Optimal error estimates of a class of system of two quasi-variational inequalities

1.   Introduction

In this paper we will study the existence of nontrivial solutions of the following nonlocal elliptic problem

where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ with a smooth boundary, $g:\Omega\times \mathbb{R}\to \mathbb{R}$ is a differential function whose properties will be given later, $\lambda_\ell$ is an eigenvalue of $\mathcal{L}_K$ and $\mathcal{L}_K$ is a non-local elliptic operator formally defined as follows

where the kernel $K: \mathbb{R}^N \setminus\{0\}\to (0, \infty)$ is a function with the properties that

The integro-differential operator $\mathcal{L}_K$ is a generalization of the fractional Laplacian $-(-\Delta)^s$ which is defined as

When one takes the kernel $K(x) = |x|^{-(N+2s)}$ then $\mathcal{L}_K = -(-\Delta)^s$. In this case the problem (1.1) becomes

The problem (1.5) can be regarded as the counterpart of the semilinear elliptic boundary value problem

where $\lambda_\ell$ is an eigenvalue of $-\Delta$ with a $0$-Dirichlet boundary value.

A weak solution for (1.1) is a function $u:\mathbb{R}^{N}\rightarrow \mathbb{R}$ such that

Here the linear space

and the functional space $X$ denotes the linear space of Lebesgue measurable functions from $\mathbb{R}^N$ to $\mathbb{R}$ such that the restriction to $\Omega$ of any function $v$ in $X$ belongs to $L^2(\Omega)$ and

where $\mathcal{C}\Omega: = \mathbb{R}^{N}\setminus\Omega$. The properties of the functional space $X_0$ will be introduced in the next section.

The non-local equations have been experiencing impressive applications in different subjects, such as the thin obstacle problem, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes and flame propagation, conservation laws, ultrarelativistic limits of quantum mechanics, quasigeostrophic flows, multiple scattering, minimal surfaces, materials science, water waves, elliptic problems with measured data, optimization, finance, etc. See ^[1] and the references therein. The non-local problems and operators have been widely studied in the literature and have attracted the attention of lot of mathematicians coming from different research areas due to the interesting analytical structure and broad applicability. Many mathematicians have applied variational methods ^[2] such as the mountain pass theorem^[3], the saddle-point theorem^[2] or other linking type of critical point theorem in the study of non-local equations with various nonlinearities that exhibit subcritical or critical growth; see ^{[1,4,5,6,7,8,9,10,11,12,13]} and references therein.

In the present paper we will apply the Morse theory to find weak solutions to (1.1). We assume, throughout the whole paper, that the nonlinear function $g\in C^1(\bar \Omega\times \mathbb{R}, \mathbb{R})$ satisfies the following growth condition

$(g)$ there is $C > 0$ and $p\in (2, \frac{2N}{N-2s})$ such that

We consider the situation that the problem (1.1) has the trivial solution $u\equiv0$ and is resonant at infinity in the sense that the function $g$ satisfies the following assumptions

We refer the reader to [12, Proposition 9 and Appendix A], [14, Propositions 2.3 and 2.4] and [11, Proposition 4] for the existence and basic properties of the eigenvalue of the linear non-local eigenvalue problem given by

that will be collected in the next section.

We make some further conditions on $g$.

$(g_{1})$ There are $c_1 > 0$ and $r \in(0, 1)$ such that

$(g^\pm_{2})$ There are $c_2 > 0$ and $r \in(0, 1)$ given in $(g_1)$ such that

We note here that $(g_{1})$ implies (1.10) which characterizes the problem (1.1) as asymptotically linear resonant near infinity at the eigenvalue $\lambda_\ell$ of the non-local operator $-\mathcal{L}_{K}$. As an example, we can take the function $g(x, t) = \pm a(x)|t|^{r-1}t$ with $a\in L^{\infty}(\Omega)$, $\inf_{{\Omega}}a > 0$ and $r\in(0, 1)$.

We will prove the following theorems. We first consider the case that $g_{t}'(x, 0)+\lambda_{\ell}$ is not an eigenvalue of (1.11). We have the following conclusions.

