Long-time dynamical behavior for a piezoelectric system with magnetic effect and nonlinear dampings

1.   Introduction

We consider an elliptic equation with nonlinear boundary condition of the form

where $\Omega \subset \mathbb{R}^{N} (N \geq 2)$ is a bounded domain with $C^{2, \alpha}$ ($0 < \alpha < 1$) boundary $\partial \Omega$, and $\partial/ \partial\eta : = \eta(x)\cdot\nabla$ denotes the outer normal derivative on the boundary $\partial \Omega$. Here $f: \partial \Omega \times \mathbb{R} \to \mathbb{R}$ is a Carathéodory function, that is, $f(\cdot, u)$ is measurable for each $u$ and $f(x, \cdot)$ is continuous for a.e. $x \in \partial \Omega$.

In this paper, we investigate the existence of maximal and minimal weak solutions (to be clarified later) between an ordered pair of sub- and supersolution of (1.1) for both monotone and nonmonotone nonlinearities. We use monotone iteration procedure when the nonlinearity is monotone. The nonmonotone case required a careful use of the surjectivity of a bounded, pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality up to the boundary. This proof, for the nonmonotone case, is motivated by the works in ^[1] and ^[2].

Elliptic equations with nonlinear boundary conditions have attracted a lot of attention over the last decades, see for instance ^{[3,4,5,6,7,8,9]} and references therein. Motivation to study equations with nonlinear boundary conditions stems from the fact that, when the reaction near the boundary depends on the density itself, linear boundary conditions (Dirichlet, Neumann, or Robin) are often inadequate to study chemical, biological, or ecological processes, see ^{[10,11,12,13]} and references therein, for specific applications.

The existence of a solution between an ordered pair of sub- and supersolution of elliptic boundary value problems has been studied extensively. For the linear boundary conditions, the sub–supersolution method for classical solutions were developed in ^[14,15,16] to study the solvability of quasi-linear and semi-linear equations using monotone iteration method. This method also yields the existence of a maximal and a minimal solution. These iterative methods can be thought as a generalization of the Perron arguments on sub- and superharmonic functions for existence of solutions of the boundary value problem. For relatively recent results on the existence of maximal and minimal solutions, for the linear boundary conditions, we refer readers to ^[2,17,18] for the Laplacian case, and ^[1,19] for the $p$-Laplacian case.

For the nonlinear boundary case, see ^[20] and [13,Ch. 4] where the existence of maximal and minimal classical solutions was established for the monotone case. To the best of our knowledge, our results concerning the existence of maximal and minimal weak solutions are new for both monotone and nonmonotone cases.

We begin with the definitions of weak solution and weak sub- and supersolution. For this, we make use of the real Lebesgue space $L^{r}(\partial \Omega)$ and the Sobolev space $H^1(\Omega)$.

Definition 1.1. We say that a function $u \in H^1(\Omega)$ is a weak solution to (1.1) whenever:

(i) $f(., u(.)) \in L^\frac{2(N-1)}{N}(\partial \Omega)$ if $N > 2$, and $f(., u(.)) \in L^{r}(\partial \Omega)$ for $r > 1$ if $N = 2$, and

(ii) $\int_{ \Omega}\left(\nabla u \nabla \psi + u\psi\right) = \int_{ \partial \Omega}f(x, u)\psi \qquad \rm{for all }\ \psi \in H^1(\Omega).$

Definition 1.2. We say that a function $\overline{u} \in H^1(\Omega)$ is a weak supersolution to (1.1) whenever:

(i) $f(., \overline{u}(.)) \in L^\frac{2(N-1)}{N}(\partial \Omega)$ if $N > 2$, and $f(., \overline{u}(.)) \in L^{r}(\partial \Omega)$ for $r > 1$ if $N = 2$, and

(ii) $\int_{ \Omega}\left(\nabla \overline{u} \nabla \psi + \overline{u}\psi\right) \geq \int_{ \partial \Omega}f(x, \overline{u})\psi\quad \mbox{ for all } 0\leq \psi \in H^1(\Omega)$.

A weak subsolution $\underline{u}$ is defined by reversing the inequality in (ii) above.

