Anisotropic $(\vec{p}, \vec{q})$-Laplacian problems with superlinear nonlinearities

1.   Introduction

Let $\Omega\subseteq \mathbb{R}^N$, $N\geq 2$ be a bounded domain with boundary $\partial\Omega$ of class $C^1$. In this paper we consider the following anisotropic differential equation involving the $(\vec{p}, \vec{q})$-Laplacian given by

where $\lambda$ is a positive parameter, $\vec{p} = (p_1, p_2, \ldots, p_N), \vec{q} = (q_1, q_2, \ldots, q_N)$, $\vec{p}, \vec{q}\in \mathbb{R}^N$ are real vectors such that

and $f\colon \Omega \times \mathbb{R} \to \mathbb{R}$ is a nonlinearity with subcritical growth that is superlinear at $\pm\infty$, see (H_f) for the precise assumptions. Moreover, for any $\vec{s} = \left(s_1, s_2, \ldots, s_N \right)\in \mathbb{R}^N$, we denote by

the anisotropic $\vec{s}$-Laplace differential operator. If $s_i = 2$ for all $i = 1, \ldots, N$, we get

that is the usual Laplacian and if $\vec{s}$ is constant (i.e. $s_i = s$ for each $i = 1, \ldots, N$) we obtain

which is called the pseudo-$s$-Laplace operator, see Belloni-Kawohl ^[1] or Brasco-Franzina ^[2]. More information about anisotropic operators and in particular about the theory of anisotropic Sobolev spaces can be found in Kufner-Rákosník ^[3], Nikol'skiĭ ^[4] and Rákosník ^[5,6].

Anisotropic differential problems have a large background in several applications, for example, the study of an epidemic disease in heterogeneous habitat can be expressed by an anisotropic nonlinear system. In general, anisotropic operators are used for modeling in which partial differential derivatives vary with the direction. Also, the anisotropic Laplacian, given by

where $F(\xi) = (\sum_{i = 1}^N |\xi_i|^2)^{\frac{1}{2}}$ for $\xi\in \mathbb{R}^N$, plays a key role in physical models of crystal growth in the context of the so-called Wulff shape of $F$ (also known as equilibrium crystal shape), see the work of Wulff ^[7]. For more information on applications in different disciplines we refer to Antontsev-Díaz-Shmarev ^[8], Bendahmane-Chrif-El Manouni ^[9], Bendahmane-Langlais-Saad ^[10], Vétois ^[11] and the references therein.

Although there are some works for $\vec{p}$-Laplacian problems, only a few exist for the anisotropic $(\vec{p}, \vec{q})$-Laplacian. Recently, Razani-Figueiredo ^[12] studied the anisotropic Dirichlet problem

in a bounded and regular domain $\Omega$ of $\mathbb{R}^N$ with $\gamma > 1$ and $\lambda > 0$. Based on a sub-supersolution approach the authors show the existence of at least one positive solution of (1.1). In Razani ^[13] nonstandard competing anisotropic $(\vec{p}, \vec{q})$-Laplacian problems with convolution of the form

have been considered, where $\phi \in L^1(\mathbb{R}^N)$. If $\mu > 0$, then the existence of a generalized solution of (1.2) is shown by using a Galerkin base for the function space and if $\mu \leq 0$, then any generalized solution turns out to be a weak solution. We also mention the recent work of Tavares ^[14] who considered existence and multiplicity of nonnegative solutions for the problem given by

