Multiple Solutions with Sign Information for a Class of Parametric Superlinear (p, 2)-Equations

Abstract

We consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a p-Laplacian (with $p>2$) and a Laplacian (a two phase equation). The reaction consists of a parametric $(p-1)$-superlinear term and a $(p-1)$-sublinear perturbation. We show that for all $\lambda >0$ big, the problem has at least three nontrivial smooth solutions, all with sign information. Also we determine their asymptotic behaviour as the parameter $\lambda \rightarrow \infty$. When we strengthen the regularity of the perturbation term, we produce a second nodal solution, for a total of four solutions, all with sign information.

1 Introduction

Let $\Omega \subset \mathbb {R}^N$ be a bounded domain with a $C^2$-boundary $\partial \Omega$. In this paper we study the following parametric (p, 2)-equation (two-phase problem):

where $p^*$ is the critical Sobolev exponent corresponding to p, namely

$$\begin{aligned} p^*=\left\{ \begin{array}{lll} \frac{Np}{N-p} &{}\quad \text {if} &{} p<N,\\ +\infty &{}\quad \text {if} &{} p\geqslant N, \end{array} \right. \end{aligned}$$

and for every $q\in (1,\infty )$ by $\Delta _q$ we denote the q-Laplace differential operator defined by

$$\begin{aligned} \Delta _q u=\mathrm {div}\,(|Du|^{q-2}Du)\quad \forall u\in W^{1,q}_0(\Omega ). \end{aligned}$$

when $q=2$, we have the usual Laplace differential operator and so we write $\Delta _2=\Delta$. In our problem $(P_{\lambda })$ the differential operator is nonhomogeneous and this is a source of difficulties in its analysis. In the reaction we have two terms. One is parametric and $(p-1)$-superlinear (since $2<p<r$) with $\lambda >0$ being the parameter. The perturbation f(z, x) is a Carathéodory function (that is, for all $x\in \mathbb {R}$, $z\longmapsto f(z,x)$ is measurable and for a.a. $z\in \Omega$, $x\longmapsto f(z,x)$ is continuous) which is $(p-1)$-sublinear. Using variational tools from the critical point theory together with suitable truncation and comparison techniques and critical groups (Morse theory), we show that for all $\lambda >0$ big, problem $(P_{\lambda })$ has at least three nontrivial smooth solutions all with sign information (two of constant sign and the third nodal (sign changing)). If we strengthen the regularity of $f(z,\cdot )$, we prove the existence of a second nodal solution, for a total of four nontrivial smooth solutions, all with sign information.

We mention that (p, 2)-equations and more generally two phase problems arise in many mathematical models of physical phenomena. In this direction we mention the works of Zhikov [36, 37] on elasticity theory and of Cherfils-Il’yasov [4] on reaction-diffusion systems. Recently there have been some existence and multiplicity results for different classes of parametric (p, 2)-equations. We mention works of Chorfi-Rădulescu [5], Gasiński-Papageorgiou [9, 10, 12, 13, 16], Papageorgiou-Rădulescu [25], Papageorgiou-Rădulescu-Repovš [27], Papageorgiou-Scapellato [29, 30], Yang-Bai [35].

Finally such sensitivity analysis for parametric equations is also important in the study of optimization and control problems. It provides information about the tolerance of the systems on the variation of the parameter and in which range we expect to find optimal solutions (see Papageorgiou [22, 23] and Sokołowski [32]).

2 Mathematical Background

In the analysis of problem $(P_{\lambda })$ we will use the Sobolev space $W^{1,p}_0(\Omega )$ and the Banach space $C_0^1(\overline{\Omega })=\{u\in C^1(\overline{\Omega }):\ u|_{\partial \Omega }=0\}$. By $\Vert \cdot \Vert$ we will denote the norm of the Sobolev space $W^{1,p}_0(\Omega )$. On account of the Poincaré inequality, we have

$$\begin{aligned} \Vert u\Vert =\Vert Du\Vert _p\quad \forall u\in W^{1,p}_0(\Omega ). \end{aligned}$$

The Banach space $C^1_0(\overline{\Omega })$ is ordered with positive (order) cone

$$\begin{aligned} C_+=\{u\in C^1_0(\overline{\Omega }): u(z)\geqslant 0\ \text {for all}\ z\in \overline{\Omega }\}. \end{aligned}$$

This cone has a nonempty interior given by

$$\begin{aligned} \mathrm {int}\, C_+=\left\{ u\in C_+: u(z)> 0\quad \text {for all}\;\ z\in \Omega ,\ \frac{\partial u}{\partial n}|_{\partial \Omega }<0\right\} , \end{aligned}$$

with n being the outward unit normal vector on $\partial \Omega$. For $q\in (1,\infty )$, let $A_q:W^{1,q}_0(\Omega )\longrightarrow W^{-1,q'}(\Omega )=W^{1,q}_0(\Omega )^*$ ($\frac{1}{q}+\frac{1}{q'}=1$) be the nonlinear map defined by

$$\begin{aligned} \langle A_q(u),h\rangle =\int _{\Omega }|Du|^{q-2}(Du,Dh)_{\mathbb {R}^N}\,dz\quad \forall u,h\in W^{1,p}_0(\Omega ). \end{aligned}$$

From Gasiński-Papageorgiou [11, Problem 2.192], we have the following properties of $A_q$.

Proposition 2.1

The map $A_q$ is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (thus maximal monotone too) and of type $(S)_+$ (that is, if $u_n{\mathop {\longrightarrow }\limits ^{w}}u$ in $W^{1,q}_0(\Omega )$ and $\limsup \limits _{n\rightarrow +\infty }\langle A_q(u_n),u_n-u\rangle \leqslant 0$, then $u_n\longrightarrow u$ in $W^{1,q}_0(\Omega )$).

Note that for $q=2$, we have $A_2=A\in \mathcal {L}(H^1_0(\Omega );H^{-1}(\Omega ))$.

Let

$$\begin{aligned} p^*=\left\{ \begin{array}{lll} \frac{Np}{N-p} &{}\quad \text {if} &{} p<N,\\ +\infty &{}\quad \text {if} &{} N\leqslant p \end{array} \right. \end{aligned}$$

(the critical Sobolev exponent corresponding to p) and let $f_0:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}$ be a Carathéodory function such that

$$\begin{aligned} |f_0(z,x)|\leqslant a_0(z)(1+|x|^{q-1})\quad \text {for a.a.}\ z\in \Omega ,\ \text {all}\ x\in \mathbb {R}, \end{aligned}$$

with $a_0\in L^{\infty }(\Omega )_+$ and $1<q\leqslant p^*$. We set

$$\begin{aligned} F_0(z,x)=\int _0^x f_0(z,s)\,ds \end{aligned}$$

and consider the $C^1$-functional $\varphi _0:W^{1,p}_0(\Omega )\longrightarrow \mathbb {R}$ defined by

$$\begin{aligned} \varphi _0(u)=\frac{1}{p}\Vert Du\Vert _p^p+\frac{1}{2}\Vert Du\Vert _2^2-\int _{\Omega }F_0(z,u)\,dz\quad \forall u\in W^{1,p}_0(\Omega ). \end{aligned}$$

The next proposition is a particular case of a more general result proved by Gasiński-Papageorgiou [8] (subcritical case) and Papageorgiou-Rădulescu [26] (critical case). The result is an outgrowth of the nonlinear regularity theory of Lieberman [19, 20]. Related regularity results can be found in the more recent works of Ragusa–Tachikawa [33, 34].

Proposition 2.2

If $u_0\in W^{1,p}_0(\Omega )$ is a local $C^1_0(\overline{\Omega })$-minimizer of $\varphi _0$, that is, there exists $\varrho _0>0$ such that

$$\begin{aligned} \varphi _0(u_0)\leqslant \varphi _0(u_0+h)\quad \forall h\in C^1_0(\overline{\Omega }),\quad \Vert h\Vert _{C^1_0(\overline{\Omega })}<\varrho _0, \end{aligned}$$

then $u_0\in C^{1,\alpha }_0(\overline{\Omega })$ for some $\alpha \in (0,1)$ and it is a local $W^{1,p}_0(\Omega )$-minimizer of $\varphi _0$, that is, there exists $\varrho _1>0$ such that

$$\begin{aligned} \varphi _0(u_0)\leqslant \varphi _0(u_0+h)\quad \forall h\in W^{1,p}_0(\Omega ),\quad \Vert h\Vert <\varrho _1. \end{aligned}$$

As we already mentioned in the Introduction our methods involve comparison arguments. In this direction, useful will be the following strong comparison principle, which is a special case of a more general result due to Gasiński-Papageorgiou [14, Proposition 3.2]. First we introduce the following notation. Given $h_1,h_2\in L^{\infty }(\Omega )$, we write $h_1\preceq h_2$ if for every $K\subseteq \Omega$ compact, we can find $\varepsilon =\varepsilon (K)>0$ such that

$$\begin{aligned} h_1(z)+\varepsilon \leqslant h_2(z)\quad \text {for a.a.}\ z\in K. \end{aligned}$$

If $h_1,h_2\in C(\Omega )$ and $h_1(z)<h_2(z)$ for all $z\in \Omega$, then $h_1\preceq h_2$.

Proposition 2.3

If $\widehat{\xi }\geqslant 0$, $h_1,h_2\in L^{\infty }(\Omega )$, $h_1\preceq h_2$ and $u\in C^1_0(\overline{\Omega })$, $v\in \mathrm {int}\, C_+$ satisfy

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta _p u-\Delta u+\widehat{\xi }|u|^{p-2}u=h_1\ \text {in}\ \Omega ,\\ -\Delta _p v-\Delta v+\widehat{\xi }v^{p-1}=h_2\ \text {in}\ \Omega , \end{array} \right. \end{aligned}$$

then $v-u\in \mathrm {int}\, C_+$.

