Solvability of Some Elliptic Equations with a Nonlocal Boundary Condition

Abstract

In this work, we study a nonlocal boundary value problem for a quasilinear elliptic equation. Using the method of regularization and parameter continuation, we prove the existence and uniqueness of a regular solution to the nonlocal boundary value problem, i.e., a solution that possesses all generalized derivatives in the sense of S. L. Sobolev, which are involved in the corresponding equation.

1. Introduction

Nonlocal problems for elliptic equations are problems in which the boundary conditions are determined not only by the values of the function and its derivatives at the boundary of the domain but also include dependencies on the values of the function in the entire domain or on its parts. Nonlocal boundary conditions can be set in the form of conditions that include integral or functional dependencies on the values of the solution at the boundary or in the domain. These problems extend classical boundary conditions, such as Dirichlet or Neumann conditions, which are local and determined only by values at the domain’s boundary.

From a mathematical point of view, the presence of a nonlocal term in an equation or in conditions causes a lot of difficulties in some standard methods for solving elliptic problems. For example, variational methods do not work when applied to prove the existence of a large class of such equations. Nonlocal problems for various partial differential equations have been considered by many authors, including N.I. Ionkin [1], A.V. Bitsadze, A.A. Samarsky [2], A.I. Kozhanov [3,4], Ya.A. Roitberg and Z.G. Sheftel [5], A.P. Soldatov [6], N.V. Zhitarashu, and S.D. Eidelman [7], where special attention was paid to the solvability of nonlocal problems, and the uniqueness issue of the boundary value problem [8,9,10,11,12,13,14,15].

In [3], the solvability of nonlocal boundary value problems with integral conditions for elliptic equations is investigated:

$$\begin{array}{l}{u}_{tt}+\Delta u=f\left(x,t\right)\\ {\left.u\left(x,t\right)\right|}_S=0,\\ u\left(x,0\right)=0, x\in \Omega ,\\ {\displaystyle \underset{0}{\overset{T}{\int }}N\left(t\right)}u\left(x,t\right)dt=0, x\in \Omega .\end{array}$$

Proof of the nonlocal problem’s solvability is based on the Fourier method. In this work, the same method was also applied to some generalized problems, including cases with more general integral conditions as well as operator-differential and quasi-elliptic equations.

In [4], the correctness of nonlocal Samarsky–Ionkin problems is investigated for elliptic equations of the second order:

$$\hspace{0.17em}{u}_{xx}+{\Delta }_{y}u+c\left(x,y\right)u=f\left(x,y\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(x,y)\in (0,1)\times \Omega ,$$

$${\left.u\right|}_S=0,$$

$$u\left(0,y\right)=\gamma \left(y\right)u\left(1,y\right), u_x\left(1,y\right)=0, y\in \Omega \hspace{0.17em}$$

$$(\mathrm{or} u_x\left(0,y\right)=\gamma \left(y\right)u_x\left(1,y\right), u\left(1,y\right)=0, y\in \Omega \hspace{0.17em}).$$

The theorem on the existence and uniqueness of regular solutions in Sobolev spaces has been proven, and spectral problems for elliptic equations with nonlocal boundary conditions have also been studied.

Nonlocal problems describe physical and biological processes occurring in domains with boundaries inaccessible for direct measurements more accurately than other models. F. Browder, in his work [16], considered a case where nonlocal terms are related to the values of the desired function throughout the entire domain; coercivity inequality and a certain condition on the adjoint operator are satisfied.

