[1,3]Andrei Stan These authors contributed equally to this work. [1] Faculty of Mathematics and Computer Science, Babeș-Bolyai University, Cluj-Napoca, 400084, Romania

2] Institute of Advanced Studies in Science and Technology, Babeş-Bolyai University, Cluj-Napoca, 400110, Romania

3] Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, 400110, Romania

MSC Classification]35J91, 35J60, 37N10, 35B45, 35B33

Existence, uniqueness, and localization results for a Dirichlet problem in geophysics

Radu Precup Email: radu.precup@ubbcluj.ro * Email: andrei.stan@ubbcluj.ro Affiliation: * Affiliation: [ Affiliation: [

Abstract

In the present paper, we investigate the existence, uniqueness, and localization of solutions for a recently derived model for gyres, formulated as a Dirichlet problem. This problem is studied from two perspectives. In the first part, we establish the existence and uniqueness of a weak solution by means of the Minty–Browder theorem. For the nonlinear term, we assume a suitable monotonicity condition together with an exponential growth condition. Here, the growth condition is admissible since the problem is posed in dimension two, which allows us to use the Trudinger–Moser inequality. In the second part, we prove the existence of a localized positive solution within a conical annular region by means of the Moser–Harnack inequality. This approach also yields multiplicity results when the required conditions are satisfied on disjoint sets. From a physical perspective, an upper bound for the solution reflects the dynamical intensity of the flow, while a lower bound indicates the persistence of ocean circulation in a given region. These bounds are determined by the behavior of the vorticity function.

keywords

Geophysical flow, ocean gyre, Minty-Browder, Trudinger-Moser inequality, Moser-Harnack inequality, exponential growth

pacs

[

1 Introduction and Preliminaries

In this paper, we study a recently derived model describing oceanic gyres, introduced in [1]. Gyres are large-scale geophysical circulatory flows generated by ocean currents that rotate around a central region. Such flows occur only north and south of the Equator, since near the Equator the dynamics of the oceanic flow have a peculiar behavior (see [2, 3, 4, 5, 6, 7, 8]). Oceanic gyres are driven by gravitational effects combined with Coriolis forces from the Earth’s rotation, and by wind stress acting on the ocean surface.

An important feature of gyre dynamics is that the ratio between the vertical and horizontal velocities is typically of order $10^{-4}$ [9], so the vertical motion can be neglected. Introducing a stream function, as in [1], to describe the horizontal component of the flow, the dynamics can be viewed as those of shallow-water flows on a rotating sphere, making a formulation in spherical coordinates natural. Using a stereographic projection, the model can subsequently be transformed from spherical coordinates into a planar partial differential equation subject to Dirichlet boundary conditions.

In the following, we briefly recall the derivation of the model presented in [1]. Let $\theta\in[0,\pi)$ denote the polar angle (so that $\theta-\pi/2$ represents the usual latitude, with $\theta=0$ corresponding to the North Pole), and let $\varphi\in[0,2\pi)$ denote the longitude (azimuthal) angle. In terms of the stream function $\psi(\theta,\varphi)$ , the horizontal gyre flow on the spherical Earth has the azimuthal and polar velocity components given by

\frac{1}{\sin\theta}\psi_{\varphi}\qquad\text{and}\qquad-\psi_{\theta}.

Letting $\Psi(\theta,\varphi)$ denote the stream function associated only with the oceanic vorticity (neglecting the effect of the Earth’s rotation), one has

\psi(\theta,\varphi)=-\omega\cos\theta+\Psi(\theta,\varphi),

where $\omega>0$ is the nondimensional Coriolis parameter. Then, the governing equation for the gyre flow takes the form

\frac{1}{\sin^{2}\theta}\Psi_{\varphi\varphi}+\Psi_{\theta}\cot\theta+\Psi_{\theta\theta}=F(\Psi-\omega\cos\theta),

(1)

where the term $F(\Psi-\omega\cos\theta)$ represents the oceanic vorticity, while $2\omega\cos\theta$ corresponds to the planetary vorticity generated by the Earth’s rotation. The total vorticity of the flow is therefore the sum of the oceanic contribution $F(\Psi-\omega\cos\theta)$ and the spin vorticity $2\omega\cos\theta$ . Typically, the oceanic vorticity is one order of magnitude larger than $\omega$ . While the planetary vorticity is prescribed, the function $F$ depends on the type of geophysical flow under consideration; for instance, $F\equiv 0$ corresponds to irrotational flows.

Equation (1) can be transformed from spherical coordinates into an equivalent planar equation (see [1]) by means of the stereographic projection of the unit sphere centered at the origin, projected from the North Pole onto the equatorial plane:

\xi=re^{i\phi},\qquad r=\cot\!\left(\frac{\theta}{2}\right)=\frac{\sin\theta}{1-\cos\theta},

where $(r,\phi)$ are polar coordinates. Under this transformation, equation (1) becomes

\psi_{\xi\bar{\xi}}+2\omega\frac{1-\xi\bar{\xi}}{(1+\xi\bar{\xi})^{3}}-\frac{F(\psi)}{(1+\xi\bar{\xi})^{2}}=0,

where $\frac{\partial}{\partial\xi}$ and $\frac{\partial}{\partial\overline{\xi}}$ are the Wirtinger derivatives. Passing to Cartesian coordinates, this equation is equivalent to the semilinear elliptic equation

\Delta\psi+8\omega\frac{1-(x_{1}^{2}+x_{2}^{2})}{(1+x_{1}^{2}+x_{2}^{2})^{3}}-\frac{4F(\psi)}{(1+x_{1}^{2}+x_{2}^{2})^{2}}=0,

(2)

where $\Delta=\partial_{x^{2}_{1}}+\partial_{x^{2}_{2}}$ denotes the Laplace operator. The flow at the surface of the gyre is determined by solutions of (2) in a planar region $\Omega$ bounded by two level sets of the stream function; therefore, equation (2) is supplemented with Dirichlet boundary conditions, i.e., $\psi=0$ on $\partial\Omega$ .