Remark 1.3. Let $\Gamma : H^{1}(\Omega)\to L^{r}(\partial \Omega)$ be the trace operator given by $\Gamma u = u|_{ \partial \Omega}$. It is known that, see e.g. ^[21], [22,Thm 2.79], and [23,Chapter 6], $\Gamma$ is continuous (compact) if

Therefore, the integrals on the right hand side of (ii) of Definition 1.1 and Definition 1.2 make sense since (i) holds, and $\frac{2(N-1)}{N}$ is the conjugate of $\frac{2(N-1)}{N-2}$ when $N > 2$.

We state and prove our results for the case $N > 2$, since the case $N = 2$ follows clearly using (1.2). We state our first result concerning maximal and minimal solutions for the monotone case.

Theorem 1.4. Suppose there exists a pair of weak sub- and supersolution $\underline{u}$ and $\overline{u}$, respectively, satisfying $\underline{u} \leq \overline{u}$ in $\overline{\Omega}$. Assume that

(H1) there exists $\ k \ge 0$ such that the map $s \mapsto f(x, s) + ks$ is nondecreasing for all $\underline{u} \leq s \leq \overline{u}$, and for all $x\in\partial \Omega$.

Then, there exist a minimal weak solution $u_{*}$ and a maximal weak solution $u^{*}$ to (1.1), in the sense that if $u$ is any weak solution to (1.1) such that $\underline{u} \leq u \leq \overline{u}$, then $u_{*} \leq u \leq u^{*}$.

Next, we note that if $f$ is locally Lipschitz with respect to the second variable $u$, and the interval $[\underline{u}, \overline{u}]$ is bounded, then $f$ satisfies the hypothesis (H1). For functions $f$ that do not satisfy the monotonicity condition given in (H1), we have the following existence result.

Theorem 1.5. Suppose there exists a pair of weak sub- and supersolution $\underline{u}$ and $\overline{u}$, respectively, satisfying $\underline{u} \leq \overline{u}$ in $\overline{\Omega}$. Assume that

(H2) there exists a $K\in L^r(\partial\Omega)$, $r > \frac{2(N-1)}{N}$, such that $|f(x, s)|\leq K(x)$ a.e. $x\in \partial\Omega$, for all $s$ satisfying $\underline{u}(x)\leq s\leq \overline{u}(x)$.

Then (1.1) has at least one weak solution $u$ such that $\underline{u} \leq u \leq \overline{u}$.

Finally, we state a result that guarantees the existence of a maximal and a minimal weak solution without assuming monotonicity condition (H1) on the nonlinearity $f$.

Theorem 1.6. Assume hypotheses of Theorem 1.5 hold. Then, there exist a minimal weak solution $u_{*}$ and a maximal weak solution $u^{*}$ to (1.1)^{, in the sense that if $u$ is any weak solution to} (1.1) such that $\underline{u} \leq u \leq \overline{u}$, then $u_{*} \leq u \leq u^{*}$.

In ^[24], Hess proved the existence of a solution, assuming that

for $q = 2$. Our result, Theorem 1.4, is sharper, needing only that condition (1.3) hold for $q = \frac{2(N-1)}{N} = 2-\frac{2}{N} < 2$.

In Section 2, we collect some known results that will be helpful in the sequel. We also state and prove a version of Kato's inequality for our setting, see Theorem 2.4 and Corollary 2.5. In Section 3, we prove Theorem 1.4 using monotone iteration method. In Section 4, we prove Theorem 1.5 by showing that an appropriately defined operator is surjective. We also prove Theorem 1.6 in Section 4 by utilizing Theorem 1.5, Zorn's Lemma and Theorem 2.4. In Section 5, we discuss applications of our results.

2.   Preliminaries and auxiliary results

Here we collect some results that we use in the sequel. First, we recall an existence and uniqueness result for a linear problem.

Proposition 2.1. (^[4,8]) Let $h \in L^q(\partial\Omega)$ for $q \geq 1$. Then, the linear problem

has a unique solution $v \in W^{1, m}(\Omega)$ and

In particular, if $q = \frac{2(N-1)}{N}$, then $u\in H^{1}(\Omega)$.