In a work by A. Ashiraliyev [17], a boundary value problem with nonlocal boundary conditions is studied for elliptic equations.

$$-u^{″}\left(t\right)+Au\left(t\right)=f\left(t\right), \left(0\le \mathrm{t}\le 1\right), u\left(0\right)=u\left(1\right), u^{\prime }\left(0\right)=u^{\prime }\left(1\right).$$

In domain $\left[0,1\right]\times \Omega$, a mixed boundary value problem for a multidimensional elliptic equation is considered:

$$\begin{array}{l}-{\displaystyle \frac{{\partial }^{2}u\left(y,x\right)}{\partial y^2}}-{\displaystyle \sum _{r=1}^n{\alpha }_r}\left(x\right){\displaystyle \frac{{\partial }^{2}u\left(y,x\right)}{\partial x_r^2}}+\delta u\left(y,x\right)=f\left(y,x\right),\\ x=\left(x_1,\dots ,x_n\right)\in \Omega , 0<y<1, \\ u\left(0,x\right)=u\left(1,x\right), {\displaystyle \frac{\partial u\left(0,x\right)}{\partial y}}={\displaystyle \frac{\partial u\left(1,x\right)}{\partial y}}, x\in \overline{\Omega },\\ f\left(0,x\right)=f\left(1,x\right), u\left(y,x\right)=0, x\in S,\end{array}$$

where $\Omega$ is a unit open cube in space ${\mathrm{R}}^n, \left(0<x_k<1, 1\le k\le n\right)$ with boundary $S,$ $\overline{\Omega }=\Omega \cup S$.

The boundary value problem on the interval $\left\{0\le y\le 1, x\in {\mathrm{R}}^n\right\}$ is also considered for multidimensional elliptic equations of order 2 m:

$$\begin{array}{l}-{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}+{\displaystyle \sum _{\left|r\right|=2m}{\alpha }_r}\left(x\right){\displaystyle \frac{{\partial }^{\left|\tau \right|}u}{\partial x_1^{r_1}\dots \partial x_n^{r_n}}}+\delta u\left(y,x\right)=f\left(y,x\right),\\ 0<y<1, x,r\in R^n, \left|r\right|=r_1+\dots +r_n,\\ u\left(0,x\right)=u\left(1,x\right), u_y\left(0,x\right)=u_y\left(1,x\right), f\left(0,x\right)=f\left(1,x\right).\end{array}$$

The correctness of the stated boundary value problems has been established in the space of smooth functions.

A significant number of papers have been devoted to the study of the solvability of quasilinear and nonlinear elliptic equations; we note some of them ([18,19,20,21,22,23,24,25,26,27,28,29,30,31]).

The analysis presented in [32,33] demonstrates the application of operator decomposition methods in problems requiring precise solutions for Stokes flow in spheroidal geometries. This approach can be valuable for modeling hydrodynamic processes in complex environments, including microfluidics, biophysical systems, and particle dynamics. These studies illustrate the application of methods for solving elliptic equations with non-local boundary conditions in hydrodynamics.

The theory of linear nonlocal elliptic boundary value problems with conditions of the Bitsadze–Samarsky type was developed in the works of A.L. Skubachevsky [23]. The nonlocal elliptic problem is reduced to elliptic differential–difference equations. In [24], convergence of the Galerkin method in Sobolev–Orlicz spaces for quasilinear elliptic equations with rapidly increasing coefficients is proved. In [25], a nonlocal boundary value problem with a p-Laplacian is investigated. This nonlinear nonlocal elliptic-type problem is a generalized Bitsadze–Samarsky type problem, and sufficient conditions for its solvability have been obtained.

In [31], the following nonlocal elliptic equation with combined nonlinearities is investigated:

$${\left({\displaystyle \underset{R^N}{\int }{\left|u(x)\right|}^{2}dx}\right)}^{\gamma }\Delta u=\lambda u+\mu {\left|u\right|}^{q-2}u+{\left|u\right|}^{p-2}u, \mathrm{x}\in { \mathrm{R}}^N.$$

Using variational methods, results regarding the existence and non-existence of solutions are obtained for various cases of $\gamma >0,$ $p>1$, and $q>1$.