Sufficient conditions for problem (3) to admit a unique solution are given in [10]. The existence of radial solutions for problem (3) has also been studied in the literature [11, 12, 13], using various fixed point techniques, such as the Leray–Schauder principle or Krasnosel’skii’s fixed point theorem.

The initial flow is considered in the Southern Ocean, within a region $\Omega^{\prime}$ bounded away from both the South Pole and the Equator. After the transformation to Cartesian coordinates described above, the domain $\Omega^{\prime}$ is mapped onto a planar domain $\Omega$ . Since the stereographic projection maps the entire southern region of the sphere into the unit disk, and the analysis is restricted to this region, there exist constants $0<r_{-}<r_{+}<1$ such that

\Omega\subset\left\{(x_{1},x_{2}):r_{-}<\sqrt{x^{2}_{1}+x^{2}_{2}}<r_{+}\right\}=:\mathcal{O}.

In this paper, we study the existence, uniqueness, and localization of a solution to equation (2) in a bounded domain $\Omega\subset\mathcal{O}$ subject to the Dirichlet boundary condition. First, in Subsection 2.1, we establish the existence and uniqueness of a weak solution by means of the Minty–Browder theorem assuming that $F$ satisfies a monotonicity property and, using the Trudinger–Moser inequality, only an exponential growth condition, which is more general than assuming polynomial type growth.

Later, in Subsection 2.2, we use the weak Moser–Harnack inequality to provide sufficient conditions for the existence of a positive solution in a conical annular region. This localization result immediately yields multiple solutions, provided that the function $F$ is oscillatory.

From a physical perspective, an upper bound for the stream function provides quantitative control on its amplitude, while a lower bound prevents it from approaching a trivial state in the region under consideration. In particular, the latter indicates the persistence of a nontrivial circulation in that region. These bounds are determined by the behaviour of the vorticity function. We emphasize, however, that bounds on the stream function do not directly yield bounds on the corresponding velocity field, which depends on its spatial derivatives.

We conclude this section by recalling some fundamental results in nonlinear analysis that will be used in the proof of our main results. See [14, 15, 16, 18, 17, 19, 20] for details and applications in other contexts.

The first one is the celebrated result of Trudinger and Moser, which essentially states that $H_{0}^{1}(\Omega)$ is continuously embedded into the Orlicz space $L_{\phi}(\Omega)$ with $\phi(t)=e^{t^{2}}$ when $\Omega\subset\mathbb{R}^{2}$ is a bounded domain with smooth boundary. More precisely, the following result holds (see [21, Theorem 2], [22, Theorem 1]).

Theorem 1 (Trudinger–Moser).

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded domain with smooth boundary. Then, for every $u\in H_{0}^{1}(\Omega)$ , one has $e^{u^{2}}\in L^{p}(\Omega)$ for all $p\geq 1$ . Moreover, there exists a constant $c=c(\Omega)>0$ such that

\sup_{\begin{subarray}{c}u\in H_{0}^{1}(\Omega)\\ \|u\|_{H_{0}^{1}(\Omega)}\leq 1\end{subarray}}\int_{\Omega}e^{\alpha u^{2}}\,dx\leq c\quad\text{for all }\alpha\leq 4\pi.

This estimate is sharp in the sense that the corresponding supremum is infinite whenever $\alpha>4\pi$ .

Our second tool is the Minty–Browder theorem for strongly monotone operators (see, for example, [23, 24]).

Theorem 2 (Minty–Browder).

Let $(X,|\cdot|_{X})$ be a separable and reflexive Banach space, and let $S\colon X\to X^{\ast}$ be a hemicontinuous and strongly monotone operator, i.e., there exists $c>0$ such that

\langle S(u)-S(v),u-v\rangle\geq c\,|u-v|_{X}^{2},\quad\text{for all }u,v\in X,

where $\langle\cdot,\cdot\rangle$ denotes the duality pairing between $X^{\ast}$ and $X$ . Then $S$ is bijective.

Another useful result in our analysis is the local Moser-Harnack inequality [25]; see also [26, Theorem 8.26] and [27, 28].

Theorem 3.

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded domain, and let $\Omega_{0}\subset\Omega$ be a subdomain such that $\overline{\Omega}_{0}\subset\Omega$ . For any $p\in[1,\infty)$ , there exists a constant $C>0$ , depending only on $p$ , $\Omega$ , and $\Omega_{0}$ , such that for every nonnegative superharmonic function $u$ in $\Omega$ (that is, $-\Delta u\geq 0$ in $\Omega$ ), the following estimate holds:

u(x)\geq C\,|u|_{L^{p}(\Omega_{0})}\quad\text{for \ }x\in\Omega_{0}.

2 Main results

In this section we present the main results concerning the existence, uniqueness, localization, and multiplicity of solutions to the boundary value problem

\begin{cases}-\Delta u(x)=g(|x|)\bigl(-F(u(x))+h(|x|)\bigr)&\text{in }\Omega,\\ u=0&\text{on }\partial\Omega,\end{cases}

(3)

where $\Omega$ is a domain from $\mathcal{O}$ , $F\in C(\mathbb{R};\mathbb{R})$ , and the functions $g,h\colon[r_{-},r_{+}]\rightarrow\mathbb{R}_{+}$ are defined by

g(r)=\frac{4}{(1+r^{2})^{2}},\qquad h(r)=2\omega\frac{1-r^{2}}{1+r^{2}}.

Let us denote

\mathcal{F}\colon\Omega\times\mathbb{R}\rightarrow\mathbb{R},\quad\mathcal{F}(x,y):=g(|x|)\left(-F(y)+h(|x|)\right).