Next, let $X$ be a reflexive Banach space and $A: X \to X^*$. We say that the operator $A$ is coercive if

We say that $A$ is pseudomonotone, whenever

We will utilize the following surjectivity result in the proof of Theorem 1.5.

Proposition 2.2. ([25,Thm. II. 2.8], [22,Thm. 2.99]) Let $X$ be a reflexive Banach space. If $A: X \to X^*$ is a bounded, pseudomonotone and coercive operator, then for each $b \in X^*$, $Au = b$ has a solution.

Finally, we say that a subset $Y$ of a partially ordered set $(X, \le)$ is a chain if $x \le y$ or $y \le x$ for every $x, y \in Y$. Then, to prove Theorem 1.6, we use the following version of Zorn's lemma (see ^[22]):

Proposition 2.3 (Zorn's lemma). If in a partially ordered set $(X, \leq)$, every chain $Y$ has an upper bound, then $X$ possesses a maximal element.

2.1.   A version of Kato's inequality

In ^[26], authors established Kato's inequality up to the boundary for a function $u \in W^{1, 1}(\Omega)$. Here, we state and prove a version of Kato's inequality up to the boundary, that is necessary in the proof of Theorem 1.6. This result can be rephrased as the maximum of two weak subsolutions is also a weak subsolution. In particular, the maximum of two weak solutions is a weak subsolution.

Theorem 2.4. Let $u_1$ and $u_2$ be functions in $H^1(\Omega)$ such that there exist $f_1$ and $f_2$ in $L^r(\partial \Omega)$, for $r\ge \frac{2(N-1)}{N}$, satisfying

for $i = 1, 2.$ Then, $u: = \max\{u_1, u_2\}$ satisfies

where $f(x): = \begin{cases} f_1(x) & \quad \mathit{\text{if}} \ u_1(x) > u_2(x)\\ f_2(x) & \quad \mathit{\text{if}} \ u_1(x)\le u_2(x), \end{cases}$ a.e. $x\in \partial \Omega$.

Proof. Define

and

Fix $0 \le \psi \in H^1(\Omega)$. Then,

Consider a sequence $\xi_n\in C^1(\mathbb{R})$ such that

and $\xi_n' > 0$ on $(0, 1/n).$ Then, define the sequence of functions

Observe that $r_n\in H^1(\Omega)$ and $r_n$ converges pointwise to $\chi_{ \Omega_1\cup \Gamma_1}$, where the characteristic function is defined as $\chi_{ \Omega_1\cup \Gamma_1}(x): = \begin{cases} 1 \quad \text{if} \ x\in \Omega_1\cup\Gamma_1\; \\ 0 \quad \text{if} \ \text{otherwise}\, . \end{cases}$ Moreover, $||r_n||_{L^{\infty}(\Omega)\cap {L^{\infty}(\partial \Omega)}}\le 1$ and $\text{supp}(\nabla r_n)\subset \overline{D}_n$, where $D_n: = \{x\in \Omega: 0 < u_1(x)-u_2(x) < \frac{1}{n}\}$. Then, using Lebesgue Dominated Convergence Theorem, we have that

Since $r_n\in H^1(\Omega)\cap L^{\infty}(\Omega)\cap {L^{\infty}(\partial \Omega)}$, it follows that $r_n\psi\in H^1(\Omega)$ for any test function $\psi\in H^1(\Omega)\cap L^{\infty}(\Omega)$. Recalling that $\nabla r_n = 0$ on $\Omega\setminus D_n$, and that $u_1$ satisfies (2.2), we can write

Taking the limit as $n\to \infty$ in the first term of the right-hand side of (2.3), using the Lebesgue Dominated Convergence Theorem, we get

Likewise, for $I_2$ we have

and

Taking the limit as $n\to \infty$ in the first term of the right-hand side of (2.4) and using the Lebesgue Dominated Convergence Theorem, we get

Using the fact that $\nabla r_n = \xi_n'(u_1-u_2)\nabla(u_1-u_2)$, the sum of the second terms of the right-hand side of (2.3) and (2.4) yields

since $\psi\ge 0\; {\text{and}}\; \xi_n'\ge 0$. Adding (2.3) and (2.4), taking the limit, and using (2.5), we get

Thus, $u: = \max\{u_1, u_2\}$ satisfies

completing the proof of Theorem 2.4.