2. Results

Statement of the Problem. Let $Q=\{(x,y):\hspace{0.17em}\hspace{0.17em}0<x<1,\hspace{0.17em}\hspace{0.17em}y\in \Omega \}$ be a cylinder; $\Omega \subset R^n,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n\ge 1$ is a limited domain with a smooth boundary $\partial \Omega \subset C^3$; and $S=(0,1)\times \partial \Omega$ is the lateral surface of the cylinder.

Let us consider a nonlinear equation in cylinder $Q$:

$$-u_{xx}-{\Delta }_{y}u-{\left|{\Delta }_{y}u\right|}^{p-2}{\Delta }_{y}u+c_{0}u=f(x,y),$$

(1)

with boundary conditions as follows:

$${\left.u\right|}_{\partial \Omega }=0, x\in [0,1],$$

(2)

$$u(0,y)=\alpha u(1,y), u_x(1,y)=0,\hspace{0.17em}\hspace{0.17em}y\in \Omega ,$$

(3)

where $f(x,y)$ is a given function; and $p>1,\hspace{0.17em}{c}_0>0$ and $\alpha >0$ are constants ${\Delta }_y={\displaystyle \frac{{\partial }^2}{\partial y_1^2}}+\dots +{\displaystyle \frac{{\partial }^2}{\partial y_n^2}}$.

The space $V_p\left(Q\right)$ is defined as follows:

$$V_p\left(Q\right)=\left\{u:u\in W_2^2(Q),\hspace{0.17em}\hspace{0.17em}{\Delta }_{y}u\hspace{0.17em}\in L_p\left(Q\right),\hspace{0.17em}\hspace{0.17em}{\Delta }_y{u}_{xx}\hspace{0.17em}\in L_2\left(Q\right)\right\}$$

with norm as follows:

$$\begin{array}{l}{‖u‖}_{V_p(Q)}={\left({\displaystyle \underset{Q}{\int }\left(|u{|}^2+|u_x{|}^2+|u_{xx}{|}^2+|{\Delta }_{y}u{|}^2+|{\Delta }_y{u}_{xx}{|}^2\right)dQ}\right)}^{{\displaystyle \frac{1}{2}}}+\\ +{\left({\displaystyle \underset{Q}{\int }|{\Delta }_{y}u{|}^{p}dQ}\right)}^{{\displaystyle \frac{1}{p}}}.\end{array}$$

In what follows, we will need the following theorem:

Theorem ([34]). 

Let $A(\lambda )$be a continuous operator on the interval $[0,1]$, i.e., for each $\lambda \in [0,1]:$$A(\lambda )\in L(X,\hspace{0.17em}Y)$, the operator $A(0)$ is continuously invertible. If for $A(\lambda )$the following condition holds:

$$\exists \gamma >0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall \lambda \in [0,1],\hspace{0.17em}\hspace{0.17em}\forall x\in X:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}‖A(\lambda )x‖\ge \gamma ‖x‖,$$

Then,  $A(1)$ is continuously invertible; moreover $‖A^{-1}(1)‖\le {\gamma }^{-1}.$

Theorem 1. 

Let the following conditions hold: $2<p<{\displaystyle \frac{2n}{n-2}},$  $0<\alpha <1,$

$$f,\hspace{0.17em}{\nabla }_{y}f\in L_2(Q),$$

(4)

then, there exists a unique solution$u\in W_2^2(Q),\hspace{0.17em}\hspace{0.17em}{\Delta }_{y}u\hspace{0.17em}\in L_p\left(Q\right)$of the nonlocal boundary value problem (1)–(3).

Proof. 