where $\alpha > 1$, $k\in L^\infty(\Omega)$ with $k(x) > 0$ for a.a. $x\in\Omega$ and a Carathéodory nonlinearity $f\colon\Omega\times \mathbb{R}\to \mathbb{R}$ with subcritical growth. Existence and regularity results for anisotropic problems driven by the $\vec{p}$-Laplacian have been obtained by several authors. Without guarantee of completeness, we mention just some of them and refer to the papers of Bonanno-D'Aguì-Sciammetta ^[15], Ciani-Figueiredo-Suárez ^[16], Ciraolo-Figalli-Roncoroni ^[17], Ciraolo-Sciammetta ^[18], DiBenedetto-Gianazza-Vespri ^[19], dos Santos-Figueiredo-Tavares ^[20], Fragalà-Gazzola-Kawohl ^[21], Perera-Agarwal-O'Regan ^[22], Ragusa-Razani-Safari ^[23], see also the references therein. Related results for the $(p, q)$-Laplacian, double phase equations, anisotropic problems or the discrete $p$-Laplacian can be found in the works of Bai-Papageorgiou-Zeng ^[24], Bohner-Caristi-Ghobadi-Heidarkhani ^[25], El Manouni-Marino-Winkert ^[26], Leggat-Miri ^[27], He-Ousbika-El Allali-Zuo ^[28], Ju-Molica Bisci-Zhang ^[29], Liu-Motreanu-Zeng ^[30], Papageorgiou ^[31], Son-Sim ^[32], Vetro-Vetro ^[33], Zeng-Bai-Gasiński-Winkert ^[34], Zeng-Papageorgiou ^[35] and Zeng-Rădulescu-Winkert ^[36].

Motivated by the above mentioned works for $\vec{p}$-Laplacian problems, we consider in our paper so-called $(\vec{p}, \vec{q})$-Laplacian problems with general right-hand sides. Our main goal is to apply a two critical point result due to Bonanno-D'Aguì ^[37, Theorem 2.1] in order to get the existence of two positive solutions for problem ($D_{\lambda}^{ \vec{p}, \vec{q}}$) with different energy sign. Note that the results in ^[37] are given in a very general setting which can be applied to a large number of problems. Our paper can be seen as an extension of the work of Bonanno-D'Aguì-Sciammetta ^[38] for $(\vec{p}, \vec{q})$-Laplacian problems. But not only the differential operator is more general than in ^[38], also the condition required on $f$ are weaker. Indeed, instead of assuming the Ambrosetti-Rabinowitz condition, we only assume that the nonlinear term on the right-hand side of ($D_{\lambda}^{ \vec{p}, \vec{q}}$) is $(p^+-1)$-superlinear at $\pm \infty$ (with $p^+$ defined in (2.2) replacing $h$ by $p$) and fulfills in addition a suitable behavior at $\pm \infty$, see (H_f). These hypotheses are weaker than the Ambrosetti-Rabinowitz condition. Under these conditions, together with the subcritical growth, we prove the existence of two weak solutions for problem ($D_{\lambda}^{ \vec{p}, \vec{q}}$) that have opposite energy sign related to the energy functional of ($D_{\lambda}^{ \vec{p}, \vec{q}}$).

The paper is organized as follows. In Section 2 we present the main properties of anisotropic Sobolev spaces and consider the main features of the anisotropic $(\vec{p}, \vec{q})$-Laplacian, see Propositions 2.3 and 2.4. Moreover, we recall the main abstract critical point theorem which will play a key role in our treatment, see Theorem 2.7. In Section 3 we first state the precise assumptions on the data of problem ($D_{\lambda}^{ \vec{p}, \vec{q}}$) and prove that the corresponding energy functional fulfills the C-condition. Then we give our main result about the existence of two nontrivial weak solutions of ($D_{\lambda}^{ \vec{p}, \vec{q}}$) (see Theorem 3.3) and some consequences in which the solutions are nonnegative, see Theorems 3.4 and 3.5. Finally, we consider the autonomous problem ($AD_{\lambda}^{ \vec{p}, \vec{q}}$), providing an existence result (see Corollary 3.6) and an example.