Then problem (3) can be written as

\begin{cases}-\Delta u(x)=\mathcal{F}(x,u(x))&\text{in }\Omega,\\ u=0&\text{on }\partial\Omega.\end{cases}

(4)

By a weak solution of (4) we mean a function $u\in H_{0}^{1}\left(\Omega\right)$ satisfying in $H^{-1}\left(\Omega\right)$ the equation

-\Delta u=\mathcal{N}_{\mathcal{F}}\left(u\right),

that is

\left(u,v\right)_{H_{0}^{1}\left(\Omega\right)}=\left\langle\mathcal{N}_{\mathcal{F}}\left(u\right),v\right\rangle,\ \ \ \text{for all \ }v\in H^{1}\left(\Omega\right),

where $\mathcal{N}_{\mathcal{F}}$ is the Nemytskii superposition operator defined by

\mathcal{N}_{\mathcal{F}}\left(u\right)=\mathcal{F}(\cdot,u(\cdot)).

Equivalently, $\ u$ is a weak solution of (4) if it solves in $H_{0}^{1}\left(\Omega\right)$ the equatin

u-N\left(u\right)=0,

(5)

that is the fixed point equation

u=N\left(u\right),

where $N$ is the composed operator

N=\left(-\Delta\right)^{-1}\mathcal{N}_{\mathcal{F}}.

Here $\left(-\Delta\right)^{-1}$ denotes the solution operator of the Dirichlet problem, $\left(-\Delta\right)^{-1}:H^{-1}\left(\Omega\right)\rightarrow H_{0}^{1}\left(\Omega\right),$ which associates to each $h\in H^{-1}\left(\Omega\right)$ the unique weak solution of

-\Delta u=h\ \ \ \text{in \ }\Omega,\ \ \ u=0\ \ \text{on \ }\partial\Omega.\

Obviously, the above operator approach requires us to specify the functional spaces with respect to which the operators are well defined and possess the properties assumed by the abstract existence results that will be applied.

2.1 Existence and uniqueness via Minty-Browder theorem

Assume that $\Omega\subset\mathbb{R}^{2}$ is a bounded domain with a smooth boundary, i.e., of class $C^{1}$ . We seek a weak solution to problem (3) in the Sobolev space $H_{0}^{1}(\Omega)$ , that is, a function $u\in H_{0}^{1}(\Omega)$ such that

\int_{\Omega}\nabla u\cdot\nabla v\,dx=\int_{\Omega}\mathcal{F}(x,u(x))v(x)\,dx\quad\text{ for all }v\in H_{0}^{1}(\Omega).

(6)

Endow $H_{0}^{1}(\Omega)$ with the inner product

(u,v)_{H_{0}^{1}(\Omega)}:=(\nabla u,\nabla v)_{L^{2}(\Omega)}=\int_{\Omega}\nabla u\cdot\nabla v\,dx,

and the corresponding norm

|u|_{H_{0}^{1}(\Omega)}=\sqrt{(u,u)_{H_{0}^{1}(\Omega)}}.

In our study, we make use of the Poincaré inequality

|u|_{L^{2}(\Omega)}\leq\frac{1}{\sqrt{\lambda_{1}}}|u|_{{H_{0}^{1}}(\Omega)}\quad\left(u\in H_{0}^{1}(\Omega)\right),

(7)

where $\lambda_{1}$ denotes the first eigenvalue of the Dirichlet problem

-\Delta u=\lambda u\ \text{\ in\ }\ \Omega,\qquad u=0\quad\text{\ on\ }\ \partial\Omega.

We also denote by $\langle\cdot,\cdot\rangle$ the duality pairing between $H_{0}^{1}(\Omega)$ and its dual space $H^{-1}(\Omega)$ .

The assumptions imposed in this section on the function $F$ consist of an exponential growth condition, related to the Trudinger–Moser inequality, and a strong monotonicity property, in connection with the Minty–Browder theorem:

(h1)

There exist $p\geq 1,\ \beta\geq 0$ and constants $C_{1},C_{2},C_{3}\geq 0$ such that

\left|F(y)\right|\leq C_{1}+C_{2}|y|^{p}+C_{3}e^{\beta y^{2}}\quad\text{ for all }y\in\mathbb{R}.

(8)

(h2)

There exists $\alpha\geq 0$ satisfying

\gamma<\frac{\lambda_{1}}{g(r_{-})},

(9)

such that

\bigl(-F(y_{1})+F(y_{2})\bigr)(y_{1}-y_{2})\leq\gamma|y_{1}-y_{2}|^{2}\quad\text{ for all\ }y_{1},y_{2}\in\mathbb{R}.

(10)

By definition of $\mathcal{F}$ , it follows that it also satisfies the growth condition (8), possibly with different constants $\mathcal{C}_{i}$ ( $i=1,2,3$ ).

Our first result concerns the well-definedness and continuity of the Nemytskii operator $\mathcal{N}_{\mathcal{F}}$ , under the exponential growth of $F$ given by (h1). The result complements the classical result on the Nemytskii operator corresponding to functions with power growth.

Lemma 1.

Let assumption (h1) hold. Then the operator $\mathcal{N}_{\mathcal{F}}$ is well defined and continuous from $H_{0}^{1}(\Omega)$ into $H^{-1}(\Omega)$ .

Proof.

Well-definedness. Since we are in dimension two, the Sobolev embedding theorem (see, e.g., [26]) ensures that $H_{0}^{1}(\Omega)$ embeds continuously into $L^{q}(\Omega)$ for every $q\in[1,\infty)$ . Combining this with Theorem 1 and the growth condition (8), we deduce that $\mathcal{F}(\cdot,u)\in L^{q}(\Omega)\subset H^{-1}\left(\Omega\right),$ for all $u\in H_{0}^{1}(\Omega)$ and all $q\in(1,\infty)$ . Thus $\mathcal{N}_{\mathcal{F}}$ is well defined.

Continuity. Let $u_{k}\rightarrow u$ in $H_{0}^{1}(\Omega)$ . Then there exists a subsequence (still denoted by $u_{k}$ ) such that $u_{k}(x)\rightarrow u(x)$ for a.e. $x\in\Omega$ . Since the embedding $L^{2}(\Omega)\hookrightarrow H^{-1}(\Omega)$ is continuous (recall that $H_{0}^{1}(\Omega)$ embeds continuously into $L^{2}(\Omega)$ and that $L^{2}(\Omega)$ is identified with its dual), it suffices to prove that $\mathcal{N}_{\mathcal{F}}(u_{k})\rightarrow\mathcal{N}_{\mathcal{F}}(u)$ in $L^{2}(\Omega)$ in order to conclude that $\mathcal{N}_{\mathcal{F}}(u_{k})\rightarrow\mathcal{N}_{\mathcal{F}}(u)$ in $H^{-1}(\Omega)$ .