We will prove the theorem using the regularization method and the method of continuation by parameter. For $\epsilon >0$ and $\lambda \in [0,1]$, consider the boundary value problem in the cylinder $Q$:

$$\epsilon {\Delta }_y{u}_{xx}-u_{xx}-{\Delta }_{y}u-{\left|{\Delta }_{y}u\right|}^{p-2}{\Delta }_{y}u+c_{0}u=f(x,y),$$

(5)

with a boundary condition (2) and

$$u(0,y)=\lambda \alpha u(1,y), u_x(1,y)=0, \hspace{0.17em}\mathrm{y}\in \Omega .$$

(6_λ)

The solvability of the boundary value problem (5), (2), (6_λ) will be proved using the theorem on the method of continuation by parameter [18,34,35]. Let us denote by $\Lambda$ the set of those values $\lambda$ from the interval $[0,1]$ for which the boundary value problem (5), (2), (6_λ) is solvable in the space $V_p(Q)$. If this set is non-empty, simultaneously open, and closed, then it will, as is known, coincide with the entire interval $[0,1]$. In this case, the boundary value problem (5), (2), (6₁) will have a solution that belongs to the space $V_p(Q)$. This will mean solvability in the space $W_2^2(Q)$ of the boundary value problem (1)–(3).

So, it is necessary to prove the non-emptiness of set $\Lambda$, and its openness and closedness.

To prove the non-emptiness of set $\Lambda$, we will show that the number 0 belongs to set $\Lambda$.

Let $\lambda =0$, then, from (5), (2), (6_λ), we obtain the boundary value problem in the cylinder$Q=\{(x,y):\hspace{0.17em}\hspace{0.17em}0<x<1,\hspace{0.17em}\hspace{0.17em}y\in \Omega \}$

$$u(0,y)=0, u_x(1,y)=0, \hspace{0.17em}y\in \Omega .$$

(6₀)

We will prove the unique solvability of the boundary value problem (5), (2), (6₀) in the space $V_p(Q)$.

Definition 1. 

A generalized solution of the boundary value problem (5), (2), (6₀) is called the function $u(x,y)\in V_p\left(Q\right)$that satisfies conditions (2), (6₀) and the integral identity.

$${\displaystyle \underset{Q}{\int }\left(\epsilon {\Delta }_y{u}_{xx}-u_{xx}-{\Delta }_{y}u-{\left|{\Delta }_{y}u\right|}^{p-2}{\Delta }_{y}u+c_{0}u\right)vdQ}={\displaystyle \underset{Q}{\int }f(x,y)vdQ},$$

(7)

for $v\in W_2^2\left(Q\right),$${\left.v\right|}_{\partial \Omega }=0, \mathrm{x}\in [0,1],$$v(0,y)=0,{ \mathrm{v}}_x(1,y)=0, \hspace{0.17em}\mathrm{y}\in \Omega$.

Theorem 2. 

Let the conditions of Theorem 1 be satisfied; then, there exists a unique generalized solution $u\in V_p(Q)$of the boundary value problem (5), (2), (6₀).

Proof. 

The solvability of the boundary value problem (5), (2), (6₀) in the space $V_p\left(Q\right)$ will be proven using the Galerkin method and compactness. The approximate solutions $u^N(x,y)$ are sought in the form as follows:

$$u^N(x,y)={\displaystyle \sum _{j=1}^N{C}_{Nj}(x){\psi }_j(y)},$$

(8)

where$\{{\psi }_j(y)\}$ is the basis of the space $W_2^2\left(\Omega \right)\cap \stackrel{0}{W_2^1}(\Omega ),$ composed of the eigenfunctions of the boundary value problem as follows:

$$\left\{\begin{array}{l}{\Delta }_y{\psi }_j(y)+{\mu }_j{\psi }_j(y)=0,\\ {\left.{\psi }_j\right|}_{\partial \Omega }=0.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\end{array}\right.$$