Next let us recall some basic facts about the spectrum of $(-\Delta ,H^1_0(\Omega ))$ which we will need in the sequel. We know that the spectrum $\widehat{\sigma }(2)$ consists of a sequence $\{\widehat{\lambda }_k(2)\}_{k\geqslant 1}$ of distinct eigenvalues such that $\widehat{\lambda }_k(2)\rightarrow +\infty$ as $k\rightarrow +\infty$. Also for every $k\in \mathbb {N}$, by $E(\widehat{\lambda }_k(2))$ we denote the corresponding eigenspace. Standard regularity theory implies that

$$\begin{aligned} E(\widehat{\lambda }_k(2))\subseteq C^1_0(\overline{\Omega })\quad \forall k\in \mathbb {N}. \end{aligned}$$

We know that $\widehat{\lambda }_1(2)>0$ and it is simple, that is, $\dim E(\widehat{\lambda }_1(2))=1$. Also we have the following variational characterization for $\widehat{\lambda }_1(2)>0$:

$$\begin{aligned} \widehat{\lambda }_1(2)=\inf \bigg \{\frac{\Vert Du\Vert _2^2}{\Vert u\Vert _2^2}:\ u\in H^1_0(\Omega ),\ u\ne 0\bigg \}. \end{aligned}$$

This infimum is realized on $E(\widehat{\lambda }_1(2))$ and from this expression it is easy to see that the element of $E(\widehat{\lambda }_1(2))\subseteq C^1_0(\overline{\Omega })$ do not change sign. Indeed note that in the above expression we can replace u by |u| (see also Gasiński-Papageorgiou [7, Theorem 6.1.21, p. 716]). By $\widehat{u}_1(2)$ we denote the positive, $L^2$-normalized (that is, $\Vert \widehat{u}_1(2)\Vert _2=1$) eigenfunction corresponding to $\widehat{\lambda }_1(2)>0$. The strong maximum principle implies that $\widehat{u}_1(2)\in \mathrm {int}\, C_+$. Note that all the other eigenvalues have nodal eigenfunctions. These properties lead to the following simple lemma (see Gasiński-Papageorgiou [11, Problem 5.67]).

Lemma 2.4

If $\vartheta _0\in L^{\infty }(\Omega )$, $\vartheta _0(z)\leqslant \widehat{\lambda }_1(2)$ for a.a. $z\in \Omega$, $\vartheta _0\not \equiv \widehat{\lambda }_1(2)$, then there exists $c_0>0$ such that

$$\begin{aligned} c_0\Vert u\Vert ^2\leqslant \Vert Du\Vert _2^2-\int _{\Omega }\vartheta _0(z)u^2\,dz\quad \forall u\in H^1_0(\Omega ). \end{aligned}$$

We will also consider a weighted eigenvalue problem for $(-\Delta ,H^1_0(\Omega ))$. So, let $\vartheta \in L^{\infty }(\Omega )$, $0\leqslant \vartheta (z)$ for a.a. $z\in \Omega$, $\vartheta \not \equiv 0$. We consider the following linear eigenvalue problem

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta y(z)=\lambda \vartheta (z)y(z)\ \text {in}\ \Omega ,\\ y|_{\partial \Omega }=0. \end{array} \right. \end{aligned}$$

The spectrum of this problem is a sequence of distinct eigenvalues $\{\widetilde{\lambda }_k(2,\vartheta )\}_{k\geqslant 1}$ which have the same properties as the sequence $\{\widehat{\lambda }_k(2)=\widetilde{\lambda }_k(2,1)\}_{k\geqslant 1}$. In particular $\widetilde{\lambda }_1(2,\vartheta )>0$, it is simple and has eigenfunctions in $C^1_0(\overline{\Omega })$ of constant sign. All other eigenvalues have nodal eigenfunctions. These properties lead to the following monotonicity property for the map $\vartheta \longmapsto \widetilde{\lambda }_1(2,\vartheta )$ (see Motreanu-Motreanu-Papageorgiou [21, Proposition 9.47]).

Lemma 2.5

If $\vartheta _1,\vartheta _2\in L^{\infty }(\Omega )$, $0\leqslant \vartheta _1(z)\leqslant \vartheta _2(z)$ for a.a. $z\in \Omega$, $\vartheta _1\not \equiv 0$, $\vartheta _1\not \equiv \vartheta _2$, then $\widetilde{\lambda }_1(2,\vartheta _2)<\widetilde{\lambda }_1(2,\vartheta _1)$.

Next let us recall some basic definitions and facts concerning critical groups which we will be used in our proofs.

Let X be a Banach space, $\varphi \in C^1(X;\mathbb {R})$ and $c\in \mathbb {R}$. We introduce the following sets

$$\begin{aligned} K_{\varphi }= & {} \{u\in X:\ \varphi '(u)=0\}\ \text {(the critical set of }\varphi \text {)},\\ K_{\varphi }^c= & {} \{u\in K_{\varphi }:\ \varphi (u)=c\},\\ \varphi ^c= & {} \{u\in X:\ \varphi (u)\leqslant c\}. \end{aligned}$$

Let $(Y_1,Y_2)$ be a topological pair such that $Y_2\subseteq Y_1\subseteq X$. For every $k\in \mathbb {N}_0$ by $H_k(Y_1,Y_2)$ we denote the k-th relative singular homology group with integer coefficients. Suppose that $u\in K_{\varphi }$ is isolated and $\varphi (u)=c$ (that is, $u\in K_{\varphi }^c$). The critical groups of $\varphi$ at u are defined by

$$\begin{aligned} C_k(\varphi ,u)=H_k(\varphi ^c\cap U,\varphi ^c\cap U\setminus \{u\})\quad \forall k\in \mathbb {N}_0. \end{aligned}$$

Here U is a neighbourhood of u such that $K_{\varphi }\cap \varphi ^c\cap U=\{u\}$. The excision property of singular homology, implies that the above definition is independent of the particular choice of the neighbourhood U.

Suppose that $\varphi \in C^1(X;\mathbb {R})$ satisfies the Palais-Smale condition (the PS-condition for short; see Gasiński-Papageorgiou [7, Definition 5.1.5]) and that $\inf \varphi (K_{\varphi })>-\infty$. Let $c<\inf \varphi (K_{\varphi })$. Then the critical groups of $\varphi$ at infinity are defined by

$$\begin{aligned} C_k(\varphi ,\infty )=H_k(X,\varphi ^c)\quad \forall k\in \mathbb {N}_0. \end{aligned}$$

The definition is independent of the choice of the level $c<\inf \varphi (K_{\varphi })$. Indeed, let $c'<c<\inf \varphi (K_{\varphi })$. From Corollary 5.3.13 of Papageorgiou-Rădulescu-Repovš [28], we have that $\varphi ^{c'}$ is a strong deformation retract of $\varphi ^c$. Then Corollary 6.1.24 of [28] implies that

$$\begin{aligned} H_k(X,\varphi ^c)=H_k(X,\varphi ^{c'})\quad \forall k\in \mathbb {N}_0. \end{aligned}$$

Suppose that $K_{\varphi }$ is finite. We introduce the following quantities:

$$\begin{aligned} M(t,u)= & {} \sum _{k\geqslant 0}\mathrm {rank}\, C_k(\varphi ,u)t^k\quad \forall t\in \mathbb {R},\ u\in K_{\varphi },\\ P(t,\infty )= & {} \sum _{k\geqslant 0}\mathrm {rank}\, C_k(\varphi ,\infty )t^k\quad \forall t\in \mathbb {R}. \end{aligned}$$

The Morse relation says that

$$\begin{aligned} \sum _{u\in K_{\varphi }}M(t,u)=P(t,\infty )+(1+t)Q(t), \end{aligned}$$

(2.1)

where $Q(t)=\sum \limits _{k\geqslant 0}\beta _k t^k$ is a formal series in $t\in \mathbb {R}$ with nonnegative coefficients.

Finally we fix our notation. For $x\in \mathbb {R}$, we let $x^{\pm }=\max \{\pm x,0\}$ and for $u\in W^{1,p}_0(\Omega )$ we define $u^{\pm }(z)=u(z)^{\pm }$ for all $z\in \Omega$. We know that

$$\begin{aligned} u^{\pm }\in W^{1,p}_0(\Omega ),\ u=u^+-u^-,\quad |u|=u^++u^-. \end{aligned}$$

Also, given a measurable function $g:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}$ (for example a Carathéodory function), we set

$$\begin{aligned} N_g(u)(\cdot )=g(\cdot ,u(\cdot ))\quad \forall u\in W^{1,p}_0(\Omega ) \end{aligned}$$

(the Nemytski map corresponding to g). By $\delta _{ki}$ we denote the Kronecker symbol defined by

$$\begin{aligned} \delta _{ki}=\left\{ \begin{array}{lll} 1 &{}\quad \text {if} &{} k=i,\\ 0 &{}\quad \text {if} &{} k\ne i. \end{array} \right. \end{aligned}$$

Finally, if $u,v\in W^{1,p}_0(\Omega )$, $v\leqslant u$, then we define

$$\begin{aligned}{}[v,u]=\{y\in W^{1,p}_0(\Omega ):\ v(z)\leqslant y(z)\leqslant u(z)\quad \text {for a.a.}\ z\in \Omega \}. \end{aligned}$$

Also by $\mathrm {int}_{C^1_0(\overline{\Omega })}[v,u]$ we define the interior in the $C^1_0(\overline{\Omega })$-norm topology of $[v,u]\cap C^1_0(\overline{\Omega })$.

