals with the multiplicity result.

2 A Sub-supersolution Theorem

Let \((X,\Vert \cdot \Vert )\) be a real Banach space and let \(X^{*}\) be its topological dual, with duality bracket \(\langle \cdot ,\cdot \rangle \). An operator \({\mathcal {A}}:X\rightarrow X^{*}\) is said to be:

  • bounded if it maps bounded sets into bounded sets.

  • coercive provided \(\displaystyle {\lim _{\Vert x\Vert \rightarrow +\infty }}\frac{\langle {\mathcal {A}}(x),x\rangle }{\Vert x\Vert }=+\infty \).

  • pseudomonotone if \(x_{n}\rightharpoonup x\) in X and \( \displaystyle {\limsup _{n\rightarrow +\infty }}\langle {\mathcal {A}} (x_{n}),x_{n}-x\rangle \le 0\) force \(\displaystyle {\liminf _{n\rightarrow +\infty }}\langle {\mathcal {A}}(x_{n}),x_{n}-z\rangle \ge \langle {\mathcal {A}}(x),x-z\rangle \) for all \(z\in X\).

Recall (see, e.g., [11, Theorem 2.99]) that

Theorem 1

If X is reflexive and \({\mathcal {A}}:X\rightarrow X^{*}\) is bounded, coercive, and pseudomonotone then \({\mathcal {A}}(X)=X^{*}\).

In the sequel, the Banach space \(W^{1,p}(\Omega )\) is equipped with the following usual norm

$$\begin{aligned} \Vert u\Vert _{1,p}:=\left( \Vert u\Vert _{p}^{p}+\Vert \nabla u\Vert _{p}^{p}\right) ^{1/p}{,\quad }u\in W^{1,p}(\Omega )\text {,} \end{aligned}$$

where, as usual,

$$\begin{aligned} \Vert v\Vert _{p}:=\left\{ \begin{array}{ll} \left( \int _{\Omega }|v(x)|^{p}\textrm{d}x\right) ^{1/p}&{}\quad \text {if }p<+\infty , \\ ess\underset{x\in \Omega }{\sup }{ }|v(x)|&{}\quad \text {otherwise.} \end{array} \right. \end{aligned}$$

The following assumptions will be posited.

  1. (H.1)

    Let \(0\le q\le p-1\). For every \(\rho >0\) there exists \(M:=M(\rho )>0\) such that

    $$\begin{aligned} |f(x,s,\xi )|\le M(1+|\xi |^{q})\text { \ in }\Omega \times [-\rho ,\rho ]\times {\mathbb {R}}^{N}. \end{aligned}$$
  2. (H.2)

    There are \({\underline{u}},{\overline{u}}\in {\mathcal {C}}^{1}({\overline{\Omega }})\) fulfilling

    $$\begin{aligned} {\overline{u}}+\delta \ge {\underline{u}}+\delta >0\text { \ a.e. in }\Omega , \end{aligned}$$
    (2)

    as well as

    $$\begin{aligned} \left\{ \begin{array}{l} \int _{\Omega }|\nabla {\underline{u}}|^{p-2}\nabla {\underline{u}}\,\nabla \varphi { }\textrm{d}x+\int _{\Omega }\frac{|\nabla {\underline{u}}|^{p}}{ {\underline{u}}+\delta }\varphi \,\textrm{d}x-\int _{\Omega }f(x,{\underline{u}},\nabla {\underline{u}})\varphi \,\textrm{d}x\le 0, \\ \int _{\Omega }|\nabla {\overline{u}}|^{p-2}\nabla {\overline{u}}\,\nabla \varphi \,\textrm{d}x+\int _{\Omega }\frac{|\nabla {\overline{u}}|^{p}}{{\overline{u}}+\delta } \varphi \,\textrm{d}x-\int _{\Omega }f(x,{\overline{u}},\nabla {\overline{u}})\varphi \,\textrm{d}x\ge 0, \end{array} \right. \end{aligned}$$
    (3)

    for all \(\varphi \in W^{1,p}(\Omega )\cap L^{\infty }(\Omega )\) with \( \varphi \ge 0\) in \(\Omega \).

The functions \({\underline{u}}\) and \({\overline{u}}\) in \((\mathrm {H.}2)\) are called subsolution and supersolution of problem \((\textrm{P})\), respectively.

2.1 An Auxiliary Problem

Let \({\underline{u}},{\overline{u}}\in {\mathcal {C}}^{1}(\overline{\Omega })\) be a sub-supersolutions of problem \((\textrm{P})\) as required in condition \(( \mathrm {H.}2)\). We consider the truncation operators \({\mathcal {T}}:W^{1,p}(\Omega )\rightarrow W^{1,p}(\Omega )\) defined by

$$\begin{aligned} {\mathcal {T}}(u(x)):=\left\{ \begin{array}{ll} {\underline{u}}(x) &{}\quad \text {when }u(x)\le {\underline{u}}(x) \\ u(x) &{}\quad \text {if }{\underline{u}}(x)\le u(x)\le {\overline{u}}(x) \\ {\overline{u}}(x) &{}\quad \text {otherwise}\end{array}\right. ,\text { for a.e. }x\in \Omega . \end{aligned}$$
(4)

Lemma 2.89 in [11] ensures that \({\mathcal {T}}\) is continuous and bounded. We introduce the cutoff function \(b:\Omega \times {\mathbb {R}}\longrightarrow {\mathbb {R}}\) defined by

$$\begin{aligned} b(x,s):=-({\underline{u}}(x)-s)_{+}^{p-1}+(s-{\overline{u}}(x))_{+}^{p-1},\quad (x,s)\in \Omega \times {\mathbb {R}}, \end{aligned}$$

The function b is a Carathéodory function satisfying the growth condition

$$\begin{aligned} |b(x,s)|\le k(x)+c|s|^{p-1},\ \text {{for a.e.} }x\in \Omega \text {, for all }s\in {\mathbb {R}}, \end{aligned}$$
(5)

where c is a positive constant and k \(\in L^{\infty }(\Omega ).\) Moreover, it holds

$$\begin{aligned} \int _{\Omega }b(\cdot ,u)u\,\textrm{d}x\ge C_{1}\Vert u\Vert _{p}^{p}-C_{2}, \text { for all }u\in W^{1,p}(\Omega ), \end{aligned}$$
(6)

with appropriate constants \(C_{1},C_{2}>0\); see, e.g., [11, pp. 95–96].

For \(\varepsilon \in (0,1)\) and for \(\mu >0\) that will be selected later on, we state the auxiliary problem

$$\begin{aligned} (\textrm{P}_{\varepsilon ,\mu })\qquad \left\{ \begin{array}{ll} -\Delta _{p}{u+}\frac{|\nabla ({\mathcal {T}}u)|^{p}}{{\mathcal {T}}u+\delta +\varepsilon }=f(x,{\mathcal {T}}u,\nabla ({\mathcal {T}}u))-\mu b(x,u) &{} \quad \text {in} \;\Omega , \\ |\nabla u|^{p-2}\frac{\partial u}{\partial \eta }=0 &{}\quad \text {on}\;\partial \Omega . \end{array} \right. \end{aligned}$$

We provide the existence of solutions \(u\in W^{1,p}(\Omega )\) for problem \(( \textrm{P}_{\varepsilon ,\mu })\). The proof is chiefly based on pseudomonotone operators theorem stated in [11, Theorem 2.99].

Next lemmas furnish useful estimates related to nonlinear terms involved in \( (\textrm{P}_{\mu })\). The first estimate deals with the nonlinearity f.

Lemma 2

Under assumption \((\mathrm {H.}1)\), there exists a constant \(C_{0}>0\) such that, for all \(u\in W^{1,p}(\Omega ),\) we have

$$\begin{aligned} \int _{\Omega }|f(\cdot ,{\mathcal {T}}u,\nabla ({\mathcal {T}}u))||u|\textrm{d} x\le \frac{1}{2}\Vert \nabla u\Vert _{p}^{p}+C_{0}(1+\left\| u\right\| _{p}+\left\| u\right\| _{p}^{p}). \end{aligned}$$

Proof

For any fixed \(\sigma \in ]0,\frac{1}{2M}[\), Young’s inequality implies

$$\begin{aligned} \begin{array}{ll} |\nabla ({\mathcal {T}}u)|^{q}|u| &{} \le \sigma |\nabla ({\mathcal {T}}u)|^{\frac{ qp}{p-1}}+c_{\sigma }|u|^{p} \\ &{} \le \sigma (1+|\nabla ({\mathcal {T}}u)|^{p})+c_{\sigma }|u|^{p}, \end{array} \end{aligned}$$
(7)

for every \(u\in W^{1,p}(\Omega )\). Moreover, by (4) we have

$$\begin{aligned} \int _{\Omega }|\nabla ({\mathcal {T}}u)|^{p}\textrm{d}x= & {} \int _{\{u\le {\underline{u}}\}}|\nabla {\underline{u}}|^{p}\textrm{d}x+\int _{\{ {\underline{u}}\le u\le {\overline{u}}\}}|\nabla u|^{p}\textrm{d} x+\int _{\{u\ge {\overline{u}}\}}|\nabla {\overline{u}}|^{p}\textrm{d}x\\\le & {} \int _{\Omega }|\nabla {\underline{u}}|^{p}\textrm{d} x+\int _{\Omega }|\nabla u|^{p}\textrm{d}x+\int _{\Omega }|\nabla {\overline{u}}|^{p}\textrm{d}x \\\le & {} |\Omega |(\left\| \nabla {\underline{u}}\right\| _{\infty }^{p}+\left\| \nabla {\overline{u}}\right\| _{\infty }^{p})+\Vert \nabla u\Vert _{p}^{p}. \end{aligned}$$