2.   Preliminaries and basic properties

Let $\Omega \subset \mathbb{R}^N$, $N\geq 2$, be a bounded domain with boundary $\partial\Omega$ of class $C^1$. For any real vector $\vec{h} = (h_0, h_1, h_2, \ldots, h_N)$ with $h_i \geq 1$ for every $i = 0, 1, \ldots, N$, we indicate with $W^{1, {\vec{h}}}(\Omega)$ the anisotropic Sobolev space defined by

equipped with the norm

Moreover, set

and suppose that $h^- < N$ and $h_0 < (h^-)^* = \frac{N h^-}{N-h^-}$. Denote by $W^{1, {\vec{h}}}_0(\Omega)$ the closure of $C^{\infty}_0(\Omega)$ endowed with the following norm

which is equivalent to the usual one given in (2.1). Indeed, taking into account that $W^{1, {h^-}}(\Omega)$ is compactly embedded in $L^{{h_0}}(\Omega)$ and using Hölder's inequality (see (2.4) in Proposition 2.1), we have that

It is well known that $W^{1, {\vec{h}}}_0(\Omega)$, endowed with the norm defined in (2.3), is a separable Banach space and it is also reflexive if $h_i > 1$ for all $i = 1, \ldots, N$, see Rákosník ^[5, Theorem 1].

Given $\vec{p} = (p_0, p_1, p_2, \ldots, p_N)$ and $\vec{q} = (q_0, q_1, q_2, \ldots, q_N)$, with $p_i, q_i\geq 2$ for every $i = 1, \ldots, N$, we suppose that

(H) $q^+ < p^- < N$, $p_0 < (p^-)^*$ and $q_0 < (q^-)^*$, where $(\cdot)^* = \frac{N (\cdot)}{N - (\cdot)}$ denotes the critical Sobolev exponent.

In the following proposition, we give a relation between the spaces $W^{1, { \vec{p}}}_0(\Omega)$ and $W^{1, { \vec{q}}}_0(\Omega)$ and their norms. In particular, we underline that $p_0$ and $q_0$ are necessary only for the definition of the anisotropic Sobolev spaces, but since we endow the spaces with the equivalent norm given in (2.3), from now on we will only deal with the components $(p_1, \ldots, p_N)$ and $(q_1, \ldots, q_N)$.

Proposition 2.1. If $q^+ < p^-$, then $W^{1, { \vec{p}}}_0(\Omega)\subseteq W^{1, { \vec{q}}}_0(\Omega)$ and

where $\left|\Omega \right|$ is the Lebesgue measure of $\Omega$.

Proof. Fix $u \in W^{1, { \vec{p}}}_0(\Omega)$ and $i \in \left\{1, \ldots, N \right\}$. In particular, $\frac{\partial u}{\partial x_i} \in L^{{p_i}}(\Omega)$ and $\left| \frac{\partial u}{\partial x_i} \right|^{p^-} \in L^{{\frac{p_i}{p^-}}}(\Omega)$. If $p_i > p^-$, by Hölder's inequality, we get

while if $p_i = p^-$, then (2.4) is an equality. Moreover, $\left| \frac{\partial u}{\partial x_i} \right|^{q^+} \in L^{{\frac{p^-}{q^+}}}(\Omega)$, then again from Hölder's inequality, we obtain

Thus, combining (2.4) and (2.5) we derive

for all $i = 1, \ldots, N$. Furthermore, if $q^+ > q_i$ and using Hölder's inequality we have

and if $q^+ = q_i$ the previous inequality becomes an equality. Then, from (2.6) and (2.7) it follows that

Hence,

and the proof is complete. □

In the sequel, we estimate the embedding constant of $W^{1, { \vec{p}}}_0(\Omega)$ into $L^{{r}}(\Omega)$ for each $r \in [1, (p^-)^*]$ with $p^- < N$, where $(p^-)^*$ is the critical Sobolev exponent to $p^-$, that is

Proposition 2.2. If $1 \le p^- < N$, then for any $r\in \left[1, (p^-)^* \right]$, $W^{1, { \vec{p}}}_0(\Omega) \hookrightarrow L^{{r}}(\Omega)$ is continuous and one has

for all $u \in W^{1, { \vec{p}}}_0(\Omega)$, where

see Talenti ^[39, formula (2)] and $\Gamma$ is the Euler function. Moreover, for any $r\in [1, (p^-)^*[$ the embedding $W^{1, { \vec{p}}}_0(\Omega) \hookrightarrow L^{{r}}(\Omega)$ is compact.