We first claim that $e^{\beta u_{k}^{2}}\to e^{\beta u^{2}}$ in $L^{2}(\Omega)$ , i.e.,

\int_{\Omega}\left(e^{\beta u_{k}^{2}}-e^{\beta u^{2}}\right)^{2}dx\to 0\quad\text{ as }k\to\infty.

(11)

Inspired by [16, Lemma 2.1] (see also [29, Proposition 17.3]), denoting $v_{k}:=u_{k}-u$ , we estimate

	$\displaystyle e^{\beta u_{k}^{2}}-e^{\beta u^{2}}$	$\displaystyle=\int_{0}^{1}\frac{d}{ds}e^{\beta(u+sv_{k})^{2}}ds$
		$\displaystyle=\int_{0}^{1}2\beta v_{k}(u+sv_{k})e^{\beta(u+sv_{k})^{2}}ds$
		$\displaystyle\leq 2\beta\left\|v_{k}(u+sv_{k})\right\|_{L^{2}(0,1)}\left\|e^{\beta(u+sv_{k})^{2}}\right\|_{L^{2}(0,1)}.$

Since

\left|v_{k}(u+sv_{k})\right|^{2}_{L^{2}(0,1)}\leq 2v_{k}^{2}(u^{2}+v_{k}^{2})

and

{\left|e^{\beta(u+sv_{k})^{2}}\right|^{2}_{L^{2}(0,1)}}\leq\int_{0}^{1}e^{4\beta(u^{2}+(sv_{k})^{2})}ds\leq e^{4\beta(u^{2}+v_{k}^{2})},

one has

(e^{\beta u_{k}^{2}}-e^{\beta u^{2}})^{2}\leq 8\beta^{2}v_{k}^{2}(u^{2}+v_{k}^{2})e^{4\beta u^{2}}e^{4\beta v_{k}^{2}}.

Using the generalized Hölder inequality with $p_{1},\ldots,p_{4}>1$ such that $1/p_{1}+\ldots+1/p_{4}=1$ , we obtain

		$\displaystyle\int_{\Omega}\left(e^{\beta u_{k}^{2}}-e^{\beta u^{2}}\right)^{2}dx$		(12)
		$\displaystyle\leq 8\beta^{2}\|v_{k}^{2}\|_{L^{p_{1}}(\Omega)}\|\left(\|u^{2}\|_{L^{p_{2}}(\Omega)}+\|v_{k}^{2}\|_{L^{p_{2}}(\Omega)}\right)\left\|e^{4\beta u^{2}}\right\|_{L^{p_{3}}(\Omega)}\left\|e^{4\beta v_{k}^{2}}\right\|_{L^{p_{4}}(\Omega)}.$		(13)

Clearly,

\left|v_{k}^{2}\right|_{L^{p_{1}}(\Omega)}\to 0\quad\text{as }k\to\infty,

and the sequence

\left(|u^{2}|_{L^{p_{2}}(\Omega)}+|v_{k}^{2}|_{L^{p_{2}}(\Omega)}\right)\left|e^{4\beta u^{2}}\right|_{L^{p_{3}}(\Omega)}

is uniformly bounded.

Denoting $w_{k}:=\frac{v_{k}}{|v_{k}|_{H_{0}^{1}(\Omega)}}$ , we have

\left|e^{4\beta v_{k}^{2}}\right|_{L^{p_{4}}(\Omega)}=\left(\int_{\Omega}e^{4\beta p_{4}|v_{k}|_{H_{0}^{1}(\Omega)}^{2}w_{k}^{2}}\,dx\right)^{\frac{1}{p_{4}}}.

(14)

Since $|v_{k}|_{H_{0}^{1}(\Omega)}\to 0$ as $k\to\infty$ , there exists $k_{0}\in\mathbb{N}$ such that

4\beta p_{4}|v_{k}|_{H_{0}^{1}(\Omega)}^{2}\leq 4\pi\quad\text{ for all }k\geq k_{0}.

Given that $|w_{k}|_{H_{0}^{1}(\Omega)}=1$ , the Trudinger–Moser inequality implies that the sequence from (14) is uniformly bounded for $k\geq k_{0}$ , whence $\left|e^{4\beta v_{k}^{2}}\right|_{L^{p_{4}}(\Omega)}$ is uniformly bounded for all $k\geq 0$ . Consequently, the right hand side of (12) converges to zero, which shows that our claim (11) holds.

We now prove that $\mathcal{N}_{\mathcal{F}}(u_{k})\rightarrow\mathcal{N}_{\mathcal{F}}(u)$ in $L^{2}(\Omega)$ . To this end, we apply Vitali’s theorem (see, e.g., [30, Lemma 9.1]). The continuity of $F$ , and hence of $\mathcal{F}$ , implies that

\mathcal{F}(x,u_{k}(x))\rightarrow\mathcal{F}(x,u(x))\quad\text{for a.e. }x\in\Omega.

(15)

Let $\varepsilon>0$ . Since $u_{k}\rightarrow u$ in $L^{q}(\Omega)$ for every $q\geq 1$ , the necessity part of Vitali’s theorem ensures that there exists $\delta_{1}>0$ such that

\max\left\{\sup_{k\geq 0}\int_{D}u_{k}^{2}\,dx,\,\sup_{k\geq 0}\int_{D}u_{k}^{2p}\,dx\right\}<\varepsilon

whenever $D\subset\Omega$ satisfies $\operatorname{meas}(D)<\delta_{1}$ . Similarly, from (11), there exists $\delta_{2}>0$ such that

\sup_{k\geq 0}\int_{D}e^{\beta u_{k}^{2}}\,dx<\varepsilon

for all $D\subset\Omega$ with $\operatorname{meas}(D)<\delta_{2}$ .