Then, using \((\mathrm {H.}1),\) (7) and the fact that \(\sigma <\frac{1 }{2\,M}\), thanks to Hölder’s inequality, we get

$$\begin{aligned} \begin{array}{l} \int _{\Omega }|f(\cdot ,{\mathcal {T}}u,\nabla ({\mathcal {T}}u))||u| \textrm{d}x\le M\int _{\Omega }(1+|\nabla ({\mathcal {T}}u)|^{q})|u| \textrm{d}x \\ \quad \le M\int _{\Omega }(|u|+\sigma (1+|\nabla ({\mathcal {T}}u)|^{p})+c_{\sigma }|u|^{p})\textrm{d}x \\ \quad \le M(|\Omega |^{\frac{p-1}{p}}\left\| u\right\| _{p}+\sigma |\Omega |(1+\left\| \nabla {\underline{u}}\right\| _{\infty }^{p}+\left\| \nabla {\overline{u}}\right\| _{\infty }^{p})+\sigma \Vert \nabla u\Vert _{p}^{p}+c_{\sigma }\left\| u\right\| _{p}^{p}) \\ \quad \le \frac{1}{2}\Vert \nabla u\Vert _{p}^{p}+C_{0}(1+\left\| u\right\| _{p}+\left\| u\right\| _{p}^{p}), \end{array} \end{aligned}$$
(8)

which completes the proof. \(\square \)

We turn to estimating the natural growth gradient term in \((\textrm{P} _{\varepsilon ,\mu })\).

Lemma 3

Assume that \((\mathrm {H.}1)\) and \((\mathrm {H.}2)\) hold. Then, for all \(\varepsilon \in (0,1)\), there exists a constant \({\hat{C}}_{\varepsilon }>0\) such that for all \(u\in W^{1,p}(\Omega )\), it holds

$$\begin{aligned} \begin{array}{l} \int _{\Omega }\frac{|\nabla ({\mathcal {T}}u)|^{p}}{{\mathcal {T}}u+\delta +\varepsilon }|u|\textrm{d}x\le {\hat{C}}_{\varepsilon }(1+\left\| u\right\| _{p}). \end{array} \end{aligned}$$
(9)

Proof

By (4) note that

$$\begin{aligned} {\mathcal {T}}u={\underline{u}}\mathbbm {1}_{\{u\le {\underline{u}}\}}+u\mathbbm {1} _{\{{\underline{u}}<u<{\overline{u}}\}}+{\overline{u}}\mathbbm {1}_{\{u\ge {\overline{u}}\}},\text { for }u\in W^{1,p}(\Omega ). \end{aligned}$$

Then

$$\begin{aligned} \int _{\Omega }\frac{|\nabla ({\mathcal {T}}u)|^{p}}{{\mathcal {T}}u+\delta +\varepsilon }|u|\textrm{d}x{} & {} =\int _{\{u\le {\underline{u}}\}}\frac{|\nabla {\underline{u}}|^{p}}{ {\underline{u}}+\delta +\varepsilon }|u|\textrm{d}x+\int _{\Omega } \frac{{|\mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}}\}}\nabla u|^{p}}}{u\mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}}\}}+\delta +\varepsilon }|u|\textrm{d}x\\{} & {} \quad \ +\int _{\{u\ge {\overline{u}}\}}\frac{|\nabla {\overline{u}}|^{p}}{{\overline{u}} +\delta +\varepsilon }|u|\textrm{d}x. \end{aligned}$$

In view of (2) we have

$$\begin{aligned} {\overline{u}}+\delta +\varepsilon \ge {\underline{u}}+\delta +\varepsilon >\varepsilon \text { \ a.e. in }\Omega . \end{aligned}$$

Hence

$$\begin{aligned}{} & {} \int _{\{u\le {\underline{u}}\}}\frac{|\nabla {\underline{u}}|^{p}}{{\underline{u}} +\delta +\varepsilon }|u|\textrm{d}x\le \frac{\left\| \nabla {\underline{u}}\right\| _{\infty }^{p}}{\varepsilon }\int _{\{u\le {\underline{u}}\}}|u|\textrm{d}x\le \frac{\left\| \nabla {\underline{u}}\right\| _{\infty }^{p}}{\varepsilon }\\{} & {} \int _{\Omega }|u|\textrm{d}x\le \frac{\left\| \nabla {\underline{u}}\right\| _{\infty }^{p}}{\varepsilon }|\Omega |^{\frac{p-1}{p}}\left\| u\right\| _{p},\int _{\{u\ge {\overline{u}}\}}\frac{|\nabla {\overline{u}}|^{p}}{{\overline{u}} +\delta +\varepsilon }|u|\textrm{d}x\le \frac{\left\| \nabla {\overline{u}}\right\| _{\infty }^{p}}{\varepsilon }\\{} & {} \int _{\{u\ge \overline{ u}\}}|u|\textrm{d}x\le \frac{\left\| \nabla {\overline{u}} \right\| _{\infty }^{p}}{\varepsilon }\int _{\Omega }|u|\textrm{d} x\le \frac{\left\| \nabla {\overline{u}}\right\| _{\infty }^{p}}{ \varepsilon }|\Omega |^{\frac{p-1}{p}}\left\| u\right\| _{p} \end{aligned}$$

and

$$\begin{aligned} \int _{\Omega }\frac{{|\mathbbm {1}_{\{{\underline{u}}<u< {\overline{u}}\}}\nabla u|^{p}}}{u\mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}} \}}+\delta +\varepsilon } |u|\textrm{d}x\le & {} \frac{\max \{| {\underline{u}}|,|{\overline{u}}|\}}{\varepsilon }\int _{\Omega } {|\mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}}\}}\nabla u)|^{p}}\textrm{d}x \\\le & {} \frac{\max \{\left\| {\underline{u}}\right\| _{\infty },\left\| {\overline{u}}\right\| _{\infty }\}}{\varepsilon }\int _{\Omega }{|\mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}}\}}\nabla u)|^{p}}\textrm{d}x \\\le & {} \frac{\max \{\left\| {\underline{u}}\right\| _{\infty },\left\| {\overline{u}}\right\| _{\infty }\}}{\varepsilon }\Vert u \mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}}\}}\Vert _{1,p}^{p}. \end{aligned}$$

Gathering the above inequalities, we obtain

$$\begin{aligned} \begin{array}{l} \int _{\Omega }\frac{|\nabla ({\mathcal {T}}u)|^{p}}{{\mathcal {T}}u+\delta +\varepsilon }|u|\textrm{d}x\le \frac{\left\| \nabla \underline{u }\right\| _{\infty }^{p}+\left\| \nabla {\overline{u}}\right\| _{\infty }^{p}}{\varepsilon }|\Omega |^{\frac{p-1}{p}}\left\| u\right\| _{p}+\frac{\max \{\left\| {\underline{u}}\right\| _{\infty },\left\| {\overline{u}}\right\| _{\infty }\}}{\varepsilon }\Vert u \mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}}\}}\Vert _{1,p}^{p}. \end{array} \nonumber \\ \end{aligned}$$
(10)

We claim that \(\Vert u\mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}}\}}\Vert _{p}\) is uniformly bounded. Indeed, test in \((\textrm{P}_{\varepsilon ,\mu }) \) with \((u+\delta )\mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}}\}}\in W^{1,p}(\Omega )\cap L^{\infty }(\Omega )\) which is possible in view of [31, Proposition 1.61]. Here, on the basis of (5) and \((\mathrm { H.}1),\) with \(-\rho \le {\underline{u}}\le {\overline{u}}\le \rho \), for \( u\in [{\underline{u}},{\overline{u}}],\) \({\underline{u}},{\overline{u}}\in L^{\infty }(\Omega )\) (see \((\mathrm {H.}2)\)), one has

$$\begin{aligned} \begin{array}{l} \left| f(x,{\mathcal {T}}u,\nabla ({\mathcal {T}}u))-\mu b(x,u)-\frac{|\nabla ( {\mathcal {T}}u)|^{p}}{{\mathcal {T}}u+\delta +\varepsilon }\right| \\ \quad \le |f(x,{\mathcal {T}}u,\nabla ({\mathcal {T}}u))|+\mu |b(x,u)|+\frac{|\nabla ( {\mathcal {T}}u)|^{p}}{{\mathcal {T}}u+\delta +\varepsilon } \\ \quad \le C_{\varepsilon }(1+|\nabla ({\mathcal {T}}u)|^{q}+|\nabla ({\mathcal {T}} u)|^{p}), \end{array} \end{aligned}$$

for a certain constant \(C_{\varepsilon }>0\) independent of u. Then, the regularity up to the boundary result in [27] ensures that \(u\in {\mathcal {C}}^{1,\tau }({\overline{\Omega }})\) for certain \(\tau \in (0,1).\) Therefore, [31, Proposition 1.61] applies.