Proof. From the Sobolev embedding theorem there exists a positive constant $T\in \mathbb{R}$ such that for all $u\in W^{1, {p^-}}_0(\Omega)$ the following holds

where $(p^-)^*$ and $T$ are defined in (2.8) and (2.10), respectively.

Fix $r\in [1, (p^-)^*]$. Since $\frac{1}{r} = \frac{1}{(p^-)^*}+\frac{(p^-)^*-r}{(p^-)^*r}$, by Hölder's inequality and (2.11), we have

that is

Arguing as in the proof of Proposition 2.1 in Bonanno-D'Aguì-Sciammetta ^[38], we derive

and, taking (2.12) into account, we get that (2.9) holds for every $r\in \left[1, (p^-)^* \right]$.

Finally, combining the continuous embedding $W^{1, { \vec{p}}}_0(\Omega) \hookrightarrow W^{1, p^-}_0(\Omega)$ with the compact embedding $W^{1, p^-}_0(\Omega) \hookrightarrow L^{r}(\Omega)$, it follows that $W^{1, { \vec{p}}}_0(\Omega) \hookrightarrow L^{r}(\Omega)$ is compact for any $r \in [1, (p^-)^*[$. □

Now, we define

and we introduce the functionals $\Phi, \Psi \colon W^{1, { \vec{p}}}_0(\Omega) \to \mathbb{R}$ given by

for every $u \in W^{1, { \vec{p}}}_0(\Omega)$. Clearly, $\Phi$ and $\Psi$ are Gâteaux differentiable and one has

for every $u, v \in W^{1, { \vec{p}}}_0(\Omega)$. Also, we consider the energy functional $I_\lambda\colon W^{1, { \vec{p}}}_0(\Omega) \to \mathbb{R}$ associated to our problem ($D_{\lambda}^{ \vec{p}, \vec{q}}$), that is given by $I_\lambda = \Phi - \lambda \Psi$ for all $\lambda > 0$.

We recall that $u \colon \Omega \to \mathbb{R}$ is a weak solution of problem ($D_{\lambda}^{ \vec{p}, \vec{q}}$) if the following condition holds for all $v \in W^{1, { \vec{p}}}_0(\Omega)$

Then, from (2.14) it follows that $u \in W^{1, { \vec{p}}}_0(\Omega)$ is a weak solution of problem ($D_{\lambda}^{ \vec{p}, \vec{q}}$) if and only if $u$ is a critical point for $I_\lambda$. Consequently, our study is based on critical point theory and in particular on a critical point theorem due to Bonanno-D'Aguì ^[37] that we state later in Theorem 2.7.

In the following, we deal with some properties of the Gâteaux derivative of the functional $\Phi$ that are needed in our investigation.

Proposition 2.3. The functional $\Phi'\colon W^{1, { \vec{p}}}_0(\Omega) \to \left(W^{1, { \vec{p}}}_0(\Omega) \right)^*$ defined in (2.14) is monotone and coercive.

Proof. First we prove that $\Phi'$ is monotone, i.e.

To this end, we use the following inequality

for each $r \geq 2$ and for some constant $C > 0$, see Simon ^[40] or Lindqvist ^[41]. Indeed, using the previous inequality we have

and (2.15) is achieved.

Now, we prove that $\Phi'$ is coercive. We observe that

Moreover, let $j \in \{1, \ldots, N\}$ be such that

Then,

Thus, we get

From (2.16) and (2.17), we derive

namely

and this implies that $\Phi'$ is coercive. □

Proposition 2.4. The map $\Phi'\colon W^{1, { \vec{p}}}_0(\Omega)\to \left(W^{1, { \vec{p}}}_0(\Omega) \right)^*$ has the $({\rm S}_+)$-property, that is

then $u_n\to u$ in $W^{1, { \vec{p}}}_0(\Omega)$.