Likewise, we have a result for the minimum of two supersolutions.

Corollary 2.5. Let $u_1$ and $u_2$ be functions in $H^1(\Omega)$ such that there exist $f_1$ and $f_2$ in $L^r(\partial \Omega)$, for $r\ge \frac{2(N-1)}{N}$, satisfying

for $i = 1, 2.$ Then, $u: = \min\{u_1, u_2\}$ satisfies

where

Proof. Using the fact that $\min\{u_1, u_2\} = \max\{-u_1, -u_2\}$, the proof follows from Theorem 2.4.

3.   Proof of Theorem 1.4

We will construct a monotone operator, and show that the iterative scheme starting with a weak subsolution (supersolution) will converge to a minimal (maximal) weak solution.

Let $J: = \{u\in H^1(\Omega):\underline{u} \leq u \leq \overline{u} \}$. Define the linear map $T: J\to H^1(\Omega)$ by $T(u) = v$, where $v$ satisfies

Step 1. $T$ is well-defined and maps $J$ into itself.

For every $u \in J$, we have $\underline{u} \leq u \leq \overline{u}$. Then using (H1) and the fact that $\underline{u}$ and $\overline{u}$ are sub and supersolutions, we get

and

Taking into account the definitions of $\underline{u}$ and $\overline{u}$, we have that $f(., \underline{u}(.)), \; f(., \overline{u}(.))$ are in $L^{\frac{2(N-1)}{N}}(\partial \Omega)$. Since $\underline{u}, \; \overline{u}\in H^1(\Omega)$, then by the continuity of the trace operator (1.2) and the embedding of $L^{\frac{2(N-1)}{N-2}}(\partial \Omega)$ into $L^{\frac{2(N-1)}{N}}(\partial \Omega)$, for every $u \in J$, we have

Therefore, $f(., u(.)) +ku(.) \in L^{\frac{2(N-1)}{N}}(\partial \Omega)$. Then, Proposition 2.1 implies that $v = T(u)\in H^1(\Omega)$ is unique. Thus, the map $T$ is well-defined.

Further, if $u, w \in J$ with $u \leq w$, then by the weak maximum principle and the fact that $f$ satisfies (H1), $T(u) \leq T(w)$, that is, the map $T$ is nondecreasing. Moreover, repeating the argument and using Definition 1.2(ii), it follows that

Hence, $T$ maps $J$ to $J$.

Step 2. There exist weakly convergent monotone sequences in $H^1(\Omega)$.

Let's construct monotone sequences $\{u_n\}$ and $\{w_n\}$ successively from the (linear) iteration process

Using (3.2) and the monotonicity of $T$, we get

We show that $\{u_n\}$ is convergent. The proof for $\{w_n\}$ is analogous. We see that $u_n = T(u_{n-1})$ satisfies

for all $\psi \in H^1(\Omega).$ Letting $u_n = T(u_{n-1})$ as a test function, we get

Since $u_{n-1}, u_n \in J$, using Hölder's inequality in (3.4), and the bound (3.1), we have

Hence, there exists a uniform constant $C' > 0$, depending on $\Omega, \ f, \ k$, $\underline{u}$ and $\overline{u}$, such that

By the reflexivity of $H^1(\Omega)$, (3.5), there is a subsequence (relabeled) $u_n$ which converges weakly to $u_*$ in $H^1(\Omega)$.

Step 3. $f(x, u_{n})+k u_{n}$ converges weakly to $f(x, u_*)+k u_*\,$ in $L^\frac{2(N-1)}{N}(\partial \Omega)$.