Then, noting that

$$\begin{aligned} b(x,u)=0\text { \ a.e. for }u\in [{\underline{u}},{\overline{u}}], \end{aligned}$$
(11)

it follows that

$$\begin{aligned} \begin{array}{l} \int _{\Omega }{|\mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}} \}}\nabla u|^{p}}\textrm{d}x+\int _{\Omega }\frac{|\nabla ({\mathcal {T}} u)|^{p}}{{\mathcal {T}}u+\delta +\varepsilon }(u+\delta )\mathbbm {1}_{\{ {\underline{u}}<u<{\overline{u}}\}}\textrm{d}x \\ \quad =\int _{\Omega }f(x,{\mathcal {T}}u,\nabla ({\mathcal {T}}u))(u+\delta )\mathbbm {1} _{\{{\underline{u}}<u<{\overline{u}}\}}\textrm{d}x. \end{array} \end{aligned}$$
(12)

Due to (2) and (4), one has

$$\begin{aligned} \int _{\Omega }\frac{|\nabla ({\mathcal {T}}u)|^{p}}{{\mathcal {T}}u+\delta +\varepsilon }(u+\delta )\mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}}\}}\textrm{d}x\ge 0. \end{aligned}$$

Therefore, from (12), we deduce that

$$\begin{aligned} \begin{array}{l} \int _{\Omega }{|\mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}} \}}\nabla u|^{p}}\textrm{d}x\le \int _{\Omega }f(x,{\mathcal {T}} u,\nabla ({\mathcal {T}}u))(u+\delta )\mathbbm {1}_{\{{\underline{u}}<u<\overline{u }\}}\textrm{d}x \\ \quad \le \int _{\Omega }|f(x,{\mathcal {T}}u,\nabla ({\mathcal {T}}u))|(|u|+\delta ) \mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}}\}}\textrm{d}x. \end{array} \end{aligned}$$
(13)

Exploiting (8) and \((\mathrm {H.}1)\), we get

$$\begin{aligned} \begin{array}{l} \int _{\Omega }|f(x,{\mathcal {T}}u,\nabla ({\mathcal {T}}u))|(|u|+\delta ) \mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}}\}}\textrm{d}x \\ \quad =\int _{\Omega }|f(x,{\mathcal {T}}u,\nabla ({\mathcal {T}}u))||u|\mathbbm {1}_{\{ {\underline{u}}<u<{\overline{u}}\}}\textrm{d}x+\delta \int _{\Omega }|f(x, {\mathcal {T}}u,\nabla ({\mathcal {T}}u))|\mathbbm {1}_{\{{\underline{u}}<u<\overline{ u}\}}\textrm{d}x \\ \quad \le \frac{1}{2}\Vert {\mathbbm {1}_{\{{\underline{u}}<u< {\overline{u}}\}}\nabla u}\Vert _{p}^{p}+C(1+\left\| u\mathbbm {1}_{\{ {\underline{u}}<u<{\overline{u}}\}}\right\| _{p}+\left\| u\mathbbm {1}_{\{ {\underline{u}}<u<{\overline{u}}\}}\right\| _{p}^{p}) \\ \quad \quad +\delta M(1+\Vert \mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}}\}}\nabla u\Vert _{p}^{q}) \\ \quad \le \frac{1}{2}\Vert {\mathbbm {1}_{\{{\underline{u}}<u< {\overline{u}}\}}\nabla u}\Vert _{p}^{p}+\delta M\Vert \mathbbm {1}_{\{ {\underline{u}}<u<{\overline{u}}\}}\nabla u\Vert _{p}^{q}+{\tilde{C}}_{0}, \end{array} \end{aligned}$$

where \({\tilde{C}}_{0}:=C(1+\rho |\Omega |^{\frac{1}{p}}+\rho ^{p}|\Omega |)+\delta M\). Combining with (13) and since \(q<p\), we conclude that there is a constant \({\tilde{C}}>0,\) independent of u,  such that

$$\begin{aligned} \Vert {\mathbbm {1}_{\{{\underline{u}}<u<{\overline{u}}\}}\nabla u}\Vert _{p}\le {\tilde{C}}. \end{aligned}$$
(14)

This proves the claim.

Consequently, in view of (10) and (14), we infer that there exists a constant \({\hat{C}}_{\varepsilon }>0\), independent of u, such that ( 9) holds true. This ends the proof. \(\square \)

The existence result for problem \((\textrm{P}_{\varepsilon ,\mu })\) is formulated as follows.

Theorem 4

Suppose \((\mathrm {H.}1)\)–\((\mathrm {H.}2)\) hold true. Then, problem \((\textrm{P}_{\mu ,\varepsilon })\) possesses a weak solution \(u\in W^{1,p}(\Omega )\), for \(\mu >0\) sufficiently large, and for all \(\varepsilon \in (0,1)\).

Proof

By (5), the Nemytskii operator \({\mathcal {B}}\) given by \({\mathcal {B}} u(x)=b(\cdot ,u)\) is well defined and \({\mathcal {B}}:W^{1,p}(\Omega )\longrightarrow W^{-1,p^{\prime }}(\Omega )\) is continuous and bounded. By the compact embedding \(W^{1,p}(\Omega )\hookrightarrow L^{p}(\Omega )\), \( {\mathcal {B}}\) is completely continuous.

Considering (2), define the function \(\pi _{\delta ,\varepsilon }:(-\delta ,+\infty )\times {\mathbb {R}}^{N}\longrightarrow {\mathbb {R}}\) by

$$\begin{aligned} \pi _{\delta }(s,\xi )=\frac{|\xi |^{p}}{s+\delta +\varepsilon } \end{aligned}$$

which satisfies the estimate

$$\begin{aligned} |\pi _{\delta }(s,\xi )|\le \frac{1}{\varepsilon }|\xi |^{p},\quad \text {for all }s>-\delta \text {, }\xi \in {\mathbb {R}}^{N}\text { and all }\varepsilon \in (0,1). \end{aligned}$$

Let \(\Pi _{\delta ,\varepsilon }:[{\underline{u}},{\overline{u}}]\subset W^{1,p}(\Omega )\longrightarrow L^{1}(\Omega )\subset W^{-1,p^{\prime }}(\Omega )\) denote the corresponding Nemytskii operator, that is \(\Pi _{\delta ,\varepsilon }u(x)=\pi _{\delta ,\varepsilon }(u(x),\nabla u(x)),\) which is bounded and continuous (see [31, Theorem 2.76] and [23, Theorem 3.4.4]). Moreover, \(\Pi _{\delta ,\varepsilon }\) is completely continuous due to the compact embedding of \(W^{1,p}(\Omega )\) into \( L^{p}(\Omega )\).

In view of \((\mathrm {H.}1),\) if \(\rho >0\) satisfies

$$\begin{aligned} -\rho \le {\underline{u}}\le {\overline{u}}\le \rho , \end{aligned}$$
(15)

the Nemitskii operator \({\mathcal {N}}_{f}:[{\underline{u}},{\overline{u}}]\subset W^{1,p}(\Omega )\rightarrow W^{-1,p^{\prime }}(\Omega )\) generated by the Carathéodory function f is bounded and completely continuous thanks to Rellich–Kondrachov compactness embedding theorem.

At this point, problem \((\textrm{P}_{\mu ,\varepsilon })\) can be equivalently expressed as

$$\begin{aligned} {\mathcal {A}}_{\mu ,\varepsilon }(u):=-\Delta _{p}u+\mu {\mathcal {B}}u+\Pi _{\delta ,\varepsilon }\circ {\mathcal {T}}(u)-{\mathcal {N}}_{f}\circ {\mathcal {T}} (u)=0\text { \ in }W^{-1,p^{\prime }}(\Omega ). \end{aligned}$$
(16)

By \((\mathrm {H.}1),\) it is readily seen that the operator \({\mathcal {A}}_{\mu ,\varepsilon }:W^{1,p}(\Omega )\rightarrow W^{-1,p^{\prime }}(\Omega )\) is well defined, bounded, and continuous.

Let us show that \({\mathcal {A}}_{\mu ,\varepsilon }\) is coercive. From (16), we have

$$\begin{aligned} \left\langle {\mathcal {A}}_{\mu ,\varepsilon }(u),u\right\rangle{} & {} =\int _{\Omega }|\nabla u|^{p}\textrm{d}x+\mu \int _{\Omega }b(x,u)u\textrm{ d }x+\int _{\Omega }\frac{|\nabla ({\mathcal {T}}u)|^{p}}{{\mathcal {T}}u+\delta +\varepsilon }u\textrm{d}x\nonumber \\{} & {} \quad \ -\int _{\Omega }f(\cdot ,{\mathcal {T}} u,\nabla ({\mathcal {T}}u))u\,\textrm{d}x \nonumber \\{} & {} \ge \int _{\Omega }|\nabla u|^{p}\textrm{d}x+\mu \int _{\Omega }b(x,u)u\textrm{d}x-\int _{\Omega }\frac{|\nabla ({\mathcal {T}}u)|^{p}}{ {\mathcal {T}}u+\delta +\varepsilon }|u|\textrm{d}x\nonumber \\{} & {} \quad \ -\int _{\Omega }f(\cdot ,{\mathcal {T}}u,\nabla ({\mathcal {T}}u))u\,\textrm{d}x. \end{aligned}$$
(17)

Bearing in mind (6) as well as the estimates in Lemmas 2 and 3, we thus arrive at

$$\begin{aligned} \begin{array}{l} \left\langle {\mathcal {A}}_{\mu ,\varepsilon }(u),u\right\rangle \ge \Vert \nabla u\Vert _{p}^{p}+\mu \left( C_{1}\Vert u\Vert _{p}^{p}-C_{2}\right) \\ \quad -{\hat{C}}_{\varepsilon }(1+\left\| u\right\| _{p})-\frac{1}{2}\Vert \nabla u\Vert _{p}^{p}-C_{0}(1+\left\| u\right\| _{p}+\left\| u\right\| _{p}^{p}). \end{array} \end{aligned}$$

In view of (14) and for \(\mu >0\) large so that \(\mu C_{1}-C_{0}>0,\) for every sequence \((u_{n})_{n}\) in \(W^{1,p}(\Omega )\), the last inequality forces

$$\begin{aligned} \lim _{n\rightarrow +\infty }\frac{\langle {\mathcal {A}}_{\mu ,\varepsilon }(u_{n}),u_{n}\rangle }{\Vert u_{n}\Vert _{1,p}}=+\infty , \end{aligned}$$

as desired.