3 Three Solutions with Sign Information

In this section without assuming any differentiability properties of $f(z,\cdot )$ we show that for all $\lambda >0$ big, problem $(P_{\lambda })$ has at least three nontrivial smooth solutions all with sign information.

The assumptions on the perturbation term f(z, x) are the following:

$\underline{H(f)_1}$ $f:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}$ is a Carathéodory function such that $f(z,0)=0$ for a.a. $z\in \Omega$ and

(i):

there exist functions $\widehat{\vartheta }_0,\vartheta _0\in L^{\infty }(\Omega )$ such that

$$\begin{aligned}&0\leqslant \widehat{\vartheta }_0(z)\leqslant \vartheta _0(z)\leqslant \widehat{\lambda }_1(2)\quad \text {for a.a.}\ z\in \Omega ,\ \widehat{\vartheta }_0\not \equiv 0,\ \vartheta _0\not \equiv \widehat{\lambda }_1(2),\\&\widehat{\vartheta }_0 (z)\leqslant \liminf _{x\rightarrow 0}\frac{f(z,x)}{x}\leqslant \limsup _{x\rightarrow 0}\frac{f(z,x)}{x}\leqslant \vartheta _0(z) \quad \text {uniformly for a.a.}\ z\in \Omega . \end{aligned}$$

(ii):

$\displaystyle \lim _{x\rightarrow \pm \infty }\frac{f(z,x)}{|x|^{p-2}x}=0$ uniformly for a.a. $z\in \Omega$;

(iii):

$f(z,x)x\geqslant 0$ for a.a. $z\in \Omega$, all $x\in \mathbb {R}$.

Evidently the function $f(z,x)=\vartheta (z)x$ with $\vartheta \in L^{\infty }(\Omega )$, $0\leqslant \vartheta (z)\leqslant \widehat{\lambda }_1(2)$, $\vartheta \not \equiv 0$, $\vartheta \not \equiv \widehat{\lambda }_1(2)$ satisfies hypotheses $H(f)_1$.

We let $F(z,x)=\int _0^x f(z,s)\,ds$.

Proposition 3.1

If hypotheses $H(f)_1$ hold, then for all $\lambda >0$ big, problem $(P_{\lambda })$ has at least two constant sign solutions $u_{\lambda }\in \mathrm {int}\, C_+$ and $v_{\lambda }\in -\mathrm {int}\, C_+$.

Proof

First we produce the positive solution.

Let $\varphi _{\lambda }^+:W^{1,p}_0(\Omega )\longrightarrow \mathbb {R}$ be the $C^1$-functional defined by

$$\begin{aligned} \varphi _{\lambda }^+(u)=\frac{1}{p}\Vert Du\Vert _p^p+\frac{1}{2}\Vert Du\Vert _2^2-\frac{\lambda }{r}\Vert u^+\Vert _r^r-\int _{\Omega }F(z,u^+)\,dz\quad \forall u\in W^{1,p}_0(\Omega ). \end{aligned}$$

On account of hypotheses $H(f)_1(i),(ii)$, given $\varepsilon >0$, we can find $c_1=c_1(\varepsilon )>0$ such that

$$\begin{aligned} F(z,x)\leqslant \frac{1}{2}(\vartheta _0(z)+\varepsilon )x^2+c_1|x|^r. \end{aligned}$$

(3.1)

Assuming that $\lambda \geqslant 1$, using (3.1), Lemma 2.4, for all $u\in W^{1,p}_0(\Omega )$, we have

$$\begin{aligned} \varphi _{\lambda }^+(u)\geqslant & {} \frac{1}{p}\Vert u\Vert ^p+\frac{1}{2}\bigg (\Vert Du\Vert _2^2-\int _{\Omega }\vartheta _0(z)u^2\,dz-\varepsilon \Vert u\Vert _{H^1_0(\Omega )}^2\bigg )-\lambda c_2\Vert u\Vert ^r\nonumber \\\geqslant & {} c_3\Vert u\Vert ^p-\lambda c_2\Vert u\Vert ^r = (c_3-\lambda c_2\Vert u\Vert ^{r-p})\Vert u\Vert ^p \end{aligned}$$

(3.2)

for some $c_2,c_3>0$ (by choosing $\varepsilon >0$ small). So, if $\varrho _{\lambda }\in (0,\frac{c_3}{\lambda c_2})$, then for $\Vert u\Vert =\varrho _{\lambda }$ we have

$$\begin{aligned} \varphi _{\lambda }^+(u)\geqslant m_{\lambda }^+>0\quad \forall \Vert u\Vert =\varrho _{\lambda }, \end{aligned}$$

(3.3)

with $\varrho _{\lambda }\rightarrow 0^+$ as $\lambda \rightarrow \infty$. Let $t\in (0,1)$ and $\overline{u}_0\in \mathrm {int}\, C_+$. We have

$$\begin{aligned} \varphi _{\lambda }^+(t\overline{u}_0) \leqslant \frac{t^p}{p}\Vert D\overline{u}_0\Vert _p^p+\frac{t^2}{2}\Vert D\overline{u}_0\Vert _2^2-\lambda \frac{t^r}{r}\Vert \overline{u}_0\Vert _r^r \leqslant c_4 t^2-\lambda c_5 t^r \end{aligned}$$

(3.4)

for some $c_4,c_5>0$ (see hypothesis $H(f)_1(iii)$ and recall that $t\in (0,1)$, $2<p$).

For fixed $t\in (0,1)$, from (3.3) we see that we can find $\widetilde{\lambda }\geqslant 1$ such that

$$\begin{aligned} \varphi _{\lambda }^+(t\overline{u}_0)<0\quad \forall \lambda \geqslant \widetilde{\lambda }, \end{aligned}$$

(3.5)

and

$$\begin{aligned} \Vert t\widetilde{u}_0\Vert >\varrho _{\lambda } \end{aligned}$$

(3.6)

(recall that $\varrho _{\lambda }\rightarrow 0^+$ as $\lambda \rightarrow \infty$).

Hypothesis $H(f)_1(ii)$ implies that given $\varepsilon >0$, we can find $M=M(\varepsilon )\geqslant 1$ such that

$$\begin{aligned} F(z,x)\leqslant \frac{\varepsilon }{p}|x|^p\leqslant \frac{\varepsilon }{p}|x|^r\quad \text {for a.a.}\ z\in \Omega ,\ \text {all}\ |x|\geqslant M\geqslant 1. \end{aligned}$$

(3.7)

We consider the Carathéodory function

$$\begin{aligned} k_{\lambda }(z,x)=\lambda |x|^{r-2}x+f(z,x). \end{aligned}$$

We set $K_{\lambda }(z,x)=\int _0^x k_{\lambda }(z,s)\,ds$ and let $q\in (p,r)$. We have

$$\begin{aligned} q K_{\lambda }(z,x)\leqslant \frac{\lambda q}{r}|x|^r+\frac{\varepsilon q}{r}|x|^r\quad \text {for a.a.}\ z\in \Omega ,\ \text {all}\ |x|\geqslant M \end{aligned}$$

(3.8)

(see (3.7)). Also using hypothesis $H(f)_1(iii)$ we have

$$\begin{aligned} k_{\lambda }(z,x)x\geqslant \lambda |x|^r\quad \text {for a.a.}\ z\in \Omega ,\ \text {all}\ x\in \mathbb {R}. \end{aligned}$$

(3.9)

From (3.8) and (3.9), we see that by choosing $\varepsilon \in (0,\lambda (r-q))$, we have

$$\begin{aligned} 0<q K_{\lambda }(z,x)\leqslant k_{\lambda }(z,x)x\quad \text {for a.a.}\ z\in \Omega ,\ \text {all}\ |x|\geqslant M. \end{aligned}$$

(3.10)

Using (3.10) (essentially the Ambrosetti–Rabinowitz condition; see Motreanu-Motreanu-Papageorgiou [21]), we can easily check that

$$\begin{aligned} \varphi _{\lambda }^+\ \text {satisfies the Palais-Smale condition}. \end{aligned}$$

(3.11)

Then (3.3), (3.5), (3.6) and (3.11) permit the use of the mountain pass theorem on the functional $\varphi _{\lambda }^+$ for all $\lambda \geqslant \widetilde{\lambda }$. So, we can find $u_{\lambda }\in W^{1,p}_0(\Omega )$ such that

$$\begin{aligned} u_{\lambda }\in K_{\varphi _{\lambda }^+}\quad \text {and}\quad \varphi _{\lambda }^+(0)=0< m_{\lambda }^+\leqslant \varphi _{\lambda }^+(u_{\lambda }) \end{aligned}$$

(3.12)

(see (3.3)). From (3.12) it follows that $u_{\lambda }\ne 0$ and

$$\begin{aligned} (\varphi _{\lambda }^+)'(u_{\lambda })=0, \end{aligned}$$

so

$$\begin{aligned} \langle A_p(u_{\lambda }),h\rangle +\langle A(u_{\lambda }),h\rangle =\int _{\Omega }\big (\lambda (u_{\lambda }^+)^{p-1}+f(z,u_{\lambda }^+)\big )h\,dz \quad \forall h\in W^{1,p}_0(\Omega ). \end{aligned}$$

(3.13)

In (3.13) we choose $h=-u_{\lambda }^-\in W^{1,p}_0(\Omega )$. We have

$$\begin{aligned} \Vert Du_{\lambda }^-\Vert _p\leqslant 0, \end{aligned}$$

so

$$\begin{aligned} u_{\lambda }\geqslant 0,\quad u_{\lambda }\ne 0. \end{aligned}$$

From (3.13) we have

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta _p u_{\lambda }(z)-\Delta u_{\lambda }(z)=\lambda u_{\lambda }(z)^{r-1}+f(z,u_{\lambda }(z))\ \text {for a.a.}\ z\in \Omega ,\\ u_{\lambda }|_{\partial \Omega }=0. \end{array} \right. \end{aligned}$$

(3.14)

From (3.14) and Theorem 7.1 of Ladyzhenskaya-Ural’tseva [18, p. 286], we have that $u_{\lambda }\in L^{\infty }(\Omega )$. Then applying Theorem 1 of Lieberman [19], we infer that

$$\begin{aligned} u_{\lambda }\in C_+\setminus \{0\}. \end{aligned}$$

From (3.14) and hypothesis $H(f)_1(iii)$, we have

$$\begin{aligned} \Delta _p u_{\lambda }(z)+\Delta u_{\lambda }(z)\leqslant 0\quad \text {for a.a.}\ z\in \Omega , \end{aligned}$$

so $u_{\lambda }\in \mathrm {int}\, C_+$ (see Pucci-Serrin [31, pp. 111, 120]).