The unknown coefficients $C_{Nj}(x)$are determined from the system of $N$ differential equations as follows:

$$\begin{array}{l}{\displaystyle \sum _{k=1}^N{\displaystyle \underset{\Omega }{\int }\left(-\epsilon {\mu }_k{C^{″}}_{Nk}(x){\psi }_k(y)-{C^{″}}_{Nk}(x){\psi }_k(y)+\right.}}\\ +{\mu }_k{C}_{Nk}(x){\psi }_k(y)+{\mu }_k{\left|{\Delta }_y{u}_N\right|}^{p-2}{C}_{Nk}(x){\psi }_k(y)+\\ +\left.c_0{C}_{Nk}(x){\psi }_k(y)\right){\psi }_j(y)dy=\\ ={\displaystyle \underset{\Omega }{\int }f(x,y){\psi }_j(y)dy},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\hspace{0.17em}2,\hspace{0.17em}\dots ,\hspace{0.17em}N,\end{array}$$

(9)

with boundary conditions as follows:

$$X_{Nj}(0)=0, {X^{\prime }}_{Nj}(1)=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,\dots ,N.$$

(10)

The solvability of the boundary value problem (9)–(10) follows from the general theory of ordinary differential equations. The convergence of the sums (8) requires a priori estimates of the solutions $u^N.$

Multiply Equation (9) respectively by ${\mu }_j{C}_{Nj}(x)$ and $C_{Nj}(x)$; sum over$j=1,\hspace{0.17em}2,\hspace{0.17em}\dots ,\hspace{0.17em}N,$; integrate over $x$ from 0 to 1; and then, we obtain the following:

$$\begin{array}{l}\epsilon {‖{\Delta }_y{u}_{Nx}‖}_{2,Q}^2+{‖{\nabla }_y{u}_{Nx}‖}_{2,Q}^2+{‖{\Delta }_y{u}_N‖}_{2,Q}^2+{‖{\Delta }_y{u}_N‖}_{p,Q}^p+\\ +c_0{‖{\nabla }_y{u}_N‖}_{2,Q}^2=-{\displaystyle \underset{Q}{\int }f{\Delta }_y{u}_{N}dQ},\end{array}$$

(11)

$$\begin{array}{l}\epsilon {‖{\nabla }_y{u}_{Nx}‖}_{2,Q}^2+{‖u_{Nx}‖}_{2,Q}^2+{‖{\nabla }_y{u}_N‖}_{2,Q}^2+c_0{\displaystyle \underset{Q}{\int }|u_N{|}^{2}dQ}=\\ ={\displaystyle \underset{Q}{\int }|{\Delta }_y{u}_N{|}^{p-2}{\Delta }_y{u}_N\cdot u_{N}dQ}+{\displaystyle \underset{Q}{\int }f{u}_{N}dQ}.\end{array}$$

(12)

From relation (11), we easily derive the following estimate:

$$\epsilon {‖{\Delta }_y{u}_{Nx}‖}_{2,Q}^2+{‖{\nabla }_y{u}_{Nx}‖}_{2,Q}^2+{‖{\Delta }_y{u}_N‖}_{2,Q}^2+{‖{\Delta }_y{u}_N‖}_{p,Q}^p+c_0{‖{\nabla }_y{u}_N‖}_{2,Q}^2\le C_1.$$

(13)

Taking into account conditions (4), the right-hand side of equality (12) will be estimated as follows:

$$\begin{array}{l}\left|{\displaystyle \underset{Q}{\int }|{\Delta }_y{u}_N{|}^{p-2}{\Delta }_y{u}_N\cdot u_{N}dQ}\right|\le {\left({\displaystyle \underset{Q}{\int }|{\Delta }_y{u}_N{|}^{p}dQ}\right)}^{{\displaystyle \frac{p-1}{p}}}{\left({\displaystyle \underset{Q}{\int }|u_N{|}^{p}dQ}\right)}^{{\displaystyle \frac{1}{p}}}\le \\ \le p^{{\displaystyle \frac{p}{1-p}}}(p-1){\displaystyle \underset{Q}{\int }|{\Delta }_y{u}_N{|}^{p}dQ}+{\displaystyle \underset{Q}{\int }|u_N{|}^{p}dQ}\le p^{{\displaystyle \frac{p}{1-p}}}(p-1)C_1+\\ +C_2^{″}{\displaystyle \underset{Q}{\int }{\left|{\nabla }_y{u}_N\right|}^{2}dQ}\le C_2^{\prime }.\end{array}$$