Thus, for any $D\subset\Omega$ with $\operatorname{meas}(D)<\delta:=\min\{\delta_{1},\delta_{2}\}$ , using the growth condition (8), one has

|\mathcal{N}_{\mathcal{F}}(u_{k})|_{L^{2}\left(\Omega\right)}\leq\mathcal{C}_{1}\operatorname{meas}(D)+\mathcal{C}_{2}|u_{k}^{\,p}|_{L^{2}(D)}+\mathcal{C}_{3}\left|e^{\beta u_{k}^{2}}\right|_{L^{2}(D)}\leq\mathcal{C}_{1}\delta+\mathcal{C}_{2}\varepsilon^{\frac{1}{2}}+\mathcal{C}_{3}\varepsilon^{\frac{1}{2}}.

(16)

By (15) and (16), the sufficient condition of Vitali’s theorem is satisfied, and therefore

\mathcal{N}_{\mathcal{F}}(u_{k})\rightarrow\mathcal{N}_{\mathcal{F}}(u)\quad\text{in }L^{2}(\Omega).

It then follows from [31, Lemma 1.1] that the entire sequence $\mathcal{N}_{\mathcal{F}}(u_{k})$ converges to $\mathcal{N}_{\mathcal{F}}(u)$ in $L^{2}(\Omega).$ Consequently, $\mathcal{N}_{\mathcal{F}}$ is continuous from $H_{0}^{1}(\Omega)$ into $H^{-1}(\Omega),$ as explained above. ∎

We return to the existence of a solution of (5), and thus of a weak solution of problem (4). More exactly, based on the Minty–Browder theorem, we have the following existence and uniqueness result.

Theorem 4.

Assume that the function $F$ satisfies conditions (h1) and (h2). Then problem (3) admits a unique weak solution in $H_{0}^{1}(\Omega)$ .

Proof.

Let $T:H_{0}^{1}\left(\Omega\right)\rightarrow H^{-1}\left(\Omega\right)$ be defined by

T=I-N,

where $I$ is the embedding operator from $H_{0}^{1}\left(\Omega\right)$ into $H^{-1}\left(\Omega\right).$ By Lemma 1, the operator $T$ is continuous and thus hemicontinuous. In view of Theorem 2, it remains to prove that $T$ is strongly monotone.

For any $u,v\in H_{0}^{1}(\Omega)$ , using (10), we obtain

		$\displaystyle\langle T(u)-T(v),u-v\rangle$
		$\displaystyle=\|u-v\|_{H_{0}^{1}\left(\Omega\right)}^{2}-\int_{\Omega}\bigl(\mathcal{F}(x,u(x))-\mathcal{F}(x,v(x))\bigr)(u(x)-v(x))\,dx$
		$\displaystyle=\|u-v\|_{H_{0}^{1}\left(\Omega\right)}^{2}-\int_{\Omega}g(\|x\|)\bigl(-F(u(x))+F(v(x))\bigr)(u(x)-v(x))\,dx$
		$\displaystyle\geq\|u-v\|_{H_{0}^{1}\left(\Omega\right)}^{2}-\gamma\int_{\Omega}g(\|x\|)(u(x)-v(x))^{2}\,dx$
		$\displaystyle\geq\|u-v\|_{H_{0}^{1}\left(\Omega\right)}^{2}-\gamma g(r_{-})\,\|u-v\|_{L^{2}(\Omega)}^{2}.$

Applying the Poincaré inequality (7), we deduce

\langle T(u)-T(v),u-v\rangle\geq c\,|u-v|_{H_{0}^{1}\left(\Omega\right)}^{2},

where, by condition (9),

c=1-\gamma\frac{g(r_{-})}{\lambda_{1}}>0.

Thus, $T$ is strongly monotone, so the Minty–Browder theorem ensures that $T$ is a bijection from $H_{0}^{1}(\Omega)$ onto $H^{-1}(\Omega)$ . Consequently, there exists a unique $u^{\ast}\in H_{0}^{1}(\Omega)$ such that $T(u^{\ast})=0$ , which is the unique weak solution to our problem (3). ∎

Remark 1.

Note that, in particular, if $F$ is nondecreasing, then condition (10) holds with $\gamma=0$ , and therefore Theorem 2 applies.

Given that the domain $\Omega$ is smooth, the unique solution to our problem (3) has a higher regularity.

Remark 2.

The unique solution $u\in H_{0}^{1}(\Omega)$ obtained in Theorem 4 satisfies $u\in W^{2,2}_{\mathrm{loc}}(\Omega)$ . This follows from the regularity result [26, Theorem 8.9], since $\mathcal{F}(\cdot,u)\in L^{2}(\Omega)$ .

If we additionally require that $\Omega$ is a $C^{\infty}$ -smooth domain and that the function $F$ is infinitely many times differentiable, then the following result holds.

Corollary 1.

Assume, in addition to the hypotheses of Theorem 4, that $\partial\Omega$ is of class $C^{\infty}$ and that $F\in C^{\infty}(\mathbb{R})$ . Then problem (3) admits a unique solution $u^{\ast}\in C^{\infty}(\Omega)$ .

Proof.

The conclusion follows immediately from Theorem 4 together with a classical regularity result (see [26, Theorem 8.14]). ∎

2.2 Existence and localization via Moser-Harnack inequality

In this subsection, we assume that the domain $\Omega$ is of class $C^{1,\beta}$ for some $\beta\in(0,1)$ , and that the function $F$ satisfies

\sup_{\mathbb{R}_{+}}F\leq h(r_{+}),

(17)

which guarantees the positivity of the right-hand side of equation (3) for every nonnegative function $u$ .