For the negative solution, we consider the $C^1$-functional $\varphi _{\lambda }^-:W^{1,p}_0(\Omega )\longrightarrow \mathbb {R}$ defined by

$$\begin{aligned} \varphi _{\lambda }^-(u)=\frac{1}{p}\Vert Du\Vert _p^p+\frac{1}{2}\Vert Du\Vert _2^2-\frac{\lambda }{r}\Vert u^-\Vert _p^p-\int _{\Omega }F(z,-u^-)\,dz \quad \forall u\in W^{1,p}_0(\Omega ). \end{aligned}$$

Reasoning as above, using this time the functional $\varphi _{\lambda }^-$, we produce a negative solution $v_{\lambda }\in -\mathrm {int}\, C_+$ for all $\lambda \geqslant \widetilde{\lambda }$ (increasing $\widetilde{\lambda }\geqslant 1$ if necessary). $\square$

The next result determines the asymptotic behaviour of the two constant sign solutions as $\lambda \rightarrow \infty$.

Proposition 3.2

If hypotheses $H(f)_1$ hold, then $u_{\lambda },v_{\lambda }\rightarrow 0$ in $C^1_0(\overline{\Omega })$ as $\lambda \rightarrow \infty$.

Proof

Recall that $u_{\lambda }\in \mathrm {int}\, C_+$ is a critical point of $\varphi _{\lambda }^+$ of mountain pass type (see the proof of Proposition 3.1). So, we have

$$\begin{aligned} \varphi _{\lambda }^+(u_{\lambda })\leqslant & {} \max _{0\leqslant s\leqslant 1}\varphi _{\lambda }^+(st\overline{u}_0)\\\leqslant & {} \max _{0\leqslant s\leqslant 1}\bigg ( \frac{s^p}{p}\Vert D(t\overline{u}_0)\Vert _p^p+\frac{s^2}{2}\Vert D(t\overline{u}_0)\Vert _2^2-\frac{\lambda s^r}{r}\Vert t\overline{u}_0\Vert _r^r \bigg )\\\leqslant & {} \max _{0\leqslant s\leqslant 1}\bigg ( \frac{s^2}{2}\big (\Vert D(t\overline{u}_0)\Vert _p^p+\Vert D(t\overline{u}_0)\Vert _2^2\big )-\frac{\lambda s^r}{r}\Vert t\overline{u}_0\Vert _r^r \bigg )\\= & {} \max _{0\leqslant s\leqslant 1}(c_6 s^2-\lambda c_7 s^r) = c_6\bigg (\frac{2c_6}{\lambda c_7 r}\bigg )^{\frac{2}{r-2}}-\lambda c_7\bigg (\frac{2c_6}{\lambda c_7 r}\bigg )^{\frac{r}{r-2}}\\= & {} \bigg (\frac{2c_6}{\lambda c_7 r}\bigg )^{\frac{2}{r-2}} c_6\frac{r-2}{r} = \frac{c_8}{\lambda ^{\frac{q}{r-2}}}, \end{aligned}$$

(3.15)

with $c_6=\frac{1}{2}(\Vert D(t\overline{u}_0)\Vert _p^p+\Vert D(t\overline{u}_0)\Vert _2^2)>0$, $c_7=\frac{1}{r}\Vert t\overline{u}_0\Vert _r^r>0$ and some $c_8>0$ (see hypothesis $H(f)_1(iii)$ and recall that $s\in [0,1]$, $2<p$).

We have

$$\begin{aligned} q\varphi _{\lambda }^+(u_{\lambda })=\frac{q}{p}\Vert u_{\lambda }\Vert ^p+\frac{q}{2}\Vert D u_{\lambda }\Vert _2^2-\int _{\Omega }q K_{\lambda }(z,u_{\lambda })\,dz \end{aligned}$$

(3.16)

and

$$\begin{aligned} 0=-\langle (\varphi _{\lambda }^+)(u_{\lambda }),u_{\lambda }\rangle =-\Vert u_{\lambda }\Vert ^p-\Vert D u_{\lambda }\Vert _2^2 +\int _{\Omega }k_{\lambda }(z,u_{\lambda })u_{\lambda }\,dz. \end{aligned}$$

(3.17)

We add (3.16) and (3.17) and use (3.15). Then

$$\begin{aligned} \bigg (\frac{q}{p}-1\bigg )\Vert u_{\lambda }\Vert ^p +\int _{\Omega }(k_{\lambda }(z,u_{\lambda })u_{\lambda }-q K_{\lambda }(z,u_{\lambda }))\,dz \leqslant \frac{q c_8}{\lambda ^{\frac{q}{r-2}}} \end{aligned}$$

(since $2<q$), so

$$\begin{aligned} \frac{q-p}{p}\Vert u_{\lambda }\Vert ^p\leqslant \frac{c_8}{\lambda ^{\frac{q}{r-2}}}+c_9, \end{aligned}$$

for some $c_9>0$ (see (3.10)), thus

$$\begin{aligned} \text {the sequence }\{u_{\lambda }\}_{\lambda \geqslant \widetilde{\lambda }}\subseteq W^{1,p}_0(\Omega )\text { is bounded.} \end{aligned}$$

(3.18)

Since $u_{\lambda }\in \mathrm {int}\, C_+$ is a solution of $(P_{\lambda })$, we have

$$\begin{aligned} \Vert u_{\lambda }\Vert ^p+\Vert Du_{\lambda }\Vert _2^2=\lambda \Vert u_{\lambda }\Vert _r^r+\int _{\Omega }f(z,u_{\lambda })u_{\lambda }\,dz, \end{aligned}$$

so

$$\begin{aligned} \lambda \Vert u_{\lambda }\Vert _r^r\leqslant c_{10}\quad \forall \lambda \geqslant \widetilde{\lambda } \end{aligned}$$

for some $c_{10}>0$ (see hypothesis $H(f)_1(iii)$ and (3.18)), thus

$$\begin{aligned} u_{\lambda }\longrightarrow 0\quad \text {in}\ L^r(\Omega )\ \text {as}\ \lambda \rightarrow 0^+. \end{aligned}$$

(3.19)

We know that

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta _p u_{\lambda }(z)-\Delta u_{\lambda }(z)=\lambda u_{\lambda }(z)^{r-1}+f(z,u_{\lambda }(z))\ \text {for a.a.}\ z\in \Omega ,\\ u_{\lambda }|_{\partial \Omega }=0,\ \lambda \geqslant \widetilde{\lambda }. \end{array} \right. \end{aligned}$$

(3.20)

From (3.18), (3.20) and Theorem 7.1 of Ladyzhenskaya-Ural’tseva [18, p. 286], we see that we can find $c_{11}>0$ such that

$$\begin{aligned} \Vert u_{\lambda }\Vert _{\infty }\leqslant c_{11}\quad \forall \lambda \geqslant \widetilde{\lambda }. \end{aligned}$$

Invoking Theorem 1 Lieberman [19], we infer that there exist $\alpha \in (0,1)$ and $c_{12}>0$ such that

$$\begin{aligned} u_{\lambda }\in C^{1,\alpha }_0(\overline{\Omega }),\quad \Vert u_{\lambda }\Vert _{C^{1,\lambda }_0(\overline{\Omega })}\leqslant c_{12}\quad \forall \lambda \geqslant \widetilde{\lambda }. \end{aligned}$$

(3.21)

From (3.21), the compactness of the embedding $C^{1,\alpha }_0(\overline{\Omega })\subseteq C^1_0(\overline{\Omega })$ and (3.19), we conclude that

$$\begin{aligned} u_{\lambda }\longrightarrow 0\quad \text {in}\ C^1_0(\overline{\Omega })\quad \text {as}\ \lambda \rightarrow +\infty . \end{aligned}$$

In a similar fashion, working this time with $\varphi _{\lambda }^-$, we show that

$$\begin{aligned} v_{\lambda }\longrightarrow 0\quad \text {in}\ C^1_0(\overline{\Omega })\quad \text {as}\ \lambda \rightarrow +\infty . \end{aligned}$$

$\square$

Next we will show that for all $\lambda \geqslant \widetilde{\lambda }$ problem $(P_{\lambda })$ has extremal constant sign solutions, that is, there is a smallest positive solution and a biggest negative solution.