Proof. Let $\{u_n\}_{n \in \mathbb{N}} \subset W^{1, { \vec{p}}}_0(\Omega)$ be such that $u_n \rightharpoonup u$ in $W^{1, { \vec{p}}}_0(\Omega)$ and

First, we observe that

since $\Phi'(u)$ is a linear operator in $W^{1, { \vec{p}}}_0(\Omega)$ and $u_n \rightharpoonup u$ in $W^{1, { \vec{p}}}_0(\Omega)$. Hence, from (2.18) and (2.19) it follows that

Now, for all $i = 1, \ldots, N$ and for all $w, v \in W^{1, { \vec{p}}}_0(\Omega)$ we set

and

Then, we can write $\Phi'$ as follows

By (2.20), we get

Moreover, we have

Also, applying Hölder's inequality one has

Hence, we derive that

which, taking (2.21) into account, implies that

Then, from Papageorgiou-Winkert ^[42, Proposition 4.1.11], since $L^{{p_i}}(\Omega)$ is uniformly convex, one has

Thus, it follows that

and our claim is proved. □

Finally, we point out the following result in order to get information on the sign of the solutions of ($D_{\lambda}^{ \vec{p}, \vec{q}}$). For this purpose, let

for all $(x, t) \in \Omega \times \mathbb{R}$ and consider the following problem

Lemma 2.5. Assume that $f(x, 0)\geq 0$ for a.a. $x \in \Omega$. Then, any weak solution of problem ($D_{\lambda, f^{+}}^{ \vec{p}, \vec{q}}$) is nonnegative and it is also a weak solution of problem ($D_{\lambda}^{ \vec{p}, \vec{q}}$).

Proof. Let $u_0$ be a weak solution of problem ($D_{\lambda, f^{+}}^{ \vec{p}, \vec{q}}$), namely

for all $v \in W^{1, { \vec{p}}}_0(\Omega)$.

In order to prove that $u_0$ is nonnegative, put $A = \left\{x \in \Omega \, :\, u_0(x) < 0 \right\}$ and $u_0^{-} = \min \left\{u_0, 0 \right\}$. Clearly, $u_0^{-} \in W^{1, { \vec{p}}}_0(\Omega)$ (see, for example, Papageorgiou-Winkert ^[42, Corollary 4.5.19]). Choosing $v = u_0^{-}$ in (2.22), one has

that is,

Hence, it holds that

which yields

for all $i = 1, \ldots, N$. Therefore, we obtain

so $u_0^- = 0$ in $\Omega$, which means $u_0\ge0$ in $\Omega$.

Now, we prove that $u_0$ is a weak solution for problem ($D_{\lambda}^{ \vec{p}, \vec{q}}$). Indeed, from (2.22) one has

for all $v \in W^{1, { \vec{p}}}_0(\Omega)$, and the conclusion is achieved. □

The proofs of our main results are based on the following two critical point theorem due to Bonanno-D'Aguì ^[37, see Theorem 2.1 and Remark 2.2]. First we recall the definition of the Cerami condition.

Definition 2.6. Let $X$ be a Banach space and $X^*$ be its topological dual space. Given $I_\lambda\in C^1(X)$, we say that $I_\lambda$ satisfies the Cerami-condition ($\text{C}$-condition for short), if every sequence $\{u_n\}_{n\in \mathbb{N}}\subseteq X$ such that

(C₁) $\left\lvert {I_\lambda(u_n)} \right\rvert \le c_1$ for some $c_1 > 0$ and for all $n \in \mathbb{N}$,

(C₂) $\left(1+\|u_n\|_X \right)I'_\lambda(u_n)\to 0$ in $X^*$ as $n\to \infty$,

admits a strongly convergent subsequence in $X$.

Theorem 2.7. Let $X$ be a real Banach space and let $\Phi$, $\Psi \colon X \to \mathbb{R}$ be two functionals of class $C^1$ such that $\inf_X \Phi(u) = \Phi(0) = \Psi(0) = 0$. Assume that there are $r \in \mathbb{R}$ and $\tilde{u} \in X$, with $0 < \Phi(\tilde{u}) < r$, such that

and, for each

the functional $I_{\lambda} = \Phi - \lambda \Psi$ satisfies the $\text{C}$-condition and it is unbounded from below. Moreover, $\Phi$ is supposed to be coercive.