Recall that [26, Theorem 8.34] (see also [32]) if $\Omega\subset\mathbb{R}^{2}$ is a bounded domain of class $C^{1,\beta}$ , with $\beta\in(0,1)$ , and $f\in L^{\infty}(\Omega)$ , then the boundary value problem

\begin{cases}-\Delta u=f&\text{in }\Omega,\\ u=0&\text{on }\partial\Omega,\end{cases}

(18)

admits a unique weak solution $u\in C_{0}^{1,\beta}(\overline{\Omega})=\left\{v\in C^{1,\beta}(\overline{\Omega})\,:\,u=0\,\,\text{ on }\partial\Omega\right\}$ . Thus, we may define the solution operator

(-\Delta)^{-1}\colon\ L^{\infty}(\Omega)\rightarrow C_{0}^{1,\beta}(\overline{\Omega}),\quad f\mapsto(-\Delta)^{-1}f,

which assigns to each $f\in L^{\infty}(\Omega)$ the unique solution $u=:(-\Delta)^{-1}f$ of problem (18). The mapping $(-\Delta)^{-1}$ is a bounded linear operator. Since the embedding $C_{0}^{1,\beta}(\overline{\Omega})\hookrightarrow C_{0}^{1}(\overline{\Omega})$ is compact (see, e.g., [26, Lemma 6.36]), it follows that $(-\Delta)^{-1}$ is completely continuous as an operator from $L^{\infty}(\Omega)$ into $C_{0}^{1}(\overline{\Omega})$ . Moreover, $(-\Delta)^{-1}$ is order preserving (monotone) with respect to the usual pointwise ordering of functions.

Letting

X:=C_{0}\left(\overline{\Omega}\right)=\left\{u\in C(\overline{\Omega})\,:\,u=0\text{ on }\partial\Omega\right\},

be endowed with the supremum norm $|u|:=|u|_{\infty}=\sup_{\overline{\Omega}}|u(x)|$ , our problem (3) is equivalent to the fixed point equation in $X$ ,

u=N(u),

where

N\colon X\rightarrow X,\qquad N(u)=(-\Delta)^{-1}\mathcal{N}_{\mathcal{F}}(u).

Observe that the operator $N$ can be written as

N=I_{2}\circ(-\Delta)^{-1}\circ I_{1}\circ\mathcal{N}_{\mathcal{F}},

where the embedding mappings

I_{1}:C(\overline{\Omega})\rightarrow L^{\infty}(\Omega),\ \ \ \ I_{2}:C_{0}^{1}(\overline{\Omega})\rightarrow X

are bounded linear operators. Since $\mathcal{N}_{\mathcal{F}}$ is continuous from $X$ to $C(\overline{\Omega})$ and bounded (maps bounded sets into bounded sets), and $(-\Delta)^{-1}$ is completely continuous from $L^{\infty}(\Omega)$ into $C_{0}^{1}(\overline{\Omega})$ , it follows that $N$ is completely continuous from $X$ to itself, as the composition of a completely continuous operator with bounded continuous operators.

For any fixed subdomain $\Omega_{0}\subset\Omega$ such that $\overline{\Omega}_{0}\subset\Omega$ , and any $p\geq 1$ , we consider the continuous seminorm on $X$

\|u\|:=\,|u|_{L^{p}(\Omega_{0})}=\left(\int_{\Omega_{0}}|u(x)|^{p}\,dx\right)^{\frac{1}{p}}.

By means of $\|\cdot\|$ , we further define the cone

K:=\left\{u\in X:\ u\geq 0\ \text{and}\ u(x)\geq C\,\|u\|\ \text{for all }x\in\Omega_{0}\right\},

where $C=C(p,\Omega,\Omega_{0})$ is the constant given by Theorem 3.

Given arbitrary numbers $0<r,R<\infty$ , we denote by $K_{r,R}$ the conical set

K_{r,R}:=\left\{u\in K:r\leq\|u\|,\;\left|u\right|\leq R\right\},

and define the quantities

\Gamma_{R}:=h(r_{-})-\frac{R}{g(r_{-})\left|(-\Delta)^{-1}1\right|},\qquad\Lambda_{r}:=h(r_{+})-\frac{r}{g(r_{+})\left\|(-\Delta)^{-1}\chi_{\Omega_{0}}\right\|},

where $\chi_{\Omega_{0}}$ is the characteristic function of $\Omega_{0}$ , i.e.,

\chi_{\Omega_{0}}(x)=\begin{cases}1,\quad\text{ if }x\in\Omega_{0},\\ 0,\quad\text{ if }x\in\Omega\setminus\Omega_{0}.\end{cases}

The following existence and localization result holds.

Theorem 5.

Let $0<r,R<\infty$ be such that the set $K_{r,R}$ is nonempty and

\min_{[0,R]}F\geq\Gamma_{R},

(19)

and

\max_{[Cr,R]}F\leq\Lambda_{r}.

(20)

Then problem (3) admits a solution $u\in K$ satisfying

r\leq\|u\|,\quad|u|\leq R.

Proof.

First, since $\Omega\subset\mathcal{O}$ , condition (17) implies that $\mathcal{F}(x,y)\geq 0$ for all $x\in\Omega$ and $y\in\mathbb{R}_{+}$ , so the order-preserving property of $(-\Delta)^{-1}$ yields $N(u)\geq 0$ for all $u\in K$ . Moreover, the Moser–Harnack inequality ensures that $N(u)\in K$ for every $u\in K$ . Therefore, the cone $K$ is invariant under the operator $N$ , that is, $N(K)\subset K$ .

We claim that

N\bigl(\overline{\text{conv}}(K_{r,R})\bigr)\subset K_{r,R}.

(21)

Since the set $K_{r,R}$ is closed, by the continuity of $N$ it suffices to guaranty that

N\bigl(\text{conv}(K_{r,R})\bigr)\subset K_{r,R}.

We prove this in two steps.

Step 1: We show that $\left|N(u)\right|\leq R$ for all $u\in K$ satisfying $\left|u\right|\leq R$ .