Since the sequence $u_n$ in Step 2 is nondecreasing and bounded (see (3.3)), it converges pointwise to $u_*$, that is,

Using the fact that $f$ is continuous in the second variable $u$ for a.e $x\in \partial \Omega$ and (3.6), we have that

By (3.1), $f(x, u_n)+ku_n$ is bounded in $L^\frac{2(N-1)}{N}(\partial \Omega)$. Then, Lebesgue Dominated Convergence Theorem yields

Therefore, $f(x, u_{n})+k u_{n}$ converges strongly (hence weakly) to $f(x, u_*)+k u_*\,$ in $L^\frac{2(N-1)}{N}(\partial \Omega)$. Thus, for all $\psi \in H^1(\Omega)\subset L^{\frac{2(N-1)}{N-2}}(\partial \Omega)$, we have

Step 4. $u_*$ is a weak solution to (1.1).

First, since $u_*\in H^1(\Omega)$, the continuity of the trace (1.2) and $L^{\frac{2(N-1)}{N-2}}(\partial \Omega) \hookrightarrow L^{\frac{2(N-1)}{N}}(\partial \Omega),$ imply that $u_*\in L^{\frac{2(N-1)}{N}}(\partial \Omega)$. Therefore, for some positive constant $C''$, we have

Second, from the monotone iteration, we know that $u_n = T(u_{n-1})$ satisfies

Observe that $u_n$ converges weakly to $u_*$ in $H^1(\Omega)$, strongly in $L^2(\partial \Omega)$ (see Step 2) and $f(x, u_n)+ku_n \; \; {\text {converges weakly to }} f(x, u_*)+ku_{*}\; \; {\text{in }}\; \; L^\frac{2(N-1)}{N}(\partial \Omega)$ (see Step 3). Then taking the limit as $n\to \infty$ and using (3.7), we get for any $\psi \in H^1(\Omega)$

Hence,

Moreover, we also have $f(x, u_*) \in L^\frac{2(N-1)}{N}(\partial \Omega)$. Thus $u_*$ is a weak solution to (1.1).

Step 5. $u_*$ is the minimal weak solution in the interval $[\underline{u}, \overline{u}]$.

Let $v$ be a weak solution to (1.1) with $\underline{u} \leq v \leq \overline{u}$. Then $v$ is a weak supersolution, and $\underline{u} \leq v$. Repeating the above iteration procedure with $u_0 = \underline{u}$, we get $\underline{u}\le u_*\le v$. Thus $u_*$ is a weak minimal solution.

Similarly, we can construct the maximal weak solution $u^*$ from the sequence $\{w_n\}$ with $w_0 = \overline{u}$. This completes the proof of Theorem 1.4.

4.   Proofs of Theorem 1.5 and Theorem 1.6

We prove Theorem 1.5 by applying Proposition 2.2 to an appropriate operator related to our problem (1.1). Then, Theorem 1.6 is proved by using Zorn's lemma and Theorem 2.4. Theorem 1.5 guarantees that the set defined for the Zorn's lemma in the proof of Theorem 1.6 is nonempty.

4.1.   Proof of Theorem 1.5

Let us consider a modified problem

where

is the truncated function. We observe that $g$ is a Carathéodory function, since $f$ is a Carathéodory function. We note that a weak solution $u$ of (4.1) is a weak solution of (1.1) whenever $\underline{u} \leq u \leq \overline{u}$.

Our plan is to establish the existence of a weak solution $u$ of (4.1), and verify that $\underline{u} \leq u \leq \overline{u}$. For the existence part, we use Proposition 2.2. For this, we define the map $B\colon H^1(\Omega)\to \big(H^{1}(\Omega)\big)^*$ given by

for all $\psi \in H^1(\Omega)\, .$

First, we show that $B$ is well-defined and bounded. The first integral of (4.3) is well-defined since $v, \psi \in H^1(\Omega)$. By the Hölder's inequality combined with the continuity of trace operator (1.2) and hypothesis (H2), we get

where $r' < \frac{2(N-1)}{N-2}$ is the conjugate of $r$. Then, the definition of $g$ given in (4.2), Definition 1.2(i), and (4.4) yield

where the last inequalities of (4.5) follow by (4.4) and (1.2), and the constant $C_2$ depends only on $K$ and $\Omega$.

Second, we show that $B$ is pseudomonotone, see definition (2.1). For this, we set $B = L-G$, where $L, G: H^1(\Omega) \to (H^1(\Omega))^*$ are defined by

for all $\psi \in H^1(\Omega)$. Then we show that $B$ is pseudomonotone in the following steps. Let $v_n\rightharpoonup v$ in $H^1(\Omega)$.