The next step is to show that the operator \({\mathcal {A}}_{\mu }\) is pseudomonotone. Toward this, suppose \(u_{n}\rightharpoonup u\) in \( W^{1,p}(\Omega )\) and

$$\begin{aligned} \limsup _{n\rightarrow +\infty }\langle {\mathcal {A}}_{\mu ,\varepsilon }(u_{n}),u_{n}-u\rangle \le 0. \end{aligned}$$

In view of the complete continuity of the operators \({\mathcal {B}}\), \(\Pi _{\delta ,\varepsilon }\) and \({\mathcal {N}}_{f}\), we get

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\langle {\mathcal {B}}(u_{n}),u_{n}-u \rangle= & {} 0, \\ \underset{n\rightarrow \infty }{\lim }\langle \Pi _{\delta ,\varepsilon }(u_{n}),u_{n}-u\rangle= & {} 0, \\ \underset{n\rightarrow \infty }{\lim }\langle {\mathcal {N}}_{f}({\mathcal {T}} u_{n}),u_{n}-u\rangle= & {} 0. \end{aligned}$$

Then, using the \((\textrm{S})_{+}\)-property of \(-\Delta _{p}\), we deduce that \(u_{n}\rightarrow u\) in \(W^{1,p}(\Omega ).\) Therefore,

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\langle {\mathcal {A}}_{\mu ,\varepsilon }(u_{n}),u_{n}-v\rangle =\langle {\mathcal {A}}_{\mu ,\varepsilon }(u),u-v\rangle , \end{aligned}$$

for all \(v\in W^{1,p}(\Omega ),\) because \({\mathcal {A}}_{\mu ,\varepsilon }\) is continuous. This proves that the operator \({\mathcal {A}}_{\mu ,\varepsilon } \) is pseudomonotone.

According to the properties above, we are in a position to apply the main theorem for pseudomonotone operators [11, Theorem 2.99] to the operator \({\mathcal {A}}_{\mu ,\varepsilon }\). It entails the existence of \( u\in W^{1,p}(\Omega )\) fulfilling

$$\begin{aligned} \left\langle {\mathcal {A}}_{\mu ,\varepsilon }(u),\varphi \right\rangle =0,\ \ \varphi \in W^{1,p}(\Omega ). \end{aligned}$$

Owing to [10, Theorem 3], one has

$$\begin{aligned} |\nabla u|^{p-2}\frac{\partial u}{\partial \eta }=0\ \text {on }\partial \Omega \text {.} \end{aligned}$$

Thus, \(u\in W^{1,p}(\Omega )\) is a weak solution of \((\textrm{P}_{\mu ,\varepsilon })\). This ends the proof. \(\square \)

2.2 A Sub-supersolution Theorem

Theorem 5

Suppose \((\mathrm {H.}1)\)–\((\mathrm {H.}2)\) hold true. Then, problem \((\textrm{P})\) possesses a solution \(u\in {\mathcal {C}}^{1}(\overline{ \Omega })\) such that

$$\begin{aligned} {\underline{u}}\le u\le {\overline{u}}. \end{aligned}$$
(18)

Proof

According to Theorem 4, problem \((\textrm{P}_{\mu ,\varepsilon })\) admits a weak solution u in \(W^{1,p}(\Omega ).\) Let us next verifythat u satisfy the inequalities (18). We provide the argument only for \( u\le {\overline{u}}\) because \({\underline{u}}\le u\) can be similarly established. First, note from Lemma 3 that \(\frac{|\nabla ({\mathcal {T}} u)|^{p}}{{\mathcal {T}}u+\delta +\varepsilon }(u-{\overline{u}})_{+}\in L^{1}(\Omega )\). Thus, test \((\textrm{P}_{\mu ,\varepsilon })\) with \((u- {\overline{u}})_{+}\in W^{1,p}(\Omega )\) and taking \((\mathrm {H.}2)\) into account, we achieve

$$\begin{aligned} \begin{array}{l} \int _{\Omega }|\nabla u|^{p-2}\nabla u{ }\nabla (u-\overline{ u })_{+}\textrm{d}x+\int _{\Omega }\frac{|\nabla ({\mathcal {T}}u)|^{p}}{ {\mathcal {T}}u+\delta +\varepsilon }(u-{\overline{u}})_{+}\textrm{d}x \\ \quad =\int _{\Omega }f(\cdot ,{\mathcal {T}}u,\nabla ({\mathcal {T}}u))(u-{\overline{u}} )_{+}\textrm{d}x-\mu \int _{\Omega }b(\cdot ,u)(u-{\overline{u}})_{+} \textrm{d}x \\ \quad =\int _{\Omega }f(\cdot ,{\overline{u}},\nabla {\overline{u}})(u-{\overline{u}} )_{+} \textrm{d}x-\mu \int _{\Omega }(u-{\overline{u}})_{+}^{p} \textrm{d}x \\ \quad \le \int _{\Omega }|\nabla {\overline{u}}|^{p-2}\nabla {\overline{u}}\,\nabla (u- {\overline{u}})_{+}\,\textrm{d}x+\int _{\Omega }\frac{|\nabla {\overline{u}} |^{p} }{{\overline{u}}+\delta +\varepsilon }(u-{\overline{u}})_{+}\,\textrm{d} x-\mu \int _{\Omega }(u-{\overline{u}})_{+}^{p}\textrm{d}x\text {.} \end{array} \end{aligned}$$

By (4) note that

$$\begin{aligned} \int _{\Omega }\frac{|\nabla ({\mathcal {T}}u)|^{p}}{{\mathcal {T}}u+\delta +\varepsilon }(u-{\overline{u}})_{+}\textrm{d}x=\int _{\Omega }\frac{ |\nabla {\overline{u}}|^{p}}{{\overline{u}}+\delta +\varepsilon }(u-{\overline{u}} )_{+}\,\textrm{d}x. \end{aligned}$$

Then, it turns out that

$$\begin{aligned} \int _{\Omega }\left( |\nabla u|^{p-2}\nabla u-|\nabla {\overline{u}} |^{p-2}\nabla {\overline{u}}\right) \nabla (u-{\overline{u}})_{+}\,\textrm{d} x\le -\mu \int _{\Omega }(u-{\overline{u}})_{+}^{p}\textrm{d}x\le 0. \end{aligned}$$
(19)

The monotonicity of \(\Delta _{p}\) directly leads to \(u\le {\overline{u}}\). Test \((\textrm{P}_{\mu ,\varepsilon })\) with \(({\underline{u}}-u)_{+}\in W^{1,p}(\Omega )\), a quite similar reasoning furnishes \({\underline{u}}\le u\). Moreover, by [34, Remark 8], one has \(u\in {\mathcal {C}}^{1,\tau }( {\overline{\Omega }})\) for some \(\tau \in ]0,1[\) as well as \(\frac{\partial u}{ \partial \eta }=0\) on \(\partial \Omega \). Consequently, u is a solution of \((\textrm{P}_{\varepsilon ,\mu })\) within \([{\underline{u}},{\overline{u}}]\) which, due to (11), reads as

$$\begin{aligned} (\textrm{P}_{\varepsilon })\qquad \left\{ \begin{array}{ll} -\Delta _{p}{u+}\frac{|\nabla ({\mathcal {T}}u)|^{p}}{{\mathcal {T}}u+\delta +\varepsilon }=f(x,{\mathcal {T}}u,\nabla ({\mathcal {T}}u)) &{}\quad \text {in}\;\Omega ,\\ |\nabla u|^{p-2}\frac{\partial u}{\partial \eta }=0 &{}\quad \text {on}\;\partial \Omega . \end{array} \right. \end{aligned}$$

The task is now to find solutions of \((\textrm{P})\) by passing to the limit in \((\textrm{P}_{\varepsilon })\) as \(\varepsilon \rightarrow 0\). To this end, set \(\varepsilon =\frac{1}{n}\) with any positive integer \(n\ge 1\), there exists \(u_{n}:=u_{\frac{1}{n}}\in {\mathcal {C}}^{1,\tau }({\overline{\Omega }})\) solution of \((\textrm{P}_{n})\) (\((\textrm{P}_{\varepsilon })\) with \(\varepsilon =\frac{1}{n}\)), that is,

$$\begin{aligned} u_{n}\in [{\underline{u}},{\overline{u}}] \end{aligned}$$
(20)

and

$$\begin{aligned} \begin{array}{l} \int _{\Omega }|\nabla u_{n}|^{p-2}\nabla u_{n}{ }\nabla \varphi \textrm{d}x+\int _{\Omega }\frac{|\nabla u_{n}|^{p}}{ u_{n}+\delta +\frac{1}{n}}\varphi \textrm{d}x=\int _{\Omega }f(x,u_{n},\nabla u_{n})\varphi \textrm{d}x, \end{array} \end{aligned}$$
(21)

for all \(\varphi \in W^{1,p}(\Omega )\cap L^{\infty }(\Omega )\). Since the embedding \({\mathcal {C}}^{1,\tau }(\overline{\Omega })\subset {\mathcal {C}}^{1}( {\overline{\Omega }})\) is compact, we can extract subsequences (still denoted by \(\{u_{n}\}\)) such that

$$\begin{aligned} u_{n}\rightarrow u\text { in }{\mathcal {C}}^{1}(\overline{\Omega })\text { with } u\in [{\underline{u}},{\overline{u}}]. \end{aligned}$$
(22)