Let $u\in K$ with $\left|u\right|\leq R$ , so that $u(x)\leq R$ for all $x\in\Omega$ . From (17), since $g$ and $h$ are decreasing, one clearly has

\sup_{x\in\Omega}\mathcal{F}(x,u(x))=\sup_{x\in\Omega}g(|x|)\bigl(-F(u(x))+h(|x|)\bigr)\leq g(r_{-})\left(-\min_{[0,R]}F+h(r_{-})\right),

which, by condition (19), yields

\sup_{x\in\Omega}\mathcal{F}(x,u(x))\leq\frac{R}{\left|(-\Delta)^{-1}1\right|}.

(22)

Also,

N(u)(x)=(-\Delta)^{-1}\mathcal{F}(x,u(x))\leq\sup_{\Omega}\mathcal{F}(\cdot,u(\cdot))\,(-\Delta)^{-1}1,

and therefore, using (22) together with the monotonicity of the norm $|\cdot|$ , we obtain

\left|N(u)\right|\leq\frac{R}{\left|(-\Delta)^{-1}1\right|}\left|(-\Delta)^{-1}1\right|=R,

as desired. Obviously, the same inequality holds for every $u\in\text{conv}(K_{r,R}).$

Step 2: We show that $\left\|N(\lambda u_{1}+(1-\lambda)u_{2})\right\|\geq r$ for all $u_{1},u_{2}\in K_{r,R}$ and $\lambda\in[0,1]$ .

Let $u_{i}\in K_{r,R}$ for $i=1,2$ , and let $\lambda\in[0,1]$ . By the definition of the cone $K$ , we have

u_{i}(x)\geq C\|u_{i}\|\geq Cr\quad\text{for all }x\in\Omega_{0},

and therefore

\lambda u_{1}(x)+(1-\lambda)u_{2}(x)\geq Cr\quad\text{for all }x\in\Omega_{0}.

(23)

Since $u_{i}(x)\leq R$ for all $x\in\Omega$ , it follows from (17) and (23) that

\min_{\Omega_{0}}\mathcal{F}\bigl(x,\lambda u_{1}(x)+(1-\lambda)u_{2}(x)\bigr)\geq g(r_{+})\left(-\max_{[Cr,R]}F+h(r_{+})\right),

which implies

\mathcal{F}\bigl(\lambda u_{1}+(1-\lambda)u_{2}\bigr)\geq g(r_{+})\left(-\max_{[Cr,R]}F+h(r_{+})\right)\chi_{\Omega_{0}}\ \ \text{on }\Omega.

(24)

Applying $(-\Delta)^{-1}$ to both sides of (24) (recall that $(-\Delta)^{-1}$ is monotone), we obtain

N\bigl(\lambda u_{1}+(1-\lambda)u_{2}\bigr)\geq g(r_{+})\left(-\max_{[Cr,R]}F+h(r_{+})\right)(-\Delta)^{-1}\chi_{\Omega_{0}}.

(25)

Now, taking the seminorm $\|\cdot\|$ in the above relation (25) (note that it is also monotone), and using condition (20), we obtain

	$\displaystyle\left\\|N\bigl(\lambda u_{1}+(1-\lambda)u_{2}\bigr)\right\\|$	$\displaystyle\geq g(r_{+})\left(-\max_{[Cr,R]}F+h(r_{+})\right)\left\\|(-\Delta)^{-1}\chi_{\Omega_{0}}\right\\|$
		$\displaystyle\geq\frac{r}{\left\\|(-\Delta)^{-1}\chi_{\Omega_{0}}\right\\|}\left\\|(-\Delta)^{-1}\chi_{\Omega_{0}}\right\\|$
		$\displaystyle=r.$

Consequently, our claim (21) follows. Since $N$ is completely continuous and $\overline{\text{conv}}(K_{r,R})$ is a bounded, closed, and convex set, Schauder’s fixed point theorem applied to the operator

N:\overline{\text{conv}}(K_{r,R})\rightarrow K_{r,R}\subset\overline{\text{conv}}(K_{r,R})

ensures the existence of a fixed point $u^{\ast}\in\overline{\text{conv}}(K_{r,R})$ for $N$ , that is, $u^{\ast}=N(u^{\ast})$ . Since $N(u^{\ast})\in K_{r,R}$ , it follows that $u^{\ast}\in K_{r,R}$ . This completes our proof, as any fixed point of $N$ is a solution of problem (3); so, $u^{\ast}\in K$ solves (3), and by the definition of $K_{rR}$ it satisfies $r\leq\|u^{\ast}\|$ and $\left|u^{\ast}\right|\leq R$ .

∎

Remark 3.

In the case $p=1$ , the proof simplifies since the set $K_{r,R}$ is convex. Indeed, when $p=1,$ the considered seminorm is linear on $K$ , and therefore the set

\left\{u\in K:\|u\|\geq r\right\}

is convex. Since

K_{r,R}=\left\{u\in K:\|u\|\geq r\right\}\cap\left\{u\in K:|u|\leq R\right\},

and both sets in the intersection are convex, it follows that $K_{r,R}$ is convex as well.

We emphasize that, in the case $p=1$ , the functional $\left\|\cdot\right\|$ is linear on the cone $K$ , although it is not linear on the whole space $X.$

Remark 4.

In the case that $F$ is a monotone function, the two conditions in Theorem 5 reduce simply to the behavior of $F$ at only two points:

(a) If $F$ is nondecreasing on $\left[0,R\right]$ , then condition (19) holds with $F(0)$ in place of $\min_{[0,R]}F$ , and condition (20) holds with $F(R)$ in place of $\max_{[Cr,R]}F$ .

(b) If $F$ is nonincreasing on $\left[0,R\right]$ , then condition (19) holds with $F(R)$ instead of $\min_{[0,R]}F$ , while condition (20) holds with $F(Cr)$ instead of $\max_{[Cr,R]}F$ .

The localization Theorem 5 immediately ensures multiple solutions in the case where the nonlinearity $F$ is oscillatory.

Theorem 6.