Step 1: $Lv_n\to Lv$ in $(H^1(\Omega))^*$.

Since $v_n\rightharpoonup v$, $\langle L(v_n) - L(v), \psi\rangle \to 0$ as $n \to \infty$ for all $\psi \in H^1(\Omega)$. Hence,

as desired.

Step 2: $G(v_n)\to G(v)$ in $(H^1(\Omega))^*$.

Suppose that $v_n\rightharpoonup v$ in $H^1(\Omega)$ but $G(v_n)\not\to G(v)$ in $(H^1(\Omega))^*$. Then there exists $\varepsilon_0 > 0$ and a subsequence $\{v_{n_j}\}$ such that

Using the fact that $\{v_{n_j}\}$ is bounded in $H^1(\Omega)$ and the compactness of the trace operator (1.2), there exists a subsequence $\{v_{n_j}'\}$ such that $v_{n_j}'\to v$ in $L^{r'}(\partial \Omega)$, where $r' < \frac{2(N-1)}{N-2}$. By [27,Theorem 4.9], there exists a subsequence $\{v_{n_j}''\}$ such that

Since $g(x, .)$ is continuous for a.e. $x\in \partial \Omega$, then $g(x, v_{n_j}''(x))\to g(x, v(x))$ a.e. $x\in \partial \Omega$ and $g(x, v_{n_j}''(x))$ is bounded in $L^{r}(\partial \Omega)$ by (H2). Using the Lebesgue Dominated Convergence Theorem, we get

By the Hölder's inequality, for all $\psi \in H^1(\Omega)$, we get

Therefore, $\|G(v_{n_j}'')-G(v)\|_{(H^1(\Omega))^*} = \sup_{\|\psi\|_{H^1(\Omega)}\le 1} |\langle G(v_{n_j}'')-G(v), \psi\rangle| \to 0$ as $j\to \infty$. Hence, $G(v_{n_j}'')\to G(v)$ in $(H^1(\Omega))^*$ as $j\to \infty$, a contradiction to (4.6).

Step 3: $B$ is pseudomonotone.

Let $v_n\rightharpoonup v$ in $H^1(\Omega)$. Using Step 1-Step 2, we get that

Therefore, $\langle B(v_n), \psi\rangle\to \langle B(v), \psi\rangle$ as $n\to \infty$ for all $\psi \in H^1(\Omega)$. Furthermore, by [27,Proposition 3.5 (iv)], $\langle B(v_{n}), v_{n}\rangle \to \langle B(v), v\rangle$ as $n\to \infty$. Hence,

establishing that $B$ is pseudomonotone.

Finally, we show that $B$ is coercive, i.e., $\langle B(\psi), \psi\rangle/\|\psi\|_{H^1(\Omega)}\to\infty$ as $\|\psi\|_{H^1(\Omega)}\to\infty$. For any $\psi \in H^1(\Omega)$, using (4.5) in the definition of the operator $B$, we have

Hence $B$ is coercive. Thus $B$ satisfies the hypotheses of Proposition 2.2 with $X = H^1(\Omega)$. Therefore, for $b = 0 \in (H^1(\Omega))^*$, there exists $u \in H^1(\Omega)$ such that

Moreover, $g(x, .)$ is bounded in $L^{\frac{2(N-1)}{N-2}}(\partial \Omega)$ by (H2), and therefore in $L^{\frac{2(N-1)}{N}}(\partial \Omega)$ by continuous embedding of $L^{\frac{2(N-1)}{N-2}}(\partial \Omega)$ into $L^{\frac{2(N-1)}{N}}(\partial \Omega)$. Hence $u$ is a weak solution of (4.1). It remains to prove that $u$ is a weak solution of (1.1). For this, we will show that $\underline{u} \le u \le \overline{u}$ in $\overline \Omega$, so that $g = f$ in (4.1).