Therefore,

$$\begin{aligned}{} & {} |\nabla u_{n}|^{p-2}\nabla u_{n}\rightarrow |\nabla u|^{p-2}\nabla u\text { \ in }{\mathcal {C}}({\overline{\Omega }}), \\{} & {} |\nabla u_{n}|^{p}\rightarrow |\nabla u|^{p}\text { \ in }{\mathcal {C}}( {\overline{\Omega }}) \end{aligned}$$

and

$$\begin{aligned} f(x,u_{n},\nabla u_{n})\rightarrow f(x,u,\nabla u)\text { \ in }{\mathcal {C}}( {\overline{\Omega }}), \end{aligned}$$

because f is a Carathéodory function. Then, on the basis of (20) and (2), owing to Lebesgue dominated convergence theorem, we may pass to the limit as \(n\rightarrow \infty \) in (21) to get

$$\begin{aligned} \begin{array}{l} \int _{\Omega }|\nabla u|^{p-2}\nabla u{ }\nabla \varphi \textrm{d}x+\int _{\Omega }\frac{|\nabla u|^{p}}{u+\delta }\varphi \textrm{d}x=\int _{\Omega }f(x,u,\nabla u)\varphi \textrm{d}x, \end{array} \end{aligned}$$

for all \(\varphi \in W^{1,p}(\Omega )\cap L^{\infty }(\Omega )\). Moreover, according to (22), we have \(u+\delta >0\) a.e. in \(\Omega \). This proves that \(u\in {\mathcal {C}}^{1}(\overline{\Omega })\) is a solution of problem \(\left( \textrm{P}\right) \) within \([{\underline{u}},{\overline{u}}]\). The proof is now completed. \(\square \)

3 Location Principle for Nodal Solutions

It this section, we focus on the location of nodal solutions for problem \( \left( \textrm{P}\right) \). We will posit the hypothesis below.

  1. (H.3)

    There exist \(\alpha ,\beta ,M>0\) such that

    $$\begin{aligned} \max \{\alpha ,\beta \}<p-1 \end{aligned}$$

    and, moreover,

    $$\begin{aligned} |f(x,s,\xi )|\le M(1+|s|^{\alpha }+|\xi |^{\beta })\text {,} \end{aligned}$$

    for all \((x,s,\xi )\in \Omega \times {\mathbb {R}} \times {\mathbb {R}}^{N}\).

Lemma 6

Assume \((\textrm{H}.2)\) and \((\textrm{H}.3)\) are fulfilled.

  1. (i)

    If \({\overline{u}}_{1},{\overline{u}}_{2}\in W^{1,p}(\Omega )\cap L^{\infty }(\Omega )\) are supersolutions for problem \( \left( \textrm{P}\right) \), then \({\overline{u}}=\min \{{\overline{u}}_{1}, {\overline{u}}_{2}\}\) is also a supersolution for problem \(\left( \textrm{P} \right) \).

  2. (ii)

    If \({\underline{u}}_{1},{\underline{u}}_{2}\in W^{1,p}(\Omega )\cap L^{\infty }(\Omega )\) are subsolutions for problem \( \left( \textrm{P}\right) \), then \({\underline{u}}=\max \{{\underline{u}}_{1}, {\underline{u}}_{2}\}\) is also a subsolution for problem \(\left( \textrm{P} \right) \).

Proof

We provide the argument only for part \(\mathrm {(}i\mathrm {)}\) because \( \mathrm {(}ii\mathrm {)}\) can be similarly established. Inspired by [11, Theorem 3.20], [9, Lemma 1] and the proof of [29, Lemma 3], for a fixed \(\varepsilon >0,\) let us define the truncation function \( \xi _{\varepsilon }(s)=\max \{-\varepsilon ,\min \{s,\varepsilon \}\}\) for \( s\in {\mathbb {R}}.\) It is shown in [28] that \(\xi _{\varepsilon }(({\overline{u}}_{1}- {\overline{u}}_{2})^{-})\in W^{1,p}(\Omega ),\)

$$\begin{aligned} \nabla \xi _{\varepsilon }(({\overline{u}}_{1}-{\overline{u}}_{2})^{-})=\xi _{\varepsilon }^{\prime }(({\overline{u}}_{1}-{\overline{u}}_{2})^{-})\nabla ( {\overline{u}}_{1}-{\overline{u}}_{2})^{-} \end{aligned}$$

For any test function \(\varphi \in C_{c}^{1}(\Omega )\) with \(\varphi \ge 0,\) it holds

$$\begin{aligned} \begin{array}{l} \left\langle -\Delta _{p}{\overline{u}}_{1}+\frac{|\nabla {\overline{u}} _{1}|^{p} }{{\overline{u}}_{1}+\delta },\xi _{\varepsilon }(({\overline{u}}_{1}- {\overline{u}}_{2})^{-})\varphi \right\rangle \\ \quad \ge \int _{\Omega }f(x,{\overline{u}}_{1},\nabla {\overline{u}}_{1})\xi _{\varepsilon }(({\overline{u}}_{1}-{\overline{u}}_{2})^{-})\varphi \textrm{d}x, \end{array} \end{aligned}$$
(23)

and

$$\begin{aligned} \begin{array}{l} \left\langle -\Delta _{p}{\overline{u}}_{2}+\frac{|\nabla {\overline{u}} _{2}|^{p} }{{\overline{u}}_{2}+\delta },(\varepsilon -\xi _{\varepsilon }(( {\overline{u}} _{1}-{\overline{u}}_{2})^{-}))\varphi \right\rangle \\ \quad \ge \int _{\Omega }f(x,{\overline{u}}_{2},\nabla {\overline{u}}_{2})\left( \varepsilon -\xi _{\varepsilon }(({\overline{u}}_{1}-{\overline{u}} _{2})^{-})\right) \varphi \textrm{d}x. \end{array} \end{aligned}$$
(24)

On the other hand, using the monotonicity of the p-Laplacian operator, we get

$$\begin{aligned} \begin{array}{l} \left\langle -\Delta _{p}{\overline{u}}_{1}+\frac{|\nabla {\overline{u}} _{1}|^{p} }{{\overline{u}}_{1}+\delta },\xi _{\varepsilon }(({\overline{u}}_{1}- {\overline{u}}_{2})^{-})\varphi \right\rangle \\ \qquad +\left\langle -\Delta _{p}{\overline{u}}_{2}+ \frac{|\nabla {\overline{u}}_{2}|^{p}}{{\overline{u}}_{2}+\delta },(\varepsilon -\xi _{\varepsilon }(({\overline{u}}_{1}-{\overline{u}}_{2})^{-}))\varphi \right\rangle \\ \quad \le \int _{\Omega }|\nabla {\overline{u}}_{1}|^{p-2}(\nabla {\overline{u}} _{1},\nabla \varphi )_{ {\mathbb {R}}^{N}}\xi _{\varepsilon }(({\overline{u}}_{1}-{\overline{u}}_{2})^{-}) \textrm{d}x \\ \qquad +\int _{\Omega }\frac{|\nabla {\overline{u}} _{1}|^{p}}{{\overline{u}}_{1}+\delta }\xi _{\varepsilon }(({\overline{u}}_{1}- {\overline{u}}_{2})^{-})\varphi \textrm{d}x \\ \quad \quad +\int _{\Omega }|\nabla {\overline{u}} _{2}|^{p-2}(\nabla {\overline{u}}_{2},\nabla \varphi )_{{\mathbb {R}}^{N}}\left( \varepsilon -\xi _{\varepsilon }(({\overline{u}}_{1}-{\overline{u}} _{2})^{-})\right) \textrm{d}x \\ \quad \quad +\int _{\Omega }\frac{|\nabla {\overline{u}} _{2}|^{p}}{{\overline{u}}_{2}+\delta }(\varepsilon -\xi _{\varepsilon }(( {\overline{u}}_{1}-{\overline{u}}_{2})^{-}))\varphi \textrm{d}x. \end{array} \end{aligned}$$
(25)

Then, gathering (23) together with (24), by means of (25), one gets

$$\begin{aligned} \begin{array}{l} \int _{\Omega }|\nabla {\overline{u}}_{1}|^{p-2}(\nabla {\overline{u}}_{1},\nabla \varphi )_{ {\mathbb {R}}^{N}}\frac{1}{\varepsilon }\xi _{\varepsilon }(({\overline{u}}_{1}-{\overline{u}} _{2})^{-})\textrm{d}x \\ \quad \quad +\int _{\Omega }\frac{|\nabla {\overline{u}} _{1}|^{p}}{{\overline{u}}_{1}+\delta }\frac{1}{\varepsilon }\xi _{\varepsilon }(({\overline{u}}_{1}-{\overline{u}}_{2})^{-}\textrm{d}x \\ \quad \quad +\int _{\Omega }|\nabla {\overline{u}} _{2}|^{p-2}(\nabla {\overline{u}}_{2},\nabla \varphi )_{ {\mathbb {R}}^{N}}\left( 1-\frac{1}{\varepsilon }\xi _{\varepsilon }(({\overline{u}}_{1}- {\overline{u}}_{2})^{-})\right) \textrm{d}x \\ \quad \quad +\int _{\Omega }\frac{|\nabla {\overline{u}} _{2}|^{p}}{{\overline{u}}_{2}+\delta }(1-\frac{1}{\varepsilon }\xi _{\varepsilon }(({\overline{u}}_{1}-{\overline{u}}_{2})^{-})\textrm{d}x \\ \quad \ge \int _{\Omega }f(x,{\overline{u}}_{1},\nabla {\overline{u}}_{1})\frac{1}{ \varepsilon }\xi _{\varepsilon }(({\overline{u}}_{1}-{\overline{u}} _{2})^{-})\varphi \textrm{d}x \\ \quad \quad +\int _{\Omega }f(x,{\overline{u}}_{2},\nabla {\overline{u}}_{2})\left( 1-\frac{1}{\varepsilon }\xi _{\varepsilon }(( {\overline{u}}_{1}-{\overline{u}}_{2})^{-})\right) \varphi \textrm{d}x, \end{array} \end{aligned}$$