To this end, we introduce the following two sets

$$\begin{aligned}&S_{\lambda }^+ \quad \text {- set of positive solutions for }(P_{\lambda }),\\&S_{\lambda }^- \quad \text {- set of negative solutions for }(P_{\lambda }). \end{aligned}$$

We know (see Proposition 3.1) that

$$\begin{aligned} \emptyset \ne S_{\lambda }^+\subseteq \mathrm {int}\, C_+\quad \text {and}\quad \emptyset \ne S_{\lambda }^-\subseteq -\mathrm {int}\, C_+\quad \forall \lambda \geqslant \widetilde{\lambda }. \end{aligned}$$

Proposition 3.3

If hypotheses $H(f)_1$ hold, then for all $\lambda >0$ big, problem $(P_{\lambda })$ has

$\bullet$ a smallest positive solution $u_{\lambda }^*\in \mathrm {int}\, C_+$;

$\bullet$ a biggest negative solution $v_{\lambda }^*\in -\mathrm {int}\, C_+$.

Proof

From Filippakis-Papageorgiou [6], we know that the set $S_{\lambda }^+$ is downward directed (that is, if $u_1,u_2\in S_{\lambda }^+$, then there exists $u\in S_{\lambda }^+$ such that $u\leqslant u_1$, $u\leqslant u_2$). Then invoking Lemma 3.10 of Hu-Papageorgiou [17, p. 178], we can find a decreasing sequence $\{u_n\}_{n\geqslant 1}\subseteq S_{\lambda }^+$ such that

$$\begin{aligned} \inf _{n\geqslant 1}u_n=\inf S_{\lambda }^+,\quad 0\leqslant u_n\leqslant u_1\quad \forall n\in \mathbb {N}. \end{aligned}$$

(3.22)

We have

$$\begin{aligned}&\langle A_p(u_n),h\rangle +\langle A(u_n),h\rangle =\lambda \int _{\Omega }u_n^{r-1}h\,dz+\int _{\Omega }f(z,u_n)h\,dz\\&\quad \forall h\in W^{1,p}_0(\Omega ),\ n\in \mathbb {N}. \end{aligned}$$

(3.23)

Choosing $h=u_n\in W^{1,p}_0(\Omega )$ and using (3.22), we infer that the sequence $\{u_n\}_{n\geqslant 1}\subseteq W^{1,p}_0(\Omega )$ is bounded. So, by passing to a suitable subsequence if necessary, we may assume that

$$\begin{aligned} u_n{\mathop {\longrightarrow }\limits ^{w}}u_{\lambda }^*\quad \text {in}\ W^{1,p}_0(\Omega )\quad \text {and}\quad u_n\longrightarrow u_{\lambda }^*\quad \text {in}\ L^r(\Omega ). \end{aligned}$$

(3.24)

In (3.23) we choose $h=u_n-u_{\lambda }^*\in W^{1,p}_0(\Omega )$, pass to the limit as $n\rightarrow \infty$ and use (3.24). We obtain

$$\begin{aligned} \lim _{n\rightarrow +\infty }\big (\big \langle A_p(u_n),u_n-u_{\lambda }^*\big \rangle +\big \langle A(u_n),u_n-u_{\lambda }^*\big \rangle \big )=0, \end{aligned}$$

so

$$\begin{aligned} \limsup \limits _{n\rightarrow +\infty }\big (\big \langle A_p(u_n),u_n-u_{\lambda }^*\big \rangle +\big \langle A(u_{\lambda }^*),u_n-u_{\lambda }^*\big \rangle \big )\leqslant 0 \end{aligned}$$

(from the monotonicity of A), thus

$$\begin{aligned} \limsup \limits _{n\rightarrow +\infty }\big \langle A_p(u_n),u_n-u_{\lambda }^*\big \rangle \leqslant 0 \end{aligned}$$

(see (3.24)) and hence we get

$$\begin{aligned} u_n\longrightarrow u_{\lambda }^*\quad \text {in}\ W^{1,p}_0(\Omega ) \end{aligned}$$

(3.25)

(see Proposition 2.1). Suppose that $u_{\lambda }^*=0$. Then from (3.25) we have

$$\begin{aligned} \Vert u_n\Vert \longrightarrow 0\quad \text {as}\ n\rightarrow \infty . \end{aligned}$$

(3.26)

We set $y_n=\frac{u_n}{\Vert u_n\Vert }$, for $n\in \mathbb {N}$. We have $\Vert y_n\Vert =1$ for all $n\in \mathbb {N}$. From (3.23), we have

$$\begin{aligned} \Vert u_n\Vert ^{p-2}\langle A_n(y_n),h\rangle +\langle A(y_n),h\rangle =\int _{\Omega }\bigg (\frac{\lambda u_n^{r-1}}{\Vert u_n\Vert }+\frac{N_f(u_n)}{\Vert u_n\Vert }\bigg )h\,dz \end{aligned}$$

for all $h\in W^{1,p}_0(\Omega )$, all $n\in \mathbb {N}$, so

$$\begin{aligned} \left\{ \begin{array}{l} -\Vert u_n\Vert ^{p-2}\Delta _p y_n(z)-\Delta y_n(z)=\frac{\lambda }{\Vert u_n\Vert }u_n(z)^{r-1}+\frac{1}{\Vert u_n\Vert }f(z,u_n(z))\\ \qquad \text {for a.a.}\ z\in \Omega ,\\ u_n|_{\partial \Omega }=0. \end{array} \right. \end{aligned}$$

(3.27)

Note that $\{\frac{N_f(u_n)}{\Vert u_n\Vert }\}_{n\geqslant 1}\subseteq L^{p'}(\Omega )$ and $\{\frac{\lambda u_n^{r-1}}{\Vert u_n\Vert }\}_{n\geqslant 1}\subseteq L^{p'}(\Omega )$ (see (3.22)). So, from (3.27) as before using the nonlinear regularity theory (see Ladyzhenskaya-Ural’tseva [18] and Lieberman [19]), at least for a subsequence, we can have

$$\begin{aligned} y_n\longrightarrow y\quad \text {in}\ C^1_0(\overline{\Omega })\quad \text {as}\ n\rightarrow +\infty . \end{aligned}$$

(3.28)

We have

$$\begin{aligned} \frac{\lambda u_n^{r-1}}{\Vert u_n\Vert }{\mathop {\longrightarrow }\limits ^{w}}0\quad \text {in}\ L^{r'}(\Omega ) \end{aligned}$$

(3.29)

and

$$\begin{aligned} \frac{N_f(u_n)}{\Vert u_n\Vert }{\mathop {\longrightarrow }\limits ^{w}}\vartheta y\quad \text {in}\ L^{p'}(\Omega ), \end{aligned}$$

(3.30)

with $\widehat{\vartheta }_0(z)\leqslant \vartheta (z)\leqslant \vartheta _0(z)$ a.e. on Z (see hypothesis $H(f)_1(i)$ and (3.26)). So, if in (3.26) we pass to the limit as $n\rightarrow \infty$ and use (3.26), (3.28), (3.29) and (3.30), we have

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta y(z)=\vartheta (z)y(z)\ \text {for a.a.}\ z\in \Omega ,\\ y|_{\partial \Omega }=0. \end{array} \right. \end{aligned}$$

(3.31)

Using (3.30) and Lemma 2.5, we have

$$\begin{aligned} 1=\widetilde{\lambda }_1(2,\widehat{\lambda }_1(2))<\widetilde{\lambda }_1(2,\vartheta ), \end{aligned}$$

so $y=0$ (see (3.31)). This is a contradiction since $\Vert y_n\Vert =1$ for all $n\in \mathbb {N}$ and we have (3.28).

Therefore $u_{\lambda }^*\ne 0$ and then using (3.25) we see that

$$\begin{aligned} u_{\lambda }^*\in S_{\lambda }^*\quad \text {and}\quad u_{\lambda }^*=\inf S_{\lambda }^*. \end{aligned}$$

The set $S_{\lambda }^-$ is upward directed (that is, if $v_1,v_2\in S_{\lambda }^-$ we can find $v\in S_{\lambda }^-$ such that $v_1\leqslant v$, $v_2\leqslant v$; see Filippakis-Papageorgiou [6]). Reasoning as above, we produce

$$\begin{aligned} v_{\lambda }^*\in S_{\lambda }^-\quad \text {and}\quad v_{\lambda }^*=\sup S_{\lambda }^-. \end{aligned}$$

$\square$

Using these extremal constant sign solutions, we can produce a nodal solution.

Proposition 3.4

If hypotheses $H(f)_1$ hold, then for all $\lambda >0$ big, problem $(P_{\lambda })$ admits a nodal solution

$$\begin{aligned} y_{\lambda }\in [v_{\lambda }^*,u_{\lambda }^*]\cap C^1_0(\overline{\Omega }). \end{aligned}$$

Proof

Using the two extremal constant sign solutions $u_{\lambda }^*\in \mathrm {int}\, C_+$ and $v_{\lambda }^*\in -\mathrm {int}\, C_+$ produced in Proposition 3.3, we introduce the following truncation of the reaction in problem $(P_{\lambda })$:

$$\begin{aligned} \widehat{k}_{\lambda }(z,x)= \left\{ \begin{array}{lll} \lambda |v_{\lambda }^*(z)|^{r-2}v_{\lambda }^*(z)+f(z,v_{\lambda }^*(z)) &{}\quad \text {if} &{} x<v_{\lambda }^*,\\ \lambda |x|^{r-2}x+f(z,x) &{}\quad \text {if} &{} v_{\lambda }^*(z)\leqslant x\leqslant u_{\lambda }^*(z),\\ \lambda u_{\lambda }^*(z)^{r-1}+f(z,u_{\lambda }^*(z)) &{}\quad \text {if} &{} u_{\lambda }^*(z)<x. \end{array} \right. \end{aligned}$$

(3.32)

We also consider the positive and negative truncations of $\widehat{k}_{\lambda }(z,\cdot )$, namely the Carathéodory functions

$$\begin{aligned} \widehat{k}_{\lambda }^{\pm }(z,x)=\widehat{k}_{\lambda }(z,\pm x^{\pm }). \end{aligned}$$