Then, for each $\lambda \in \tilde{\Lambda}$, the functional $I_{\lambda}$ admits at least two nontrivial critical points $u_{\lambda, 1}$, $u_{\lambda, 2}\in X$ such that $I(u_{\lambda, 1}) < 0 < I(u_{\lambda, 2})$.

3.   Main results

In this section we formulate and prove our main results. We suppose the following assumptions on the nonlinearity.

(H_f) $f\colon\Omega\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function such that the conditions below are satisfied:

(f₁) there exist $\alpha < (p^-)^*$ and constants $a_1, a_2 > 0$ such that

for a.a. $x\in\Omega$ and for all $t\in \mathbb{R}$;

(f₂) if $F(x, s) = \int_0^sf(x, \xi)\, {\rm d} \xi$, then

uniformly for a.a. $x\in\Omega$;

(f₃) there exist $\beta, \gamma \in \mathbb{R}$, with

such that

uniformly for a.a. $x \in \Omega$, and

uniformly for a.a. $x \in \Omega$.

Remark 3.1. We observe that from hypotheses (f₁) and (f₂) it follows that

Furthermore, we emphasize that such $\beta$ and $\gamma$ in (f₃) exists, since

Finally, we underline that in this context it is possible to choose two different exponents $\beta$ and $\gamma$ for going to $+\infty$ and $- \infty$, respectively.

In the following we give a preliminary result on the energy functional associated to our problem.

Proposition 3.2. Let hypotheses (H) and (H_f) be satisfied. Then the functional $I_{\lambda}\colon W^{1, { \vec{p}}}_0(\Omega) \to \mathbb{R}$ satisfies the $\text{C}$-condition for each $\lambda > 0$.

Proof. Let $\{u_n\}_{n\in \mathbb{N}}\subseteq W^{1, { \vec{p}}}_0(\Omega)$ be a sequence such that (C₁) and (C₂) hold, see Definition 2.6. First, we prove that $\{u_n\}_{n \in \mathbb{N}}$ is bounded in $L^{{\beta}}(\Omega)$.

From (C₂), we get

for all $v\in W^{1, { \vec{p}}}_0(\Omega)$ and with $\varepsilon_n\to 0^+$. Fix $n \in \mathbb{N}$ and choose $v = u_n\in W^{1, { \vec{p}}}_0(\Omega)$. Substituting in (3.1), we derive

for all $n\in \mathbb{N}$. From (C₁) we have

Adding (3.2) and (3.3) and taking into account that $q^+ < p^+$, we obtain

for some $c_2 > 0$ and for all $n\in \mathbb{N}$.

Without loss of generality, we may assume $\beta \le \gamma$. Then, assumptions (f₁) and (f₃) imply that there exist $c_3 \in (0, m)$ and $c_{4} > 0$ such that

for a.a. $x\in\Omega$ and for all $s\in \mathbb{R}$. Exploiting this in (3.4), we derive

namely $\left\{u_n \right\}_{n\in \mathbb{N}}$ is bounded in $L^{{\beta}}(\Omega)$.

Now, we prove that $\{u_n\}_{n \in \mathbb{N}}$ is bounded in $W^{1, { \vec{p}}}_0(\Omega)$.

From hypotheses (f₁) and (f₃) it follows that $\beta < \alpha < (p^-)^*$. Hence, there exists $t\in (0, 1)$ such that

By the interpolation inequality (see Papageorgiou-Winkert ^[42, Proposition 2.3.17]), we get

Since $\left\{u_n \right\}_{n\in \mathbb{N}}$ is bounded in $L^{{\beta}}(\Omega)$, using also Proposition 2.2, one has

with some $c_{6} > 0$. Choosing $v = u_n\in W^{1, { \vec{p}}}_0(\Omega)$ in (3.1), we have

for all $n\in \mathbb{N}$. By (2.17), (f₁) and (3.6) we obtain that there exists $j \in \{1, \ldots, N\}$ such that

for some $c_7 > 0$ and for all $n\in \mathbb{N}$.