Theorem 1.1. Assume (1.3), (1.9) and $(g_{1})$. Then the problem (1.1) admits at least one nontrivial weak solution in each of the following cases:

For the case that $g_{t}'(x, 0)+\lambda_{\ell} = \lambda_{m}$, an eigenvalue of (1.11), i.e., the trivial solution $u = 0$ of (1.1), is degenerate. In this case the problem (1.1) is double resonant at both infinity and zero. We have the following conclusions.

Theorem 1.2. Assume (1.3), (1.9) and $(g_{1})$. Then the problem (1.1) admits at least one nontrivial weak solution in each of the following cases:

Notice that in Theorem 1.2 there is a large difference between $\lambda_\ell$ and $\lambda_m$. This can be reduced by imposing on $g$ some local sign conditions near zero. We denote $f(x, t): = \lambda_\ell t + g(x, t)$ and $F(x, t) = \int_0^t f(x, s)ds$. We assume the following

$(F_{0}^\pm)$ $f_{t}'(x, 0)\equiv \lambda_{m}$ and there is $\delta > 0$ such that

Theorem 1.3. Assume (1.3), (1.9) and $(g_{1})$. Then the problem (1.1) admits at least one nontrivial weak solution in each of the following cases:

We give some remarks and comparisons. The non-local equations with resonance at infinity have been studied in some recent works. In ^[6,7], the famous saddle-point theorem ^[2] has been applied in the existence of solutions of the non-local problem related to (1.1) for the Landesman-Lazer resonance condition ^[15]. In ^[6], the authors treated a case in which one version of the Landesman-Lazer resonance condition ^[15] was formulated as follows:

In ^[7], the authors treated an autonomous case in which another version of the Landesman-Lazer resonance condition ^[15] was formulated as follows:

where $E(\lambda_\ell)$ is the linear space generated by the eigenfunctions corresponding to $\lambda_\ell$. In ^[7], there is a crucial assumption that all functions in $E(\lambda_\ell)$ having a nodal set with the zero Lebesgue measure, which is valid for the fractional Laplacian $(-\Delta)^s$ (see ^[16]) and is still open for the general non-local elliptic operator $-\mathcal{L}_K$ (see [7, Equation (1.12)] and remarks therein).

We note here that the common feature in (1.12) and (1.13) is that the nonlinear term $g$ is bounded. Motivated by previous works ^[6,7], we treat, in the present paper, the completely resonant case via the application of Morse theory and critical groups. The results in this paper are new in two aspects. On one hand, the nonlinear term $g$ is indeed unbounded and by imposing on $g$ the global conditions $(g_1)$ and $(g_2^\pm)$, we do not make the same assumption on the eigenfunctions of (1.11) as that in ^[7]. On the other hand, we explore a new application of the abstract results about critical groups at infinity that were built in ^[17] and modified in ^[18]. The conditions on $g$ used here were first constructed in ^[19] for semilinear elliptic problems at resonance. Some of the above theorems may be regarded as the natural extension of local setting (1.6) to the non-local fractional setting.

We prove the main results via Morse theory ^[20,21] and critical group computations. Precisely, we will work under the abstract framework built in ^[17] and modified in ^[18]. In Section 2, we collect some preliminaries about the variational formulas related to (1.1). In Section 3, we give the proofs of the main theorems including some technical lemmas.

2.   Preliminaries

In this section we will give the preliminaries for the variational structure of (1.1) and preliminary results in Morse theory.

2.1.   The functional setting

We first recall some basic results on the functional $X_{0}$ mentioned in Section 1. The functional space $X_{0}$ is non-empty because $C_0^2(\Omega) \subset X_0$ (see [22, Lemma 11]), and it is endowed with the norm defined as

Furthermore, $(X_{0}, \|\cdot\|_{X_{0}})$ is a Hilbert space with a scalar product (see [10, Lemmas 6 and 7]) defined by

The norm (2.1) on $X_0$ is related to the so-called Gagliardo norm

of the usual fractional Sobolev space $H^s(\Omega)$. For further details related to the fractional Sobolev spaces one can see ^[1,10,13] and the references therein.