(3.33)

We set $\widehat{K}_{\lambda }(z,x)=\int _0^x \widehat{k}_{\lambda }(z,s)\,ds$, $\widehat{K}_{\lambda }^{\pm }(z,x)=\int _0^x \widehat{k}_{\lambda }^{\pm }(z,s)\,ds$ and consider the $C^1$-functionals $\widehat{\varphi }_{\lambda },\widehat{\varphi }_{\lambda }^{\pm }:W^{1,p}_0(\Omega )\longrightarrow \mathbb {R}$ defined by

$$\begin{aligned} \widehat{\varphi }_{\lambda }(u)= & {} \frac{1}{p}\Vert Du\Vert _p^p+\frac{1}{2}\Vert Du\Vert _2^2-\int _{\Omega }\widehat{K}_{\lambda }(z,u)\,dz\quad \forall u\in W^{1,p}_0(\Omega ),\\ \widehat{\varphi }_{\lambda }^{\pm }(u)= & {} \frac{1}{p}\Vert Du\Vert _p^p+\frac{1}{2}\Vert Du\Vert _2^2-\int _{\Omega }\widehat{K}_{\lambda }^{\pm }(z,u)\,dz\quad \forall u\in W^{1,p}_0(\Omega ). \end{aligned}$$

Using (3.32) and (3.33) and the nonlinear regularity theory (see Ladyzhenskaya-Ural’tseva [18] and Lieberman [19]), we easily check that

$$\begin{aligned} K_{\widehat{\varphi }_{\lambda }} \subseteq [v_{\lambda }^*,u_{\lambda }^*]\cap C^1_0(\Omega ),\quad K_{\widehat{\varphi }_{\lambda }^+}\subseteq [0,u_{\lambda }^*]\cap C_+,\quad K_{\widehat{\varphi }_{\lambda }^-}\subseteq [v_{\lambda }^*,0]\cap (-C_+). \end{aligned}$$

The extremality of $u_{\lambda }^*$ and $v_{\lambda }^*$ implies that

$$\begin{aligned} K_{\widehat{\varphi }_{\lambda }^+}=\{0,u_{\lambda }^*\},\quad K_{\widehat{\varphi }_{\lambda }^-}=\{0,v_{\lambda }^*\}. \end{aligned}$$

(3.34)

From (3.32) and (3.33) we see that $\widehat{\varphi }_{\lambda }^+$ is coercive. Also using the Sobolev embedding theorem, we have that $\widehat{\varphi }_{\lambda }^+$ is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find $\widetilde{u}_{\lambda }^*\in W^{1,p}$, such that

$$\begin{aligned} \widetilde{\varphi }_{\lambda }^+(\widetilde{u}_{\lambda }^*)=\inf \big \{\widehat{\varphi }_{\lambda }^+(u):\ u\in W^{1,p}_0(\Omega )\big \}. \end{aligned}$$

(3.35)

Let $\overline{u}_0\in \mathrm {int}\, C_+$. Using Proposition 4.1.22 of Papageorgiou-Rădulescu-Repovš [28], we can find $t\in (0,1)$ small such that $0\leqslant t\overline{u}_0\leqslant u_{\lambda }^*$. Then using (3.32), (3.33) and hypothesis $H(f)_1(iii)$, we have

$$\begin{aligned} \widehat{\varphi }_{\lambda }^+(t\overline{u}_0)\leqslant & {} \frac{t^p}{p}\Vert D\overline{u}_0\Vert _p^p +\frac{t^2}{2}\Vert D\overline{u}_0\Vert _2^2 -\lambda \frac{t^r}{r}\Vert \overline{u}_0\Vert _r^r\\\leqslant & {} c_{13}t^2-\lambda c_{14}t^r, \end{aligned}$$

for some $c_{13},c_{14}>0$ (recall that $t\in (0,1)$, $2<p$).

Fixing $t\in (0,1)$, from the above inequality we see that for $\lambda \geqslant 1$ big, we have

$$\begin{aligned} \widehat{\varphi }_{\lambda }^+(t\overline{u}_0)<0, \end{aligned}$$

so

$$\begin{aligned} \widehat{\varphi }_{\lambda }^+(\widetilde{u}_{\lambda }^*)<0=\widehat{\varphi }_{\lambda }^+(0) \end{aligned}$$

(see (3.35)) and thus

$$\begin{aligned} \widetilde{u}_{\lambda }^*\ne 0. \end{aligned}$$

(3.36)

Note that $\widetilde{u}_{\lambda }^*\in K_{\widehat{\varphi }_{\lambda }^+}$ (see (3.35)). Then from (3.34) and (3.36) we infer that

$$\begin{aligned} \widetilde{u}_{\lambda }^*=u_{\lambda }^*\in \mathrm {int}\, C_+. \end{aligned}$$

(3.37)

From (3.32) and (3.33) it is clear that

$$\begin{aligned} \widehat{\varphi }_{\lambda }|_{C_+}=\widehat{\varphi }^+|_{C_+}, \end{aligned}$$

so $u_{\lambda }^*$ is a local $C^1_0(\overline{\Omega })$-minimizer of $\widehat{\varphi }_{\lambda }$ (see (3.37)), and by Proposition 2.2, we get that

$$\begin{aligned} u_{\lambda }^*\ \text {is a local}\ W^{1,p}_0(\Omega )\text {-minimizer of}\ \widehat{\varphi }_{\lambda }. \end{aligned}$$

(3.38)

Similarly, using this time the functional $\widehat{\varphi }_{\lambda }^-$, we show that

$$\begin{aligned} v_{\lambda }^*\ \text {is a local}\ W^{1,p}_0(\Omega )\text {-minimizer of}\ \widehat{\varphi }_{\lambda }. \end{aligned}$$

(3.39)

We may assume that

$$\begin{aligned} \widehat{\varphi }_{\lambda }(v_{\lambda }^*)\leqslant \widehat{\varphi }_{\lambda }(u_{\lambda }^*). \end{aligned}$$

The reasoning is the same if the opposite inequality holds, using this time (3.39) instead of (3.38).

On account of (3.34) we see that we may assume that

$$\begin{aligned} K_{\widehat{\varphi }_{\lambda }}\ \text {is finite}. \end{aligned}$$

(3.40)

Otherwise on account of the extremality of $u_{\lambda }^*$ and $v_{\lambda }^*$, we see that we already have an infinity of smooth nodal solutions (see (3.34)) and we are done.

From (3.38), (3.40) and Theorem 5.7.6 of Papageorgiou-Rădulescu-Repovš [28], we can find $\varrho \in (0,1)$ small such that

$$\begin{aligned} \widehat{\varphi }_{\lambda }(v_{\lambda }^*)\leqslant \widehat{\varphi }_{\lambda }(u_{\lambda }^*) <\inf \{\widehat{\varphi }_{\lambda }(u):\ \Vert u-u_{\lambda }^*\Vert =\varrho \}=\widehat{m}_{\lambda },\quad \Vert v_{\lambda }^*-u_{\lambda }^*\Vert >\varrho . \end{aligned}$$

(3.41)

Note that $\widehat{\varphi }_{\lambda }$ is coercive (see (3.32)). Then Proposition 5.1.15 of Papageorgiou-Rădulescu-Repovš [28] implies that

$$\begin{aligned} \widehat{\varphi }_{\lambda }\ \text {satisfies the Palais-Smale condition.} \end{aligned}$$

(3.42)

From (3.41) and (3.42) we see that we can apply the mountain pass theorem. So, there exists $y_{\lambda }\in W^{1,p}_0(\Omega )$ such that

$$\begin{aligned} y_{\lambda }\in K_{\widehat{\varphi }_{\lambda }}\subseteq [v_{\lambda }^*,u_{\lambda }^*]\cap C^1_0(\overline{\Omega }) \quad \text {and}\quad \widehat{m}_{\lambda }\leqslant \widehat{\varphi }_{\lambda }(y_{\lambda }). \end{aligned}$$

(3.43)

From (3.41) and (3.43) we see that

$$\begin{aligned} y_{\lambda }\not \in \{u_{\lambda }^*,v_{\lambda }^*\} \end{aligned}$$

(3.44)

So, if we show that $y_{\lambda }\ne 0$, then $y_{\lambda }$ will be the desired nodal solution. Since $y_{\lambda }$ is a critical point of $\widehat{\varphi }_{\lambda }$ of mountain pass type, we have

$$\begin{aligned} C_1(\widehat{\varphi }_{\lambda },y_{\lambda })\ne 0 \end{aligned}$$

(3.45)

(see Papageorgiou-Rădulescu-Repovš [28, Theorem 6.5.8]).

From hypotheses $H(f)_1(i),(ii)$, we see that given $\varepsilon >0$, we can find $c_{15}=c_{15}(\varepsilon )>0$ such that

$$\begin{aligned} F(z,x)\leqslant \frac{1}{2}(\vartheta _0(z)+\varepsilon )x^2+c_{15}|x|^r\quad \text {for a.a.}\ z\in \Omega ,\ \text {all}\ x\in \mathbb {R}. \end{aligned}$$

(3.46)

Then taking $\lambda \geqslant 1$ and using (3.46), for $u\in W^{1,p}_0(\Omega )$, we have

$$\begin{aligned} \widehat{\varphi }_{\lambda }(u)\geqslant & {} \frac{1}{p}\Vert Du\Vert _p^p +\frac{1}{2}\bigg (\Vert Du\Vert _2^2-\int _{\Omega }\vartheta _0(z)u^2\,dz-\varepsilon c_{16}\Vert u\Vert ^2\bigg ) -\lambda c_{17}\Vert u\Vert ^r\\\geqslant & {} \frac{1}{p}\Vert u\Vert ^p+\frac{1}{2}(c_0-\varepsilon c_{16})\Vert u\Vert ^2-\lambda c_{17}\Vert u\Vert ^r \end{aligned}$$

for some $c_{16},c_{17}>0$ (see Lemma 2.4).