Let $r_{i}$ and $R_{i}$ , $i=1,2,\ \ldots,n$ , be positive real numbers such that the sets $K_{r_{i},R_{i}}$ are nonempty and pairwise disjoint. If

\min_{[0,R_{i}]}F\geq\Gamma_{R_{i}},\quad\max_{[Cr_{i},R_{i}]}F\leq\Lambda_{r_{i}},\quad\text{for }i=1,2,\ \ldots,n,

then problem (3) admits at least $n$ distinct solutions $u_{i}\in K$ satisfying

r_{i}\leq\|u_{i}\|,\quad\left|u_{i}\right|\leq R_{i},\quad\text{for }i=1,2,\ \ldots,n.

Proof.

The application of Theorem 5 to each set $K_{r_{i},R_{i}}$ yields a solution in that set. The sets being pairwise disjoint, these $n$ solutions are distinct. ∎

Remark 5.

(a) A necessary condition for a set $K_{r,R}$ be nonempty is $rC\leq R.$ Indeed, if $u\in K_{r,R},$ then $u\geq C\left\|u\right\|$ on $\Omega_{0},$ $\left\|u\right\|\geq r$ and $u\leq\left|u\right|\leq R.$ These inequalities immediately yields $rC\leq R.$

(b) A sufficient condition for the set $K_{r,R}$ to be nonempty is $r\leq R\left\|\phi_{1}\right\|,$ where $\phi_{1}$ is the first eigenfunction of the Dirichlet problem for the Laplacian, normalized by $\left|\phi_{1}\right|=1.$ Indeed, under this condition there exists a positive constant $\mu$ such that $\mu\phi_{1}\in$ $K_{r,R}.$ In fact, $\mu$ must satisfy $\mu\left|\phi_{1}\right|\leq R$ and $\mu\left\|\phi_{1}\right\|\geq r.$ Such a number $\mu$ exists provided that $r\leq R\left\|\phi_{1}\right\|.$

(c) If the numbers $r_{i},\ R_{i}$ in Theorem 6 satisfy $\ r_{i}C\leq R_{i}<r_{i+1}C$ for $i=1,2,\ ...,n-1,$ then the sets $K_{r_{i},R_{i}}$ are pairwise disjoint. To see this, from the above condition, we have $r_{i}<r_{j}$ for $1\leq i<j\leq n.$ If now $u\in K_{r_{i},R_{i}},$ then

r_{i+1}C>R_{i}\geq u\geq C\left\|u\right\|\geq r_{i}C\ \ \ \text{in }\Omega_{0},

whence $\left\|u\right\|<r_{i+1}<r_{i+2}<\ ...<r_{n}.$ Hence $u\notin K_{r_{j},R_{j}}$ for every $j>i.$ Similarly, the conditions on $r_{i}$ and $R_{i}$ imply $R_{j}<R_{i}$ for $1\leq j<i\leq n.$ Then $\left|u\right|\geq r_{i}C>R_{i-1}>R_{i-2}>\ ...>R_{1},$ which shows that $u\notin K_{r_{j},R_{j}}$ for every $j<i.$ Therefore, the sets $K_{r_{i},R_{i}}$ are pairwise disjoint.

\bmhead

AcknowledgementsThe second author A. Stan acknowledges the support provided by the project ”Nonlinear Studies of Stratified Oceanic and Atmospheric Flows”, funded by the European Union through the Next Generation EU initiative and the Romanian Government under the National Recovery and Resilience Plan for Romania. The project is contracted under number 760040/23.05.2023, cod PNRR-C9-I8-CF 185/22.11.2022, through the Romanian Ministry of Research, Innovation, and Digitalization, within Component 9, Investment I8. He is also grateful to Professor Călin I. Martin for valuable discussions and helpful suggestions throughout the course of this project.
Both authors thank the anonymous reviewer for the careful reading of the manuscript and for the suggested modifications.

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References

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		$\displaystyle\langle T(u)-T(v),u-v\rangle$
		$\displaystyle=\|u-v\|_{H_{0}^{1}\left(\Omega\right)}^{2}-\int_{\Omega}\bigl(\mathcal{F}(x,u(x))-\mathcal{F}(x,v(x))\bigr)(u(x)-v(x))\,dx$
		$\displaystyle=\|u-v\|_{H_{0}^{1}\left(\Omega\right)}^{2}-\int_{\Omega}g(\|x\|)\bigl(-F(u(x))+F(v(x))\bigr)(u(x)-v(x))\,dx$
		$\displaystyle\geq\|u-v\|_{H_{0}^{1}\left(\Omega\right)}^{2}-\gamma\int_{\Omega}g(\|x\|)(u(x)-v(x))^{2}\,dx$
		$\displaystyle\geq\|u-v\|_{H_{0}^{1}\left(\Omega\right)}^{2}-\gamma g(r_{-})\,\|u-v\|_{L^{2}(\Omega)}^{2}.$

	$\displaystyle\left\\|N\bigl(\lambda u_{1}+(1-\lambda)u_{2}\bigr)\right\\|$	$\displaystyle\geq g(r_{+})\left(-\max_{[Cr,R]}F+h(r_{+})\right)\left\\|(-\Delta)^{-1}\chi_{\Omega_{0}}\right\\|$
		$\displaystyle\geq\frac{r}{\left\\|(-\Delta)^{-1}\chi_{\Omega_{0}}\right\\|}\left\\|(-\Delta)^{-1}\chi_{\Omega_{0}}\right\\|$
		$\displaystyle=r.$

Existence, Uniqueness, and Localization Results for a Dirichlet Problem in Geophysics

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Existence, uniqueness, and localization results for a Dirichlet problem in geophysics

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1 Introduction and Preliminaries

Theorem 1 (Trudinger–Moser).

Theorem 2 (Minty–Browder).

Theorem 3.

2 Main results

2.1 Existence and uniqueness via Minty-Browder theorem

Lemma 1.

Proof.

Theorem 4.

Proof.

Remark 1.

Remark 2.

Corollary 1.

Proof.

2.2 Existence and localization via Moser-Harnack inequality

Theorem 5.

Proof.

Remark 3.

Remark 4.

Theorem 6.

Proof.

Remark 5.

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