Passing to the limit as \(\varepsilon \rightarrow 0\) and noticing that

$$\begin{aligned} \frac{1}{\varepsilon }\xi _{\varepsilon }(({\overline{u}}_{1}-{\overline{u}} _{2})^{-}\rightarrow \mathbbm {1}_{\{{\overline{u}}_{1}<{\overline{u}}_{2}\}}(x) \text {, \ a.e. in }\Omega \text { as }\varepsilon \rightarrow 0, \end{aligned}$$

we obtain

$$\begin{aligned} \int _{\Omega }|\nabla {\overline{u}}|^{p-2}\nabla {\overline{u}}\nabla \varphi \textrm{d}x+\int _{\Omega }\frac{|\nabla {\overline{u}}|^{p}}{{\overline{u}}+\delta }\varphi \textrm{d}x\ge \int _{\Omega }f(x,{\overline{u}},\nabla {\overline{u}})\varphi \textrm{d}x, \end{aligned}$$

for all \(\varphi \in C_{c}^{1}(\Omega ),\) \(\varphi \ge 0\) a.e. in \(\Omega \). Since \(C_{c}^{1}(\Omega )\) is dense in \(W^{1,p}(\Omega ),\) we achieve the desired conclusion. \(\square \)

We set \(r_{\pm }:=\max \{\pm r,0\}\) and so \(w_{+}\) and \(w_{-}\) denote the positive and the negative part of a function w, respectively (that is, \(w=w_{+}-w_{-}\)). Inspired by [30], next we set forth a result addressing location of solutions and a priori estimates for problem \(\left( \textrm{P}\right) \).

Theorem 7

Assume that condition \((\textrm{H}.2){-}(\textrm{H}.3)\) are fulfilled.

  1. (i)

    If \(f(x,0,0)\ge 0\) for a.e. \(x\in \Omega \), then for every nodal solution \(u_{0}\in [{\underline{u}},{\overline{u}}]\) of problem \(\left( \textrm{P}\right) \) there exists a nontrivial solution \( u_{+}\) of \(\left( \textrm{P}\right) \) such that \(u_{0}\le u_{+}\le {\overline{u}}_{+}\) and \(u_{+}\ge 0\) on \(\Omega \).

  2. (ii)

    If \(f(x,0,0)\le 0\) for a.e. \(x\in \Omega \), then for every nodal solution \(u_{0}\in [{\underline{u}},{\overline{u}}]\) of problem \(\left( \textrm{P}\right) \) there exists a nontrivial solution \( u_{-}\) of \(\left( \textrm{P}\right) \) such that \(u_{0}\ge u_{-}\ge {\overline{u}}_{-}\) and \(u_{-}\le 0\) on \(\Omega \).

  3. (iii)

    If \(f(x,0,0)=0\) for a.e. \(x\in \Omega \), then for every nodal solution \(u_{0}\in [{\underline{u}},{\overline{u}}]\) of problem \(\left( \textrm{P}\right) \) there exist two other nontrivial solutions \(u_{+}\) and \(u_{-}\) of \(\left( \textrm{P}\right) \) such that \( u_{-}\le u_{0}\le u_{+}\), \(u_{+}\ge 0\) and \(u_{-}\le 0\) on \(\Omega \).

Proof

(i) Let \(u_{0}\) be a nodal solution of problem \(\left( \textrm{P}\right) \) within \([{\underline{u}},{\overline{u}}]\). The assumption \( f(x,0,0)\ge 0\) for a.e. \(x\in \Omega \) ensures that 0 is a subsolution of problem \(\left( \textrm{P}\right) \). By Lemma 6, part \(\mathrm {(} i\mathrm {),}\) we infer that \(u_{0,+}:=\max \{0,u_{0}\}\ \)is a subsolution of problem \(\left( \textrm{P}\right) \) which obviously satisfies \(u_{0,+}\le {\overline{u}}_{+}\), with \({\overline{u}}_{+}:=\max \{0,{\overline{u}}\}\). So, in view of Theorem 5, there exists a solution \(u_{+}\in {\mathcal {C}}^{1}( {\overline{\Omega }})\) of \(\left( \textrm{P}\right) \) within \([u_{0,+}, {\overline{u}}_{+}]\). Since the solution \(u_{0}\) of \(\left( \textrm{P}\right) \) is nodal, its positive part \(u_{0,+}\) is strictly positive on a subset of \( \Omega \) of positive measure. Hence, \(u_{+}\) is positive.

(ii) Let \(u_{0}\) be a nodal solution of problem \(\left( \textrm{P}\right) \) within \([{\underline{u}},{\overline{u}}]\). The assumption \( f(x,0,0)\le 0\) for a.e. \(x\in \Omega \) ensures that 0 is a supersolution of problem \(\left( \textrm{P}\right) \). By Lemma 6, part \( \mathrm {(}ii\mathrm {)}\), we infer that \(u_{0,-}:=\min \{0,u_{0}\}\ \)is a supersolution of problem \(\left( \textrm{P}\right) \) which clearly satisfies \(u_{0,-}\ge {\overline{u}}_{-}\), with \({\overline{u}}_{-}:=\min \{0,{\overline{u}} \}\). Then, Theorem 5 implies that there exits a solution \(u_{-}\in {\mathcal {C}}^{1}({\overline{\Omega }})\) of \(\left( \textrm{P}\right) \) within \( [ {\overline{u}}_{-},u_{0,-}]\). Recalling that the solution \(u_{0}\) of \(\left( \textrm{P}\right) \) is nodal, its negative part \(u_{0,-}\) is strictly negative on a subset of \(\Omega \) of positive measure. Therefore, \(u_{-}\le 0\) and \(u_{-}\ne 0\).

(iii) If \(f(x,0,0)=0\) for a.e. \(x\in \Omega \), then the assertions (i) and (ii) can be applied simultaneously, giving rise to two nontrivial opposite constant-sign solutions \(u_{+}\) and \(u_{-}\) of problem \(\left( \textrm{P}\right) \) with the properties required in the statement. \(\square \)

4 Nodal Solutions

In this section, beside \((\textrm{H}_{1}),\) we will posit the hypothesis below.

  1. (H.4)

    With appropriate \(m>0\) one has

    $$\begin{aligned} \lim _{|s|\rightarrow 0}\inf \{f(x,s,\xi ):\xi \in {\mathbb {R}}^{N}\}>m\text {,} \end{aligned}$$

    uniformly in \(x\in \Omega \).

Our first goal is to construct sub-and-supersolution pairs of \(\left( \textrm{P}\right) \). With this aim, consider the homogeneous Dirichlet problem

$$\begin{aligned} -\Delta _{p}z=1,\text { in }\Omega ,z=0\text { on }\partial \Omega \text {,} \end{aligned}$$
(26)

which admits a unique solution \(z\in {\mathcal {C}}^{1,\tau }({\overline{\Omega }} )\) satisfying

$$\begin{aligned}{} & {} \Vert z\Vert _{C^{1,\tau }({\overline{\Omega }})}\le L\text {,} \end{aligned}$$
(27)
$$\begin{aligned}{} & {} \frac{d(x)}{c}\le z\le cd(x)\text {\ in\ }\Omega \text {,\ \ \ \ } \frac{\partial z}{\partial \eta }<0\text {\ on\ }\partial \Omega \text {,} \end{aligned}$$
(28)

for certain constant \(c>1\).

Now, given \(0<\delta <\textrm{diam}(\Omega )\), denote by \(z_{\delta }\in {\mathcal {C}}^{1,\tau }({\overline{\Omega }})\) the solution of the Dirichlet problem

$$\begin{aligned} -\Delta _{p}u=\left\{ \begin{array}{cl} 1 &{}\quad \text {if }x\in \Omega \backslash {\overline{\Omega }}_{\delta }, \\ -1 &{}\quad \text {otherwise}, \end{array} \right. \quad u=0\text { on }\partial \Omega , \end{aligned}$$
(29)

where \(\Omega _{\delta }:=\left\{ x\in \Omega :d(x)<\delta \right\} \) with \(\delta >0\) sufficiently small.

Existence and uniqueness directly stem from Minty–Browder’s Theorem [6] while \({\mathcal {C}}^{1,\tau }({\overline{\Omega }})\) regularity follows from Lieberman’s regularity Theorem [27]. Moreover, the weak comparison principle implies that

$$\begin{aligned} z_{\delta }\le z\ \text {in }\Omega , \end{aligned}$$
(30)

while, for \(\delta >0\) small enough, it holds

$$\begin{aligned} \frac{\partial z_{\delta }}{\partial \eta }<\frac{1}{2}\frac{\partial z}{ \partial \eta }<0\text { on }\partial \Omega \text { \ and \ }z_{\delta }\ge \frac{1}{2}{ }z\text { \ in }\Omega \end{aligned}$$
(31)

(see [25]).