$$\left|{\displaystyle \underset{Q}{\int }f{u}_{N}dQ}\right|\le {\displaystyle \frac{1}{2}}{\displaystyle \underset{Q}{\int }|u_N{|}^{2}dQ}+{\displaystyle \frac{1}{2}}{\displaystyle \underset{Q}{\int }|f{|}^{2}dQ}.$$

Then, from relation (13), we obtain the estimate as follows:

$$\epsilon {‖{\nabla }_y{u}_{Nx}‖}_{2,Q}^2+{‖u_{Nx}‖}_{2,Q}^2+{‖{\nabla }_y{u}_N‖}_{2,Q}^2+c_0{‖u_N‖}_{2,Q}^2\le C_2.$$

(14)

Now, multiplying Equation (10) by ${\mu }_j{C}_{\mathit{Nj}}^{″}(x)$ and summing over $j=1,\hspace{0.17em}2,\hspace{0.17em}\dots ,\hspace{0.17em}N,$ and integrating over $x$ from 0 to 1, we obtain the following:

$$\begin{array}{l}\epsilon {‖{\Delta }_y{u}_{Nxx}‖}_{2,Q}^2+{‖{\nabla }_y{u}_{Nxx}‖}_{2,Q}^2+{‖{\Delta }_y{u}_{Nx}‖}_{2,Q}^2+(p-1){\displaystyle \underset{Q}{\int }\left|{\Delta }_y{u}_N{|}^{p-2}\right|{\Delta }_y{u}_{Nx}{|}^{2}dQ}=\\ =-c_0{\displaystyle \underset{Q}{\int }{\nabla }_y{u}_N{\nabla }_y{u}_{Nxx}dQ}+{\displaystyle \underset{Q}{\int }{\nabla }_{y}f{\nabla }_y{u}_{Nxx}dQ}.\end{array}$$

(15)

Estimating the right-hand side of identity (15), taking into account estimate (15), we obtain the following:

$$\begin{array}{l}\left|c_0{\displaystyle \underset{Q}{\int }{\nabla }_y{u}_N{\nabla }_y{u}_{Nxx}dQ}\right|\le \\ \le c_0{\left({\displaystyle \underset{Q}{\int }|{\nabla }_y{u}_N{|}^{2}dQ}\right)}^{{\displaystyle \frac{1}{2}}}{\left({\displaystyle \underset{Q}{\int }|{\nabla }_y{u}_{Nxx}{|}^{2}dQ}\right)}^{{\displaystyle \frac{1}{2}}}\le {\displaystyle \frac{1}{4}}{‖{\nabla }_y{u}_{Nxx}‖}_{2,Q}^2+C_3^{″}.\end{array}$$

$$\left|{\displaystyle \underset{Q}{\int }{\nabla }_{y}f{\nabla }_y{u}_{Nxx}dQ}\right|\le {\displaystyle \frac{1}{4}}{‖{\nabla }_y{u}_{Nxx}‖}_{2,Q}^2+{‖{\nabla }_{y}f‖}_{2,Q}^2.$$

Thus, from relation (16), we obtain the estimate as follows:

$$\epsilon {‖{\Delta }_y{u}_{Nxx}‖}_{2,Q}^2+{‖{\nabla }_y{u}_{Nxx}‖}_{2,Q}^2+{‖{\Delta }_y{u}_{Nx}‖}_{2,Q}^2+{\displaystyle \underset{Q}{\int }\left|{\Delta }_y{u}_N{|}^{p-2}\right|{\Delta }_y{u}_{Nx}{|}^{2}dQ}\le C_3.$$

(16)