Choosing $\varepsilon \in (0,\frac{c_0}{c_{16}})$, we see that

$$\begin{aligned} \widehat{\varphi }_{\lambda }(u)\geqslant \frac{1}{p}\Vert u\Vert ^p-\lambda c_{17}\Vert u\Vert ^r. \end{aligned}$$

Since $r>p$, we can find $\varrho _{\lambda }\in (0,\delta )$ such that

$$\begin{aligned} \widehat{\varphi }_{\lambda }(u)\geqslant 0\quad \forall \Vert u\Vert \leqslant \varrho _{\lambda }, \end{aligned}$$

so

$$\begin{aligned} u=0\ \text {is a local minimizer of}~ \ \widehat{\varphi }_{\lambda }, \end{aligned}$$

thus

$$\begin{aligned} C_k(\widehat{\varphi }_{\lambda },0)=\delta _{k,0}\mathbb {Z}\quad \forall k\in \mathbb {N}_0. \end{aligned}$$

(3.47)

From (3.47), (3.45) and (3.44), we infer that

$$\begin{aligned} y_{\lambda }\not \in \{0,u_{\lambda }^*,v_{\lambda }^*\}, \end{aligned}$$

so $y_{\lambda }\in [v_{\lambda }^*,u_{\lambda }^*]\cap C^1_0(\overline{\Omega })$ (see (3.43)) is nodal. $\square$

If we strengthen the hypotheses on the perturbation $f(z,\cdot )$ we can improve the conclusion of Proposition 2.2. The new hypotheses on f are the following:

$\underline{H(f)_2}$ $f:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}$ is a Carathéodory function such that $f(z,0)=0$ for a.a. $z\in \Omega$, hypotheses $H(f)_2(i),(ii),(iii)$ are the same as the corresponding hypotheses hypotheses $H(f)_1(i),(ii),(iii)$ and

(iv):

for every $\varrho >0$, there exists $\widehat{\xi }_{\varrho }>0$ such that for a.a. $z\in \Omega$, the function $x\longmapsto f(z,x)+\widehat{\xi }_{\varrho }|x|^{p-2}x$ is nondecreasing on $[-\varrho ,\varrho ]$.

Remark 3.5

Evidently hypothesis $H(f)_2(iv)$ implies a lower local Lipschitz condition for $f(z,\cdot )$.

Proposition 3.6

If hypotheses $H(f)_2$ hold, then for all $\lambda >0$ big, problem $(P_{\lambda })$ has a nodal solution

$$\begin{aligned} y_{\lambda }\in \mathrm {int}_{C^1_0(\overline{\Omega })} [v_{\lambda }^*,u_{\lambda }^*]. \end{aligned}$$

Proof

From Proposition 3.4, we know that for all $\lambda >0$ big, problem $(P_{\lambda })$ has a nodal solution

$$\begin{aligned} y_{\lambda }\in [v_{\lambda }^*,u_{\lambda }^*]\cap C^1_0(\overline{\Omega }). \end{aligned}$$

(3.48)

Let $\varrho =\max \{\Vert u_{\lambda }^*\Vert _{\infty },\Vert v_{\lambda }^*\Vert _{\infty }\}$ and let $\widehat{\xi }_{\varrho }>0$ be as postulated by hypotheses $H(f)_2(iv)$. Let $\widetilde{\xi }_{\varrho }>\widehat{\xi }_{\varrho }$. We have

$$\begin{aligned}&-\Delta _p y_{\lambda }-\Delta y_{\lambda }+\widetilde{\xi }_{\varrho }|y_{\lambda }|^{p-2}y_{\lambda }\\&\quad \leqslant \lambda (u_{\lambda }^*)^{r-1}+f(z,u_{\lambda }^*)+\widehat{\xi }_{\varrho }(u_{\lambda }^*)^{p-1} +(\widetilde{\xi }_{\varrho }-\widehat{\xi }_{\varrho })(u_{\lambda }^*)^{p-1}\\&\quad \leqslant -\Delta _p u_{\lambda }^*-\Delta u_{\lambda }^*+\widetilde{\xi }_{\varrho }(u_{\lambda }^*)^{p-1}\quad \text {for a.a.}\ z\in \Omega \end{aligned}$$

(3.49)

(see hypothesis $H(f)_2(iv)$ and (3.48)).

Let $a:\mathbb {R}^N\longrightarrow \mathbb {R}^N$ be defined by

$$\begin{aligned} a(y)=|y|^{p-2}y+y\quad \forall y\in \mathbb {R}^N. \end{aligned}$$

Evidently $a\in C^1(\mathbb {R}^N;\mathbb {R}^N)$ (recall that $2<p$) and

$$\begin{aligned} \nabla a(y)=|y|^{p-2}\bigg (\mathrm {id}+(p-2)\frac{y\otimes y}{|y|^2}\bigg )+\mathrm {id}\quad \forall y\in \mathbb {R}^N, \end{aligned}$$

so

$$\begin{aligned} \big (\nabla a(y)\xi ,\xi \big )_{\mathbb {R}^N}\geqslant |\xi |^2\quad \forall y,\xi \in \mathbb {R}^N. \end{aligned}$$

Note that

$$\begin{aligned} \mathrm {div}\,a(Du)=\Delta _p u+\Delta u\quad \forall u\in W^{1,p}_0(\Omega ). \end{aligned}$$

So, invoking the tangency principle of Pucci-Serrin [31, Theorem 2.5.2], we obtain

$$\begin{aligned} y_{\lambda }(z)<u_{\lambda }^*(z)\quad \forall z\in \Omega . \end{aligned}$$

Since $y_{\lambda },u_{\lambda }^*\in C^1_0(\overline{\Omega })$, we have

$$\begin{aligned} (\widetilde{\xi }_{\varrho }-\widehat{\xi }_{\varrho })(y_{\lambda })^{p-2}y_{\lambda }\preceq (\widetilde{\xi }_{\varrho }-\widehat{\xi }_{\varrho })(u_{\lambda }^*)^{p-1}. \end{aligned}$$

Then using Proposition 2.3, we have

$$\begin{aligned} u_{\lambda }^*-y_{\lambda }\in \mathrm {int}\, C_+. \end{aligned}$$

Similarly we show that

$$\begin{aligned} y_{\lambda }-v_{\lambda }^*\in \mathrm {int}\, C_+. \end{aligned}$$

We conclude that

$$\begin{aligned} y_{\lambda }\in \mathrm {int}_{C^1_0(\overline{\Omega })}[v_{\lambda }^*,u_{\lambda }^*]. \end{aligned}$$

$\square$

We can now state our first multiplicity theorem.

Theorem 3.7

• (a) If hypotheses $H(f)_1$ hold, then for all $\lambda >0$ big, problem $(P_{\lambda })$ has at least three nontrivial solutions

  $$\begin{aligned} u_{\lambda }\in \mathrm {int}\, C_+,\quad v_{\lambda }\in -\mathrm {int}\, C_+,\quad y_{\lambda }\in [v_{\lambda },u_{\lambda }]\cap C^1_0(\overline{\Omega })\ \text {nodal} \end{aligned}$$

  and $u_{\lambda },v_{\lambda },y_{\lambda }\longrightarrow 0$ in $C^1_0(\overline{\Omega })$ as $\lambda \rightarrow +\infty$.

• (b) If hypotheses $H(f)_2$ hold, then for all $\lambda >0$ big, problem $(P_{\lambda })$ has at least three nontrivial solutions

  $$\begin{aligned} u_{\lambda }\in \mathrm {int}\, C_+,\quad v_{\lambda }\in -\mathrm {int}\, C_+,\quad y_{\lambda }\in \mathrm {int}_{C^1_0(\overline{\Omega })}[v_{\lambda },u_{\lambda }]\ \text {nodal} \end{aligned}$$

  and $u_{\lambda },v_{\lambda },y_{\lambda }\longrightarrow 0$ in $C^1_0(\overline{\Omega })$ as $\lambda \rightarrow +\infty$.

4 Four Solutions with Sign Information

In this section by strengthening the regularity of $f(z,\cdot )$, we can improve the above multiplicity theorem and produce a second nodal solution, for a total of four nontrivial smooth solutions, all with sign information.

The new hypotheses on the perturbation f(z, x) are the following:

$\underline{H(f)_3}$ $f:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}$ is a measurable function such that $f(z,0)=0$, $f(z,\cdot )\in C^1(\mathbb {R})$ for a.a. $z\in \Omega$ and

(i):

$|f_x'(z,x)|\leqslant a_0(z)(1+|x|^{q-1})$ for a.a. $z\in \Omega$, all $x\in \mathbb {R}$, with $a_0\in L^{\infty }(\Omega )$, $1<q<p^*$;

(ii):

$f_x'(z,0)=\lim \limits _{x\rightarrow 0}\frac{f(z,x)}{x}$ uniformly for a.a. $z\in \Omega$ and

$$\begin{aligned} 0\leqslant f_x'(z,0)\leqslant \widehat{\lambda }_1(2)\ \text {for a.a.}\ z\in \Omega ,\ f_x'(\cdot ,0)\not \equiv 0,\ f_x'(\cdot ,0)\not \equiv \widehat{\lambda }_1(2); \end{aligned}$$

(iii):

$\lim \limits _{x\rightarrow \pm \infty }\frac{f(z,x)}{|x|^{p-2}x}=0$ uniformly for a.a. $z\in \Omega$;

(iv):

$f(z,x)x\geqslant 0$ for a.a. $z\in \Omega$, all $x\in \mathbb {R}$;

(v):

for every $\varrho >0$, there exists $\widehat{\xi }_{\varrho }>0$ such that for a.a. $z\in \Omega$ the function $x\longmapsto f(z,x)+\widehat{\xi }_{\varrho }|x|^{p-2}x$ is nondecreasing on $[-\varrho ,\varrho ]$.