By [10, Lemma 8] and [13, Lemma 9], we have following embedding results.

Proposition 2.1. For each $q\in [1, \frac{2N}{N-2s}]$, the embedding $X_0\hookrightarrow L^q(\mathbb{R}^N)$ is continuous and there is $C_q > 0$ such that

This embedding is compact whenever $q\in [1, \frac{2N}{N-2s})$.

2.2.   An eigenvalue problem for $-\mathcal{L}_{K}$

Next, we recall some basic facts about the eigenvalue problem associated with the integro-differential operator $-\mathcal{L}_{K}$

The number $\lambda \in \mathbb{R}$ is an eigenvalue of (2.3) if there is a nontrivial function $v: \mathbb{R}^{N}\rightarrow \mathbb{R}$ such that for all $\varphi\in X_0$

We denote by $\{\lambda_{k}\}_{k\in \mathbb{N}}$ the sequence of the eigenvalue of the problem (2.3), with

We denote by $\phi_k$ the eigenfunction corresponding to $\lambda_k$. The sequence $\{\phi_k\}_{k\in \mathbb{N}}$ can be normalized in such a way that the sequence provides an orthonormal basis of $L^2(\Omega)$ and an orthogonal basis of $X_0$. By [14, Proposition 2.4] one has that all $\phi_k \in L^\infty(\Omega)$. One can refer to [12, Proposition 9 and Appendix A], [14, Proposition 2.3] and [11, Proposition 4] for a complete study of the spectrum of the integro-differential operator $-\mathcal{L}_K$.

The first eigenvalue $\lambda_{1}$ is simple and can be characterized as

Each eigenvalue $\lambda_k$, $k\geqslant2$, has finite multiplicity. More precisely, we say that $\lambda_k$ has the finite multiplicity $\nu_k\in \mathbb{N}$ if

The set of all of the eigenfunctions corresponding to $\lambda_k$ agrees with

The eigenvalue $\lambda_1$ is achieved at a positive function $\phi_1$ with $\|\phi_1\|_{L^2(\Omega)} = 1$. For each $k\geqslant2,$ the eigenvalue $\lambda_k$ can be characterized as follows:

where

Corresponding to the eigenvalue $\lambda_{k}$ of $-\mathcal{L}_{K}$ with multiplicity $\nu_{k}$, the space $X_0$ can be split as follows:

where

For each eigenvalue $\lambda_k$, we can define a linear operator $\mathcal{A}_k:X_0\to X_0^*$ by

By the continuous embedding from $X_0$ into $L^2(\Omega)$ in Proposition 2.1, one can deduce that $\mathcal{A}_k$ is a bounded self-adjoint linear operator so that $\langle \mathcal{A}_k \phi, \phi \rangle = 0$ for all $\phi \in V_k = :\ker(\mathcal{A}_k).$

Finally, we conclude this subsection with the following variational inequalities which can be deduced by the variational characterization of the eigenvalues and the standard Fourier decomposition:

3.   Proofs of main results

In this section we give the proofs of the main results in this paper via some abstract results on Morse theory ^[20,21] for a $C^2$ functional $\mathcal{J}$ defined on a Hilbert space. These results come from ^{[17,18,20,21,23,24,25]}, etc. We refer the readers to ^[26] for a brief summary of the concepts, definitions and the abstract results about critical groups and Morse theory.

First of all, we observe that the problem (1.1) has a variational structure; indeed, it is the Euler-Lagrange equation of the functional $\mathcal{J}: X_{0}\to \mathbb{R}$ defined as

where $G(x, t) = \int_0^t g(x, \varsigma) d \varsigma$. Since the nonlinear function $g$ satisfies the assumption $(g)$, by Proposition 2.1, the functional $\mathcal{J}$ is well defined on $X_{0}$ and is of class $C^2$ (see a detailed proof in ^[26]) with the derivatives given by

From (3.2) and (1.7), one sees that critical points of $\mathcal{J}$ are exactly weak solutions to (1.1).