Clearly, $(u-\overline{u})_+: = \max\{0, u-\overline{u}\} \in H^1(\Omega)$ and $(\underline{u}-u)_+: = \max\{0, \underline{u}-u\} \in H^1(\Omega)$. Then, using the weak formulation of (4.1) with the test function $\psi: = (u-\overline{u})_+\geq 0$ in $H^1(\Omega)$, and the facts that $\overline{u}$ is a supersolution of (1.1) and $(u-\overline{u})_+ = 0$ in $\{u\le\overline{u}\}$, we have

Then, (4.7) yields

which implies that $\|(u-\overline{u})_+\|_{H^1(\Omega)} = 0$. That is, $u\leq \overline{u}$ a.e. in $\Omega$. Using the the continuity of the trace operator (1.2), we get that $\|(u-\overline{u})_+\|_{L^{\frac{2(N-1)}{N-2}}(\partial \Omega)} = 0$. Hence, $u\leq \overline{u}$ a.e. in $\overline{ \Omega}$.

Analogously, taking the test function $\psi: = (\underline{u}-u)_+\geq 0$ and using the fact that $\underline{u}$ is a subsolution of (1.1), we obtain that

Therefore, $\underline{u}\leq u$ a.e. in $\overline \Omega$, and hence $\underline{u}\leq u \leq \overline{u}$ a.e. in $\overline \Omega$. Thus, $u$ is a weak solution of (1.1), completing the proof of Theorem 1.5.

4.2.   Proof of Theorem 1.6

We will use Zorn's Lemma and Proposition 2.3, to prove our result. Consider the set

and we note that $\mathcal{A}$ is nonempty by Theorem 1.5. Let $\{u_i\}_{i\in I}\subseteq \mathcal{A}$ be a family of chain. Since $u_i$ is a weak solution of (1.1), taking $u_i$ as the test function and using (4.4), we get

where $C$ depends on $\underline{u}, \, \overline{u}, \, K, \, \partial\Omega$ but independent of $i \in I$. By the separability and reflexivity of $H^1(\Omega)$, there exists an increasing sequence $u_n$ such that

Clearly, $u$ is an upper bound of the chain $\{u_i\}_{i \in I}$. It suffices to show that $u \in \mathcal{A}$. Since $\{u_n\}$ is nondecreasing and $\underline{u} \le u_n \leq \overline{u}$, we have that $u_n(x) \to u(x)$, and $u_n(x)\le u(x)$ for all $n$, and $\underline{u}(x) \le u(x) \leq \overline{u}(x)$ pointwise a.e. in $\overline{ \Omega}$. Furthermore, since $f$ is Carathéodory, we have that

This, in conjunction with (H2), and the Lebesgue Dominated Convergence Theorem yields $\|f(x, u_n)-f(x, u)\|_{L^r(\partial \Omega)}\to 0$ as $n\to \infty$. Therefore, using Hölder's inequality, we deduce that

which yields

Taking the limit as $n\to \infty$, we get for any $\psi \in H^1(\Omega)$

Hence, $u$ is a weak solution of (1.1), thus concluding that $u\in \mathcal{A}$.

By Zorn's Lemma, there exists a maximal element $u^*\in \mathcal{A}$. It remains to show that $u^*$ is maximal in the sense that if $\hat u$ is any other weak solution of (1.1) between $\underline{u}$ and $\overline{u}$, then $\hat{u} \le u^*$. So, let $\hat{u}$ be a weak solution of (1.1) between $\underline{u}$ and $\overline{u}$, and $u^*$ is the maximal element of $\mathcal{A}$. By Proposition 2.4, $u = \max\{\hat{u}, u^*\}$ is a subsolution of (1.1). Then, by Theorem 1.5, there exists a weak solution $u_0$ of (1.1) satisfying

Thus, $u_0\in \mathcal{A}$. On the other hand, $u^*\leq \max\{\hat{u}, u^*\} = u\leq u_0$. But $u^*$ is maximal element of $\mathcal{A}$, so necessarily $u^* = u_0$. Therefore, we readily see that $\hat{u}\leq u\leq u_0 = u^*$, and hence

as desired. The existence of a minimal element $u_*$ of $\mathcal{A}$ is proved analogously. This completes the proof of Theorem 1.6.