Define

$$\begin{aligned} {\underline{u}}:=\delta ^{\frac{1}{p}}z_{\delta }^{\omega }-\delta \text {, \ \ \ }{\overline{u}}:=\delta ^{-p}z^{\overline{\omega }}-\delta \text {,} \end{aligned}$$
(32)

where

$$\begin{aligned} \frac{\omega -1}{\omega }>\frac{1}{p-1}>\frac{{\overline{\omega }}-1}{ {\overline{\omega }}}\text { \ with }\omega>{\overline{\omega }}>1 \end{aligned}$$
(33)

and

$$\begin{aligned} {\overline{\omega }}<1+p(1-\frac{\max \{\alpha ,\beta \}}{p-1}). \end{aligned}$$
(34)

From (32), (27), (30) and (28), it follows

$$\begin{aligned} {\overline{u}}\le \delta ^{-p}(Ld)^{{\overline{\omega }}}\text { \ and \ }\Vert \nabla {\overline{u}}\Vert _{\infty }\le \delta ^{-p}{\hat{L}}, \end{aligned}$$
(35)

with \({\hat{L}}:={\overline{\omega }}L^{{\overline{\omega }}}\). Moreover,

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial {\underline{u}}}{\partial \eta }=\delta ^{\frac{1}{p}}\frac{ \partial (z_{\delta }^{\omega })}{\partial \eta }=\delta ^{\frac{1}{p} }\omega z_{\delta }^{\omega -1}\frac{\partial z_{\delta }}{\partial \eta }=0 \\ \frac{\partial {\overline{u}}}{\partial \eta }=\delta ^{-p}\frac{\partial (z^{ {\overline{\omega }}})}{\partial \eta }=\delta ^{-p}{\overline{\omega }}z^{ \overline{\omega }-1}\frac{\partial z}{\partial \eta }=0 \end{array} \right. \text { on }\partial \Omega , \end{aligned}$$
(36)

because z,  \(z_{\delta }\) solve (26), (29), respectively, and \( \omega ,{\overline{\omega }}>1\).

We claim that \({\underline{u}}\le {\overline{u}}\) with a small \(\delta >0\). Indeed, since \(\omega >{\overline{\omega }}\) and \(z_{\delta }\le z\) for all \( \delta <\textrm{diam}(\Omega ),\) it follows that

$$\begin{aligned} \begin{array}{l} {\overline{u}}(x)-{\underline{u}}(x)=\left( \delta ^{-p}z^{{\bar{\omega }}}-\delta \right) -\left( \delta ^{\frac{1}{p}}z_{\delta }^{\omega }-\delta \right) \\ \ge \delta ^{-p}z^{{\bar{\omega }}}-\delta ^{\frac{1}{p}}z^{\omega }=z^{\omega }(\delta ^{-p}z^{{\bar{\omega }}-\omega }-\delta ^{\frac{1}{p}}) \\ \ge z^{\omega }(\delta ^{-p}(cd(x))^{{\bar{\omega }}-\omega }-\delta ^{\frac{1 }{p}})\ge 0, \end{array} \end{aligned}$$

provided that \(\delta >0\) is small. Thus, \({\underline{u}}\le {\overline{u}}\) in \(\Omega \), as desired.

Theorem 8

Let \((\mathrm {H.}3)\)–\((\mathrm {H.}4)\) be satisfied. Then, for \(\delta >0\) small enough in (32), problem \( \left( \textrm{P}\right) \) admits a nodal solution \(u_{0}\in {\mathcal {C}}^{1}( {\overline{\Omega }})\) within \([{\underline{u}},{\overline{u}}]\) such that \(u_{0}(x)\) is negative once \(d(x)\rightarrow 0\). Furthermore, there exists a positive solution \(u_{+}\in {\mathcal {C}}^{1}({\overline{\Omega }} )\) of \(\left( \textrm{P}\right) \) with \(u_{+}(x)\) is zero once \( d(x)\rightarrow 0\).

Proof

Let us show that the function \({\overline{u}}\) given by (32) satisfies (3). With this aim, pick \(u\in W^{1,p}(\Omega )\) such that \(-{\overline{u}}\le u\le {\overline{u}}\). From \((\mathrm {H.}3),\) (35), it follows

$$\begin{aligned} \begin{aligned} |f(\cdot ,{\overline{u}},\nabla {\overline{u}})|&\le M(1+|{\overline{u}} |^{\alpha }+|\nabla {\overline{u}}|^{\beta }) \\&\le M(1+(\delta ^{-p}(Ld)^{{\overline{\omega }}})^{\alpha }+(\delta ^{-p}{\hat{L}})^{\beta } \\&\le C\delta ^{-p\max \{\alpha ,\beta \}}\text {,} \end{aligned} \end{aligned}$$
(37)

for some constant \(C>0\) and for \(\delta >0\) sufficiently small. A direct computation gives

$$\begin{aligned}{} & {} -\Delta _{p}z^{{\bar{\omega }}}+\lambda \frac{|\nabla z^{{\bar{\omega }}}|^{p}}{ z^{ {\bar{\omega }}}}={\bar{\omega }}^{p-1}\left( 1-({\bar{\omega }}-1)\left( p-1\right) \frac{|\nabla z|^{p}}{z}\right) z^{({\bar{\omega }}-1)(p-1)}+\bar{ \omega }^{p} \frac{z^{({\bar{\omega }}-1)p}|\nabla z|^{p}}{z^{{\bar{\omega }}}} \\{} & {} \quad ={\bar{\omega }}^{p-1}\left( 1-({\bar{\omega }}-1)\left( p-1\right) \frac{|\nabla z|^{p}}{z}\right) z^{({\bar{\omega }}-1)(p-1)}+{\bar{\omega }}^{p}z^{({\bar{\omega }} -1)(p-1)}\frac{z^{{\bar{\omega }}-1}|\nabla z|^{p}}{z^{{\bar{\omega }}}} \\{} & {} \quad ={\bar{\omega }}^{p-1}\left[ 1+{\bar{\omega }}(1-\frac{({\bar{\omega }}-1)\left( p-1\right) }{{\bar{\omega }}})\frac{|\nabla z|^{p}}{z}\right] z^{({\bar{\omega }} -1)(p-1)}. \end{aligned}$$

Using (32) and (33) one has

$$\begin{aligned} \begin{array}{l} -\Delta _{p}{\overline{u}}+\frac{|\nabla {\overline{u}}|^{p}}{{\overline{u}} +\delta }=\delta ^{-p(p-1)}(-\Delta _{p}z^{{\bar{\omega }}}+\frac{|\nabla z^{ {\bar{\omega }}}|^{p}}{z^{{\bar{\omega }}}}) \\ \quad =\delta ^{-p(p-1)}{\bar{\omega }}^{p-1}\left[ 1+{\bar{\omega }}(1-\frac{(\bar{ \omega }-1)\left( p-1\right) }{{\bar{\omega }}})\frac{|\nabla z|^{p}}{z}\right] z^{({\bar{\omega }}-1)(p-1)} \\ \quad \ge \delta ^{-p(p-1)}{\bar{\omega }}^{p-1}\left\{ \begin{array}{ll} z^{({\bar{\omega }}-1)(p-1)} &{} \quad \text {in}\;\Omega \backslash {\overline{\Omega }} _{\delta }, \\ {\bar{\omega }}(1-\frac{({\bar{\omega }}-1)\left( p-1\right) }{{\bar{\omega }}})z^{( {\bar{\omega }}-1)(p-1)-1}|\nabla z|^{p} &{} \quad \text {in}\;\Omega _{\delta }\text {.} \end{array} \right. \end{array} \end{aligned}$$
(38)

Thus, after decreasing \(\delta \) if necessary, we achieve

$$\begin{aligned} \begin{array}{l} \delta ^{-p(p-1)}{\bar{\omega }}^{p-1}z^{({\bar{\omega }}-1)(p-1)} \\ \quad \ge \delta ^{-p(p-1)}{\bar{\omega }}^{p-1}(c^{-1}d(x))^{({\bar{\omega }}-1)(p-1)} \\ \quad \ge \delta ^{-p(p-1)}{\bar{\omega }}^{p-1}(c^{-1}\delta )^{({\bar{\omega }} -1)(p-1)} \\ \quad =\delta ^{({\bar{\omega }}-1-p)(p-1)}{\bar{\omega }}^{p-1}c^{-({\bar{\omega }} -1)(p-1)} \\ \quad \ge \delta ^{-p\max \{\alpha ,\beta \}}\text { \ in }\Omega \backslash {\overline{\Omega }}_{\delta }, \end{array} \end{aligned}$$
(39)

because of (34). Thus, (37)–(39) yield

$$\begin{aligned} -\Delta _{p}{\overline{u}}+\frac{|\nabla {\overline{u}}|^{p}}{{\overline{u}} +\delta }\ge f(\cdot ,{\overline{u}},\nabla {\overline{u}})\text {\ \ in\ } \Omega \backslash {\overline{\Omega }}_{\delta }\text {.} \end{aligned}$$

Let now \(x\in {\Omega }_{\delta }\). From (28) and (33), one can find a constant \({\bar{\mu }}>0\) such that

$$\begin{aligned} (1-\frac{({\bar{\omega }}-1)\left( p-1\right) }{{\bar{\omega }}})|\nabla z|>\bar{ \mu }\text { \ in }{\Omega }_{\delta }. \end{aligned}$$