Define the functional $\mathcal{F}:X_0\to \mathbb{R}$ by

According to (2.7) with $\lambda_\ell$, the functional $\mathcal{J}$ can be written as

Using the assumption $(g_{1})$ we can deduce that

Therefore $\mathcal{J}$ fits the basic assumptions in the abstract framework required by [26, Proposition 2.5] with respect to $X_0 = V_\ell \oplus W_\ell$.

Next we prove one technical lemma that will be used to verify the angle conditions required by [26, Proposition 2.5] for computation of the critical groups at infinity.

Lemma 3.1. Assume $(g_{1})$ and $(g^\pm_{2})$. Then there exist $M > 0$, $\epsilon\in (0, 1)$ and $\beta > 0$ such that

for any $u = v+w\in X_0 = V_{\ell}\oplus W_{\ell}$ with $\|u\|_{X_0} \geqslant M$ and $\|w\|_{X_0}\leqslant \epsilon \|u\|_{X_0}$.

Proof. We give the proof for the case that $(g_{1})$ and $(g^+_{2})$ hold.

For $u = v+w \in X_0 = V_{\ell}\oplus W_{\ell}$, we set

where $M > 0$ and $\epsilon\in(0, 1)$ will be chosen below.

For $u\in \mathcal{C}(M, \epsilon),$ we have

It follows from $(g_{1})$ and $(g^+_{2})$ that

By Proposition 2.1 and the Hölder inequality we have

Since $V_{\ell}$ is finite dimensional, by the elementary inequality $|a+b|^{q}\leqslant 2^{q-1}(|a|^{q}+|b|^{q})$ for all $a, b\in \mathbb{R}$, we have that

here $\hat c$ is the embedding constant of $L^{1+r}(\Omega)\hookrightarrow V_\ell$. Therefore for $u\in \mathcal{C}(M, \epsilon),$ it follows from (3.9)–(3.11) that

Now we can take $M > 0$ large enough and $0 < \epsilon < 1$ small enough so that

hence,

The proof is complete. □

In order to apply [26, Proposition 2.5] and Morse theory to prove our results, we have to verify that $\mathcal{J}$ satisfies the Palais-Smale condition.

Lemma 3.2. Assume $(g_{1})$ and $(g^\pm_{2})$. Then the functional $\mathcal{J}$ defined by (3.1) satisfies the Palais-Smale condition.

Proof. Let the sequence $\{u_{n}\} \subset X_0$ be such that

We show that $\{u_n\}$ is bounded in $X_0$. Suppose, by the way of contradiction, that

Write $u_{n} = v_{n}+w_{n}$, where $v_{n}\in V_{\ell}$ and $w_{n}\in W_{\ell}.$ By the variational inequalities (2.8) and (2.9), we have

where

By (3.11), there is $N_1 \in \mathbb{N}$ such that

By (3.2) and (3.5), we have

By (3.6) and (3.15) we have

It follows that, for any given $\delta > 0$ sufficiently small and all $n$ sufficiently large,

Since $\delta > 0$ was chosen arbitrarily, from (3.15) and (3.20) we deduce that

It follows that there is $N_2 \in \mathbb{N}$ with $N_2\geqslant N_1$ such that

where $M > 0$ and $\epsilon \in (0, 1)$ was given in Lemma 3.1. Therefore by Lemma 3.1 we have that

On the other hand, by (3.14), we get

This contradicts (3.23). Hence $\{u_{n}\}$ is bounded in $X_0$.

Since $X_0$ is a Hilbert space, there is a subsequence of $\{u_{n}\}$, still denoted by $\{u_{n}\}$, and there exists $u^* \in X_0$, such that

By Proposition 2.1, up to a subsequence, it holds that

as $n\rightarrow \infty$. By $(g_1)$, Proposition 2.1 and (3.26) we get

From (3.14) we deduce that

as $n\to\infty$, and

as $n\to\infty$. Now by (3.14) and (3.26)–(3.29), we deduce from

that

This completes the proof for verifying the Palais-Smale condition. □

Notice here that only (3.14) is used for verifying the Palais-Smale condition, it follows that the critical point set of $\mathcal{J}$ is compact and is then bounded.