5.   Examples

In this section, we apply our existence results, Theorem 1.4 and Theorem 1.5, to problems involving sublinear nonlinearities. In particular, in each case we construct an ordered pair of weak sub- and supersolution. We apply Theorem 1.4 to establish Theorem 5.1 and Theorem 1.5 in Remark 5.2 below.

Theorem 5.1. Consider

where $\lambda > 0$ parameter and $f:[0, \infty) \to [0, \infty)$ is locally Lipschitz continuous function satisfying

(i) $f(0) = 0$ with $f'(0) > 0$, and

(ii) $\lim\limits_{s \to \infty}\frac{f(s)}{s} = 0$.

Then (5.1) has a positive weak solution for $\lambda > \frac{\mu_1}{f'(0)}$, where $\mu_1 > 0$ is the first eigenvalue of the Steklov eigenvalue problem

and $0 < \varphi_1 \in H^1(\Omega)$ is the corresponding eigenfunction.

Proof. Let $\lambda > \frac{\mu_1}{f'(0)}$ be fixed. Using hypothesis (i), we verify that $\underline{u}: = \epsilon \varphi_1$ is a subsolution of (5.1) for $\epsilon \approx 0$. Indeed, we observe that since $\lambda > \frac{\mu_1}{f'(0)}$ is fixed, $\xi(s): = \mu_1 s -\lambda f(s)$ satisfies $\xi(0) = 0$ and $\xi'(0) < 0$, then $\xi(s) < 0$ for $s \approx 0$. Therefore, for all $0 \leq \psi \in H^1(\Omega)$, the following holds for $\epsilon \approx 0$

Next, using hypothesis (ii), we show that there exists $M_{\lambda} > 0$ such that $\overline{u}: = M e$ is a weak supersolution of (5.1) for all $M \geq M_{\lambda}$, where $e$ is the unique positive solution of

We observe that while $f$ is not assumed to be nondecreasing, $\overline{f}(t): = \max\limits_{s \in [0, t]}f(s)$ is nondecreasing, and $f(t) \leq \overline{f}(t)$ for all $t \geq 0$. Moreover, due to hypothesis (ii), $\overline{f}$ satisfies the sublinear condition at infinity

Therefore, there exists $M_{\lambda} > 0$ such that for all $M \geq M_{\lambda}$

Then $\overline{u} = Me \in H^1(\Omega)$ satisfies

for all $0 \le \phi \in H^1(\Omega)$. Therefore, $\overline{u}$ is a weak supersolution of (1.1) for each $\lambda > \frac{\mu_1}{f'(0)}$. Clearly $\overline{u} = Me \geq \epsilon(\lambda)\varphi_1 = \underline{u}$ a.e. in $\overline{\Omega}$. We remark that since $f$ is locally Lipschitz and $[\underline{u}, \overline{u}]$ is bounded, $f$ satisfies hypothesis (H1) of Theorem 1.4. Hence, there exists a positive weak solution $u$ of (5.1) such that $\epsilon\varphi_1 \leq u \leq M e$ a.e. in $\overline \Omega$ for any $\lambda > \frac{\mu_1}{f'(0)}$. This completes the proof.

Remark 5.2. On the other hand, if $f$ is continuous (not necessarily Lipschitz), satisfies hypothesis (ii) of Theorem 5.1 and $f(s) > 0$ for $s \geq 0$, the problem (5.1) has a positive weak solution for each $\lambda > 0$. Indeed, it is easy to see that $\underline{u}\equiv 0$ is a strict weak subsolution and for each $\lambda > 0$, there exists $M_{\lambda} > 0$ such that $\overline{u} = Me$ is a weak supersolution for all $M \geq M_{\lambda}$, as in the proof of Theorem 5.1. Then, the result follows by Theorem 1.5.

Acknowledgements

We would like to thank the referee for carefully reading the manuscript and providing valuable suggestions. We would like to thank Professor Jesús Jaramillo for helpful discussions about Step 2 in the proof of Theorem 1.5. The fifth author is supported by grants PID2019-103860GB-I00, MICINN, Spain, and by UCM-BSCH, Spain, GR58/08, Grupo 920894. All authors acknowledge MSRI for bringing this group together for collaboration.

Conflict of interest

The authors declare there is no conflicts of interest.