Then, (28) and (34) entail

$$\begin{aligned} \begin{array}{l} \delta ^{-p(p-1)}{\bar{\omega }}^{p}(1-\frac{({\bar{\omega }}-1)\left( p-1\right) }{{\bar{\omega }}})z^{({\bar{\omega }}-1)(p-1)-1}|\nabla z{|}^{p} \\ \quad \ge \delta ^{-p(p-1)}{\bar{\omega }}^{p}(cd(x))^{({\bar{\omega }}-1)(p-1)-1}\bar{ \mu }^{p} \\ \quad \ge \delta ^{-p(p-1)}{\bar{\omega }}^{p}(c\delta )^{({\bar{\omega }}-1)(p-1)-1} {\bar{\mu }}^{p} \\ \quad \ge \delta ^{-p\max \{\alpha ,\beta \}}\text { \ in }\Omega _{\delta }\text {, } \end{array} \end{aligned}$$

for \(\delta >0\) small enough, that is

$$\begin{aligned} -\Delta _{p}{\overline{u}}+\frac{|\nabla {\overline{u}}|^{p}}{{\overline{u}} +\delta }\ge f(\cdot ,{\overline{u}},\nabla {\overline{u}})\text {\ \ in\ } \Omega _{\delta }\text {.} \end{aligned}$$

Summing up,

$$\begin{aligned} -\Delta _{p}{\overline{u}}+\frac{|\nabla {\overline{u}}|^{p}}{{\overline{u}} +\delta }\ge f(\cdot ,{\overline{u}},\nabla {\overline{u}})\text {\ on the whole\ }\Omega \text {.} \end{aligned}$$

Finally, test with \(\varphi \in W_{b}^{1,p}(\Omega ),\) \(\varphi \ge 0\) a.e. in \(\Omega \), and recall (36) to get

$$\begin{aligned} \begin{array}{l} \int _{\Omega }|\nabla {\overline{u}}|^{p-2}\nabla {\overline{u}}\nabla \varphi \, \textrm{d}x+\int _{\Omega }\frac{|\nabla {\overline{u}}|^{p}}{{\overline{u}} +\delta }\varphi \textrm{d}x-\left\langle \frac{\partial {\overline{u}}}{ \partial \eta _{p}},\gamma _{0}(\varphi )\right\rangle _{\partial \Omega } \\ \quad \ge \int _{\Omega }f(\cdot ,{\overline{u}},\nabla {\overline{u}})\varphi { \text {d}}x\text {,} \end{array} \end{aligned}$$

as desired. Here, \(\gamma _{0}\) is the trace operator on \(\partial \Omega \),

$$\begin{aligned} \frac{\partial w}{\partial \eta _{p}}:=|\nabla w|^{p-2}\frac{\partial w}{ \partial \eta },\quad \forall \,w\in W^{1,p}(\Omega )\cap C^{1}(\overline{ \Omega }), \end{aligned}$$
(40)

while \(\left\langle \cdot ,\cdot \right\rangle _{\partial \Omega }\) denotes the duality brackets for the pair

$$\begin{aligned} (W^{1/p^{\prime },p}(\partial \Omega ),W^{-1/p^{\prime },p^{\prime }}(\partial \Omega )). \end{aligned}$$

Next, we show that the function \({\underline{u}}\) in (32) satisfies ( 3). In \(\Omega \backslash {\overline{\Omega }}_{\delta }\), a direct computation gives

$$\begin{aligned} \begin{array}{l} -\Delta _{p}z_{\delta }^{\omega }+\frac{|\nabla z_{\delta }^{\omega }|^{p}}{ z_{\delta }^{\omega }}=\omega ^{p-1}\left( 1-(\omega -1)\left( p-1\right) \frac{|\nabla z_{\delta }|^{p}}{z_{\delta }}\right) z_{\delta }^{(\omega -1)(p-1)}+\omega ^{p}\frac{z_{\delta }^{(\omega -1)p}|\nabla z_{\delta }|^{p} }{z_{\delta }^{\omega }} \\ \quad =\omega ^{p-1}\left[ 1+\omega (1-\frac{(\omega -1)\left( p-1\right) }{\omega })\frac{|\nabla z_{\delta }|^{p}}{z_{\delta }}\right] z_{\delta }^{(\omega -1)(p-1)}, \end{array} \end{aligned}$$

while in \(\Omega _{\delta },\) we get

$$\begin{aligned} \begin{array}{l} -\Delta _{p}z_{\delta }^{\omega }+\frac{|\nabla z_{\delta }^{\omega }|^{p}}{ z_{\delta }^{\omega }}=\omega ^{p-1}\left[ -1+\omega (1-\frac{(\omega -1)\left( p-1\right) }{\omega })\frac{|\nabla z_{\delta }|^{p}}{z_{\delta }} \right] z_{\delta }^{(\omega -1)(p-1)}\text {.} \end{array} \end{aligned}$$

Thus, by (32) and due to (33), we have

$$\begin{aligned} -\Delta _{p}{\underline{u}}+\frac{|\nabla {\underline{u}}|^{p}}{{\underline{u}} +\delta }=\delta ^{\frac{1}{p^{\prime }}}(-\Delta _{p}z_{\delta }^{\omega }+ \frac{|\nabla z_{\delta }^{\omega }|^{p}}{z_{\delta }^{\omega }})\le \left\{ \begin{array}{ll} \delta ^{\frac{1}{p^{\prime }}}\omega ^{p-1}z_{\delta }^{(\omega -1)(p-1)} &{} \quad \text {in }\Omega \backslash {\overline{\Omega }}_{\delta } \\ 0 &{} \quad \text {in }\Omega _{\delta }. \end{array} \right. \end{aligned}$$

Hence, on account of (30), (27) and for an appropriate constant m in \((\mathrm {H.}4)\) chosen so that

$$\begin{aligned} m>\delta ^{\frac{1}{p^{\prime }}}\omega ^{p-1}L^{(\omega -1)(p-1)}\text { \ for }\delta >0\text { sufficiently small,} \end{aligned}$$

we get

$$\begin{aligned} -\Delta _{p}{\underline{u}}+\frac{|\nabla {\underline{u}}|^{p}}{{\underline{u}} +\delta }\le f(\cdot ,{\underline{u}},\nabla {\underline{u}}). \end{aligned}$$
(41)

Finally, test (41) with \(\varphi \in W_{b}^{1,p}(\Omega ),\) \(\varphi \ge 0\) a.e. in \(\Omega ,\) and recall (36), besides Green’s formula [10], to arrive at

$$\begin{aligned} \begin{array}{l} \int _{\Omega }|\nabla {\underline{u}}|^{p-2}\nabla {\underline{u}}\nabla \varphi \,\textrm{d}x+\int _{\Omega }\frac{|\nabla {\underline{u}}|^{p}}{{\underline{u}} +\delta }\varphi \,\textrm{d}x \\ \quad \le \int _{\Omega }|\nabla {\underline{u}}|^{p-2}\nabla {\underline{u}}\nabla \varphi \,\textrm{d}x-\left\langle \frac{\partial {\underline{u}}}{\partial \eta _{p}},\gamma _{0}(\varphi )\right\rangle _{\partial \Omega }+\int _{\Omega }\frac{|\nabla {\underline{u}}|^{p}}{{\underline{u}}+\delta } \varphi \,\textrm{d}x \\ \quad =\int _{\Omega }(-\Delta _{p}{\underline{u}}\ +\frac{|\nabla {\underline{u}}|^{p} }{{\underline{u}}+\delta })\varphi \,\textrm{d}x\le \int _{\Omega }f(\cdot , {\underline{u}},\nabla {\underline{u}})\varphi \,\textrm{d}x, \end{array} \end{aligned}$$

because \(\gamma _{0}(\varphi )\ge 0\) whatever \(\varphi \in W^{1,p}(\Omega )\), \(\varphi \ge 0\) a.e. in \(\Omega \) (see [11, p. 35]).

Therefore, \({\underline{u}}\) and \({\overline{u}}\) satisfy assumption \((\mathrm { H. }2)\), whence Theorem 7 can be applied, and there exists a solution \(u_{0}\in {\mathcal {C}}^{1,\tau }({\overline{\Omega }}),\) \(\tau \in ]0,1[,\) of problem \(\left( \textrm{P}\right) \) such that

$$\begin{aligned} {\underline{u}}\le u_{0}\le {\overline{u}}. \end{aligned}$$
(42)

Moreover, \(u_{0}\) is nodal. In fact, through (32) and (28) we obtain

$$\begin{aligned} {\overline{u}}=\delta ^{-p}z^{{\overline{\omega }}}-\delta \le \delta ^{-p}(cd(x))^{{\overline{\omega }}}-\delta , \end{aligned}$$

which actually means

$$\begin{aligned} {\overline{u}}(x)<0\;\;\text {provided}\;\;d(x)<c^{-1}\delta ^{\frac{p+1}{ {\overline{\omega }}}}. \end{aligned}$$
(43)

Again, by (32)–(28) together with (31), it follows that

$$\begin{aligned} {\underline{u}}=\delta ^{\frac{1}{p}}z_{\delta }^{\omega }-\delta \ge \delta ^{\frac{1}{p}}\left( \frac{d(x)}{2c}\right) ^{\omega }-\delta , \end{aligned}$$

which implies that

$$\begin{aligned} {\underline{u}}(x)>0\;\;\text {as soon as}\;\;d(x)>2c\delta ^{\frac{1}{\omega p^{\prime }}}\text {.} \end{aligned}$$
(44)

On account of (42)–(44), the conclusion follows.

On the other hand, on the basis of \((\mathrm {H.}4)\) and bearing in mind Theorem 7, there exists a nontrivial solution \(u_{+}\) of \(\left( \textrm{P}\right) \) such that \(u_{0}\le u_{+}\le {\overline{u}}_{+}\) and \( u_{+}\ge 0\) on \(\Omega \). In view of (32), \({\overline{u}}_{+}=0\) once \( d(x)\rightarrow 0\) and so \(u_{+}\) vanishes as \(d(x)\rightarrow 0\). This completes the proof. \(\square \)