Now we are ready to give the proofs of the main results in this paper.

Proof of Theorem 1.1. We give the proof of the case (ⅰ). Since

it follows from Lemma 3.1 that $\mathcal{J}$ satisfies the angle condition ($\mathcal{AC}_{\infty}^{-}$) in [26, Proposition 2.5] at infinity with respect to $X_0 = V_\ell \oplus W_\ell$. Thus by [26, Proposition 2.5(ⅱ)] we have

where

Therefore $\mathcal{J}$ has a critical point $u_*$ satisfying

The second derivative of $\mathcal{J}$ at the trivial solution $u = 0$ can be written as

By the condition we see that $u = 0$ is a nondegenerate critical point of $\mathcal{J}$ with the Morse index

Hence

Since $\lambda_{m}\neq\lambda_{\ell},$ we get that $\mu_\ell+\nu_\ell\neq\bar{\mu}_0,$ and we see from (3.33) and (3.34) that $u_*\ne0$. The case (ⅱ) can be proved in the same way. The proof is complete.□

Proof of Theorem 1.2 We give the proof of the case (ⅱ). It follows from Lemma 3.1 that $\mathcal{J}$ satisfies the angle condition ($\mathcal{AC}_{\infty}^{+}$) in [26, Proposition 2.5] at infinity with respect to $X_0 = V_\ell \oplus W_\ell$. Thus by [26, Proposition 2.5(ⅱ)] we have

and $\mathcal{J}$ has a critical point $u_*$ satisfying

Now $\mathcal{J}''(0)$ takes the form

It follows that $0$ is a degenerate critical point of $\mathcal{J}$ with the Morse index $\mu_0$ and the nullity $\nu_0$ given by

By the Gromoll-Meyer result^[27], we have that

It follows from $\lambda_{m} < \lambda_{\ell-1} < \lambda_{\ell}$ or $\lambda_{\ell-1} < \lambda_{m-1}$ that $\mu_0+\nu_0 < \mu_\ell$ or $\mu_0 > \mu_\ell,$ and we see from (3.36) and (3.39) that $u_*\ne0.$ The case (ⅰ) can be proved in the same way. The proof is complete. □

Lemma 3.3. Assume (1.3), (1.9), $(g_1)$ and $(F_0^\pm)$. Then

(ⅰ) $C_q(\mathcal J, 0) \cong \delta_{q, \mu_0 + \nu_0} \mathbb{Z}$ for $(F^+_0)$ holds,

(ⅱ) $C_q(\mathcal J, 0) \cong \delta_{q, \mu_0} \mathbb{Z}$ for $(F^-_0)$ holds,

where $\mu_0$ and $\nu_0$ are given by (3.38).

Proof. We will apply [24, Proposition 2.3] to prove the results. We first note that by $(g_1)$ and the last part in the proof of Lemma 3.2, the functional $\mathcal{J}$ verifies the bounded Palais-Smale condition which ensures the deformation property for computing $C_q(\mathcal{J}, 0)$ (see ^[20,21]).

We treat the case (ⅱ) for which $(F^-_0)$ holds. We will prove that $\mathcal J$ has the local linking structure at $0$ as with respect to $X_0 = E^-\oplus E^+$, where $E^- = W_m^-$ and $E^+ = V_m \oplus W_m^+$. We refer the readers to ^[28,29] for the concept of the local linking.