From (3.5) and (f₃) follows that $t\alpha < p_j$, indeed

Then, (3.7) and (3.8) imply that $\{u_n\}_{n\in \mathbb{N}}\subseteq W^{1, { \vec{p}}}_0(\Omega)$ is bounded.

Finally, we prove that $\left\{u_n \right\}_{n\in \mathbb{N}}$ admits a strongly convergent subsequence in $W^{1, { \vec{p}}}_0(\Omega)$. Because of the boundedness of $\{u_n\}_{n\in \mathbb{N}}\subseteq W^{1, { \vec{p}}}_0(\Omega)$, there exists a subsequence, not relabeled, such that

We choose $v = u_n-u\in W^{1, { \vec{p}}}_0(\Omega)$ in (3.1). Passing to the limit as $n\to \infty$ and using (3.9), we derive

Since $\Phi'$ has the $({\rm S}_+)$-property (see Proposition 2.4), it follows that $u_n\to u$ in $W^{1, { \vec{p}}}_0(\Omega)$ and this completes the proof. □

Now, we present our main result. To this aim, put

Standard computations show that there exists $x_0 \in \Omega$ such that $B(x_0, R) \subseteq \Omega$ and we denote by

the measure of the $N$-dimensional ball of radius $R$. Finally, we set

Theorem 3.3. Let hypotheses (H) and (H_f) be satisfied. Assume that there exist two constants $r, d > 0$ such that

(h₁) $\mathcal{K}\sum_{i = 1}^N(d^{p_i}+d^{q_i}) < r$,

(h₂) $F(x, s)\geq 0$ for a.a. $x\in\Omega$ and for all $s\in [0, d]$,

(h₃) $\dfrac{a_1 T_1 \delta + \dfrac{a_2}{\alpha} T_{\alpha}^{\alpha} \delta^\alpha}{r} < \dfrac{1}{\mathcal{K}} \dfrac{ \int_{B \left(x_0, \frac{R}{2} \right)}F(x, d)\, {\rm d} x}{\sum_{i = 1}^N(d^{p_i}+d^{q_i})}$,

where $a_1, a_2, \alpha$ are given in (f₁), $T_1$ and $T_{\alpha}$ are defined by formula (2.10), $R$ is given in (3.10), and $\mathcal{K}$ as well as $\delta$ are defined by (3.11) and (3.12), respectively.

Then, for each

problem ($D_{\lambda}^{ \vec{p}, \vec{q}}$) has at least two nontrivial weak solutions $u_\lambda, v_\lambda \in W^{\, \vec{p}}_0(\Omega)$ such that $I_\lambda(u_\lambda) < 0 < I_\lambda(v_\lambda)$.

Proof. Our aim is to apply Theorem 2.7 for $X = W^{1, { \vec{p}}}_0(\Omega)$ and $\Phi$, $\Psi$ defined as in (2.13). The functionals $\Phi$ and $\Psi$ satisfy all the required regularity properties, since $\Phi$ is coercive by construction (see Proposition 2.3), the energy functional $I_\lambda$ satisfies the $\text{C}$-condition due to Proposition 3.2 and it is unbounded from below by (f₂) and finally

Moreover, the interval $\Lambda$ is nonempty because of assumption (h₃). Thus, it remains to verify hypothesis (2.23). First, we observe that

Then, from (f₁), (2.9) and (3.13) we estimate that

On the other hand, we introduce the following function

where $R$ is given in (3.10). Clearly, $\tilde{u} \in W^{1, { \vec{p}}}_0(\Omega)$. From assumption (h₂), we get

Furthermore, it holds

Hypothesis (h₁) implies that $0 < \Phi(\tilde{u}) < r$ and combining (3.14), (3.15) as well as (3.16) we have

This proves (2.23). Hence, since $\Lambda \subseteq \tilde{\Lambda}$, Theorem 2.7 ensures that problem ($D_{\lambda}^{ \vec{p}, \vec{q}}$) admits at least two nontrivial weak solutions $u_\lambda, v_\lambda \in W^{1, { \vec{p}}}_0(\Omega)$ with opposite energy sign. □

The following result is a consequence of Lemma 2.5 and of Theorem 3.3.