Evidently the function $f(z,x)=\vartheta (z)x+|x|^{q-2}x$ with $0\leqslant \vartheta (z)\leqslant \widehat{\lambda }_1(2)$ for a.a. $z\in \Omega$, $\vartheta \not \equiv 0$, $\vartheta \not \equiv \widehat{\lambda }_1(2)$ and $2<q<p$, satisfies hypotheses $H(f)_3$.

Proposition 4.1

If hypotheses $H(f)_3$ hold, then for all $\lambda >0$ big, problem $(P_{\lambda })$ has at least two nodal solutions

$$\begin{aligned} y_{\lambda },\widehat{y}_{\lambda }\in \mathrm {int}_{C^1_0(\overline{\Omega })} [v_{\lambda }^*,u_{\lambda }^*]. \end{aligned}$$

Proof

From Theorem 3.7(b), we already have a nodal solution

$$\begin{aligned} y_{\lambda }\in \mathrm {int}_{C^1_0(\overline{\Omega })}[v_{\lambda }^*,u_{\lambda }^*]. \end{aligned}$$

(4.1)

We consider the energy (Euler) functional $\psi _{\lambda }:W^{1,p}_0(\Omega )\longrightarrow \mathbb {R}$ for problem $(P_{\lambda })$ defined by

$$\begin{aligned} \varphi _{\lambda }(u)=\frac{1}{p}\Vert Du\Vert _p^p+\frac{1}{2}\Vert Du\Vert _2^2-\frac{\lambda }{r}\Vert u\Vert _r^r-\int _{\Omega }F(z,u)\,dz \quad \forall u\in W^{1,p}_0(\Omega ). \end{aligned}$$

Also, we consider the function $\widehat{\varphi }_{\lambda }:W^{1,p}_0(\Omega )\longrightarrow \mathbb {R}$ from the proof of Proposition 3.4. Hypotheses $H(f)_3$ imply that

$$\begin{aligned} \varphi _{\lambda }\in C^2(W^{1,p}_0(\Omega )),\quad \widehat{\varphi }_{\lambda }\in C^{2-0}(W^{1,p}_0(\Omega )). \end{aligned}$$

(4.2)

We consider the homotopy

$$\begin{aligned} h(t,u)=(1-t)\varphi _{\lambda }(u)+t\widehat{\varphi }_{\lambda }(u)\quad \forall t\in [0,1],\ \text {all}\ u\in W^{1,p}_0(\Omega ). \end{aligned}$$

Suppose we could find $\{t_n\}_{n\geqslant 1}\subseteq [0,1]$ and $\{u_n\}_{n\geqslant 1}\subseteq W^{1,p}_0(\Omega )$ such that

$$\begin{aligned} t_n\longrightarrow t\ \text {in}\ [0,1],\quad u_n\longrightarrow y_n\ \text {in}\ W^{1,p}_0(\Omega ) \quad \text {and}\quad h_u'(t_n,u_n)=0\ \text {for all}\ n\in \mathbb {N}. \end{aligned}$$

(4.3)

From the equality in (4.3), we have

$$\begin{aligned}&\langle A_p(u_n),h\rangle +\langle A(u_n),h\rangle +\int _{\Omega }\big ((1-t_n)k_{\lambda }(z,u_n)+t_n\widehat{k}_{\lambda }(z,u_n)\big )h\,dz \\&\quad \forall h\in W^{1,p}_0(\Omega ),\ \text {all}\ n\in \mathbb {N}\end{aligned}$$

(see the proofs of Propositions 3.1 and 3.4 ), so

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta _p u_n(z)-\Delta u_n(z)=(1-t)k_{\lambda }(z,u_n(z))+t\widehat{k}(z,u_n(z))\quad \text {for a.a.}\ z\in \Omega ,\\ u_n|_{\partial \Omega }=0,\ \text {for all}\ n\in \mathbb {N}. \end{array} \right. \end{aligned}$$

(4.4)

As before (see the proof of Proposition 3.2), from (4.4), (4.3) and the nonlinear regularity theory, we have

$$\begin{aligned} u_n\longrightarrow y_{\lambda }\quad \text {in}\ C^1_0(\overline{\Omega })\quad \text {as}\ n\rightarrow +\infty , \end{aligned}$$

so

$$\begin{aligned} u_n\in [v_{\lambda }^*,u_{\lambda }^*]\cap C^1_0(\overline{\Omega })\quad \forall n\geqslant n_0. \end{aligned}$$

(4.5)

Again without any loss of generality we assume that $K_{\widehat{\varphi }_{\lambda }}$ is finite (see (3.40)). Then finiteness of $K_{\widehat{\varphi }_{\lambda }}$ and (4.5), (3.32) lead to a contradiction. So, (4.3) cannot occur and then the homotopy invariance property of the critical groups (see Theorem 6.3.8 of Papageorgiou-Rădulescu-Repovš [28]) implies that

$$\begin{aligned} C_k(\varphi _{\lambda },y_{\lambda })=C_k(\widehat{\varphi }_{\lambda },y_{\lambda })\quad \forall k\in \mathbb {N}_0. \end{aligned}$$

(4.6)

Recall that $C_1(\widehat{\varphi }_{\lambda },y_{\lambda })\ne 0$ (see (3.45)). Hence $C_1(\varphi _{\lambda },y_{\lambda })\ne 0$ (see (4.6)). Then (4.2) and Claim 3 of Papageorgiou-Rădulescu [24, p. 412], imply that

$$\begin{aligned} C_k(\varphi _{\lambda },y_{\lambda })=\delta _{k,1}\mathbb {Z}\quad \forall k\in \mathbb {N}_0, \end{aligned}$$

so

$$\begin{aligned} C_k(\widehat{\varphi }_{\lambda },y_{\lambda })=\delta _{k,1}\mathbb {Z}\quad \forall k\in \mathbb {N}_0 \end{aligned}$$

(4.7)

(see (4.6)). We know that $u_{\lambda }^*,v_{\lambda }^*,0$ are local minimizers of $\widehat{\varphi }_{\lambda }$ (see (3.38), (3.39), (3.39)). Hence we have

$$\begin{aligned} C_k(\widehat{\varphi }_{\lambda },u_{\lambda }^*)=C_k(\widehat{\varphi }_{\lambda },v_{\lambda }^*)=C_k(\widehat{\varphi }_{\lambda },0)=\delta _{k,0}\mathbb {Z}\quad \forall k\in \mathbb {N}_0. \end{aligned}$$

(4.8)

Since $\widehat{\varphi }_{\lambda }$ is coercive, we have

$$\begin{aligned} C_k(\widehat{\varphi }_{\lambda },\infty )=0\quad \forall k\in \mathbb {N}_0 \end{aligned}$$

(4.9)

(see Papageorgiou-Rădulescu-Repovš [28, Proposition 6.2.24]).

If $K_{\widehat{\varphi }_{\lambda }}=\{0,u_{\lambda }^*,v_{\lambda }^*,y_{\lambda }\}$, then from (4.7), (4.8), (4.9) and the Morse reaction (see (2.1)) with $t=-1$, we have

$$\begin{aligned} 3(-1)^0+(-1)^1=(-1)^0, \end{aligned}$$

so $(-1)^0=0$, a contradiction. So, there exists $\widehat{y}_{\lambda }\in K_{\widehat{\lambda }_{\lambda }}$, $\widehat{y}_{\lambda }\not \in \{0,u_{\lambda }^*,v_{\lambda }^*,y_{\lambda }\}$. Then $\widehat{y}_{\lambda }\in C^1_0(\overline{\Omega })$ is a second nodal solution of $(P_{\lambda })$ (see (3.34)) district from $y_{\lambda }$. Moreover, using Proposition 2.3, we have $\widehat{y}_{\lambda }\in \mathrm {int}_{C^1_0(\overline{\Omega })}[v_{\lambda }^*,u_{\lambda }^*]$ (see the proof of Proposition 3.6). $\square$

Now we can state our second multiplicity theorem for problem $(P_{\lambda })$.

Theorem 4.2

If hypotheses $H(f)_3$ hold, then for all $\lambda >0$ big, problem $(P_{\lambda })$ has at least four nontrivial solutions

$$\begin{aligned} u_{\lambda }\in \mathrm {int}\, C_+,\quad v_{\lambda }\in -\mathrm {int}\, C_+,\quad y_{\lambda },\widehat{y}_{\lambda }\in \mathrm {int}_{C^1_0(\overline{\Omega })}[v_{\lambda },u_{\lambda }]\ \text {nodal} \end{aligned}$$

and $u_{\lambda },v_{\lambda },y_{\lambda },\widehat{y}_{\lambda }\longrightarrow 0$ in $C^1_0(\overline{\Omega })$ as $\lambda \rightarrow +\infty$.

Remark 4.3

It will be interesting to extend the results of this work to problems with convection (that is, f depends also on Du). Helpful in that respect can be the recent work of Bai-Gasiński-Papageorgiou [2] (see also Bai-Gasiński-Papageorgiou [1], Candito-Gasiński-Papageorgiou [3] and Gasiński-Papageorgiou [15]).