1) Take $u\in E^- = W^-_m$. Since $W_m^-$ is finite dimensional, there is $\rho > 0$ such that

Consequently, thanks to (2.8) with $\lambda_m$ and $(F^-_0)$, for any $u \in E^-$ with $\|u\| \leqslant \rho$, we get

2) For $u \in E^+ = V_m \oplus W_m^+$, we write $u = w+z$, where $w \in W_m^+$ and $z\in V_m$. Then

Since $V_{m}$ is finite dimensional, there is $\rho > 0$ such that

Consequently,

By $(F_0^-)$, we have

By $(g_1)$, we get that, for each given $\sigma\in (2, \frac{2N}{N-2s}]$, there is $\kappa = \kappa(\sigma, {\delta}) > 0$ such that

Hence

Now, by (3.41), (3.42) and (3.44) we get

Since $\sigma > 2$, one sees from (3.42) and (3.45) that for $\rho > 0$ small enough once again, it holds that

For $z \in V_m$ with $\|{z}\|\leqslant \rho$, we have by $({F}^-_{0})$ that

Thus for all ${ {z}}\in B_\rho\cap V_m$,

To apply [24, Proposition 2.3], we need to show that the above inequality holds strictly for $z\neq 0$. Assume, for contradiction, that for any $0 < \epsilon\leqslant \rho$, there is ${{z}}_\epsilon \in V_m$ such that $0 < \|z_\epsilon\| < \epsilon$ and $\mathcal{J}(z_\epsilon) = 0$. Then, the following holds:

and then

Given that $z_{\epsilon}\in V_m$, going back to (1.1), we see that $z_\epsilon$ is a nontrivial solution of (1.1). This contradicts the conventional assumption that ${0}$ is an isolated solution of (1.1). In summary, we obtain by (3.46) and (3.47) that

Therefore, $\mathcal{J}$ has a local linking structure with respect to $E = E^-\oplus E^+$ with $\mu_0 = \dim E^-$. It follows from [24, Proposition 2.3] that $C_q(\mathcal{J}, {0}) \cong \delta_{q, \mu_0} \mathbb{Z}$.

The case (ⅰ) is proved in a similar and simpler way. The proof is complete. □

Proof of Theorem 1.3. We give the proof of the case (ⅳ). As in the proof of Theorem 1.1(ⅱ), we have gotten the following conclusion that $\mathcal{J}$ satisfies the angle condition ($\mathcal{AC}_{\infty}^{+}$) in [26, Proposition 2.5] at infinity with respect to $X_0 = V_\ell \oplus W_\ell$, and then that $\mathcal{J}$ has a critical point $u_*$ satisfying

By $(F_0^-)$ and Lemma 3.3, $J$ has a local linking at $0$ with respect to $X_0 = E^-\oplus E^+$. Thus it follows from [24, Proposition 2.3] that

By $\lambda_{\ell-1} < \lambda_{m-1}$, we have that $\lambda_{\ell} < \mu_0$. It follows from (3.48) and (3.49) that $u_*\ne0.$ The other cases can be proved in the same way. The proof is complete. □

Remark 3.4. We conclude the paper with some remarks.

1) In Theorem 1.3, the result for one nontrivial solution is valid for $f$ that is locally Lipschitz continuous with $f'(x, 0) \equiv \lambda_m$ being replaced by satisfying $\lim_{t\to 0} \frac{f(x, t)}{t} = \lambda_m$. In this case, we have only $\mathcal{J}\in C^{2-0}(X_0, \mathbb{R})$ and no Morse index is involved. We have the critical groups at zero by applying the local linking theorem in ^[23] as follows:

2) In the case that $\lambda_\ell = \lambda_1$ and $(g_2^-)$ holds, we have that $\mu_\ell = 0$ and

Thus $\mathcal{J}$ has a critical point $u^*$ with

It follows that

Indeed, (3.50) is equivalent to $\mathcal{J}$ being bounded from below and (3.51) is equivalent to $u^*$ being a local minimizer of $\mathcal{J}$. Furthermore, in the case that $C_q(\mathcal{J}, 0)\ne0$ for some $q\geqslant 1$, we can apply [30, Theorem 2.1], i.e., the most general version of the three critical point theorem, to get two nontrivial solutions of (1.1).

Use of AI tools declaration

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

Acknowledgments

The authors appreciate the reviewers for carefully reading the manuscript and giving valuable comments to improve the exposition of the paper. This work was supported by the NSFC (12001382, 12271373, 12171326).

Conflict of interest

The authors declare there is no conflicts of interest.