Theorem 3.4. Let hypotheses (H) and (H_f) be satisfied. Assume (h₁)–(h₃) and that $f(x, 0)\ge 0$ for a.a. $x \in \Omega$. Then, for every

problem ($D_{\lambda}^{ \vec{p}, \vec{q}}$) has at least two nonnegative weak solutions $u_\lambda, v_\lambda \in W^{\, \vec{p}}_0(\Omega)$ such that $I_\lambda(u_\lambda) < 0 < I_\lambda(v_\lambda)$.

Now, we consider the particular case in which the nonlinearity is nonnegative.

Theorem 3.5. Let hypotheses (H) and (H_f) be satisfied. Assume that $f$ is nonnegative and

(h₄) $\limsup\limits_{t \to 0^+} \dfrac{\inf\limits_{x \in \Omega}F(x, t)}{t^{q^-}} = + \infty$

Then, for each $\lambda \in ]0, \lambda^*[$, with

problem ($D_{\lambda}^{ \vec{p}, \vec{q}}$) admits at least two nonnegative weak solutions $u_\lambda$, $v_\lambda \in W^{1, { \vec{p}}}_0(\Omega)$ such that $I_\lambda(u_\lambda) < 0 < I_\lambda(v_\lambda)$.

Proof. Fix $\lambda \in ]0, \lambda^*[$, then there exists $r > 0$ such that

From (h₄) follows that

since

Then, in correspondence of $\frac{1}{\lambda}$, there exists $t > 0$ small enough such that

namely assumption (h₃) is satisfied. Since (h₂) follows from the sign assumption on the nonlinearity, we can apply Theorem 3.3 and Lemma 2.5 to complete the proof. □

Finally, we deal with the autonomous case and we present an existence result which is a consequence of Theorem 3.5. Consider the autonomous problem

where $g\colon \mathbb{R}\to \mathbb{R}$ is a nonnegative continuous function. From Lemma 2.5 it follows that we can consider the nonlinearity only in $[0, + \infty)$. We assume the following:

(H_g) (g₁) there exist $\alpha < (p^-)^*$ and constants $a_1, a_2 > 0$ such that

for all $t\ge 0$;

(g₂) if $G(s) = \int_0^s g(\xi)\, {\rm d} \xi$, then

(g₃) there exists $\beta \in \mathbb{R}$, with

such that

The following result holds.

Corollary 3.6. Let hypotheses (H) and (H_g) be satisfied. Assume that

(h₄')        $\limsup\limits_{t \to 0^+} \dfrac{G(t)}{t^{q^-}} = + \infty$.

Then, for each $\lambda \in ]0, \lambda^*[$, with $\lambda^*$ defined in (3.17), the problem $(AD_{\lambda}^{ \vec{p}, \vec{q}})$ admits at least two nonnegative weak solutions $u_\lambda$, $v_\lambda \in W^{1, { \vec{p}}}_0(\Omega)$ such that $I_\lambda(u_\lambda) < 0 < I_\lambda(v_\lambda)$.

In conclusion, we provide an example.

Example 3.7. Consider two constants $c, \kappa$ such that

Let $g \colon [0, + \infty) \to \mathbb{R}$ be a function defined by

Then, $g$ satisfies assumptions (H_g) with $\beta = \kappa$ and $\alpha = \kappa+\sigma$, with $\sigma > 0$ small enough such that

Moreover, the function $g$ satisfies assumption (h₄’), hence we can apply Corollary 3.6 to get the existence of two nonnegative weak solutions of problem ($AD_{\lambda}^{ \vec{p}, \vec{q}}$) with opposite energy sign.

Acknowledgments

The first two authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The authors have been partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The paper is partially supported by PRIN 2017 – Progetti di Ricerca di rilevante Interesse Nazionale, "Nonlinear Differential Problems via Variational, Topological and Set-valued Methods" (2017AYM8XW) and by FFR-2023-Sciammetta.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare there is no conflict of interest.