Similarly, by multiplying identity (9) in a scalar sense in $L_2(0,1)$ by the function $C_{\mathit{Nj}}^{″}(x)$ and summing over$j=1,\hspace{0.17em}2,\hspace{0.17em}\dots ,\hspace{0.17em}N,$ we obtain the following:

$$\begin{array}{l}\epsilon {‖{\nabla }_y{u}_{Nxx}‖}_{2,Q}^2+{‖u_{Nxx}‖}_{2,Q}^2+{‖{\nabla }_y{u}_{Nx}‖}_{2,Q}^2+c_0{\displaystyle \underset{Q}{\int }{u}_{Nx}^{2}dQ}=\\ =-{\displaystyle \underset{Q}{\int }|{\Delta }_y{u}_N{|}^{p-2}{\Delta }_y{u}_N\cdot u_{Nxx}dQ}+{\displaystyle \underset{Q}{\int }f{u}_{Nxx}dQ}.\end{array}$$

(17)

Let us estimate the right-hand side of equality (17) by applying Hölder’s and Young’s inequalities, and taking into account estimates (13) and (16), we obtain the following:

$$\begin{array}{l}\left|{\displaystyle \underset{Q}{\int }|{\Delta }_y{u}_N{|}^{p-2}{\Delta }_y{u}_N\cdot u_{Nxx}dQ}\right|\le {\left({\displaystyle \underset{Q}{\int }|{\Delta }_y{u}_N{|}^{p}dQ}\right)}^{{\displaystyle \frac{p-1}{p}}}{\left({\displaystyle \underset{Q}{\int }|u_{Nxx}{|}^{p}dQ}\right)}^{{\displaystyle \frac{1}{p}}}\le \\ \le p^{{\displaystyle \frac{p}{1-p}}}(p-1){\displaystyle \underset{Q}{\int }|{\Delta }_y{u}_N{|}^{p}dQ}+{\displaystyle \underset{Q}{\int }|{\nabla }_y{u}_{Nxx}{|}^{2}dQ}\le p^{{\displaystyle \frac{p}{1-p}}}(p-1)C_1+\\ +C_2^{″}{\displaystyle \underset{Q}{\int }{\left|{\nabla }_y{u}_{Nxx}\right|}^{2}dQ}\le C_4^{″}.\end{array}$$

$$\left|{\displaystyle \underset{Q}{\int }f{u}_{Nxx}dQ}\right|\le {\displaystyle \frac{1}{4}}{\displaystyle \underset{Q}{\int }|u_{Nxx}{|}^{2}dQ}+{\displaystyle \underset{Q}{\int }|f{|}^{2}dQ}.$$

Substituting into relation (16), we obtain the estimate as follows:

$$\epsilon {‖{\nabla }_y{u}_{Nxx}‖}_{2,Q}^2+{‖u_{Nxx}‖}_{2,Q}^2+{‖{\nabla }_y{u}_{Nx}‖}_{2,Q}^2+c_0{\displaystyle \underset{Q}{\int }{u}_{Nx}^{2}dQ}\le C_4.$$

(18)

In the a priori estimates (14), (15), (17), and (19), constants $C_i,\hspace{0.17em}\hspace{0.17em}i=1,2,3,4$ do not depend on $N$.

Thus, we have obtained the following estimate:

$${‖u_N(x,y)‖}_{V_p(Q)}\le C_5.$$

(19)

From the estimate (19), it follows that the sequence of approximate solutions $\left\{u^N(x,y)\right\}$is bounded in the space $V_p(Q)$. Then, it is possible to select a subsequence $\left\{u^{N_k}\right\}$ and pass to the limit over $N$ for $k\to \infty$ in system (9). The limit function belongs to the space $V_p(Q)$. Since the system $\left\{{\psi }_j(y)\right\}$ is dense in $L_2(\Omega ),$ we obtain that on the limiting function $u(x,t)$, the Equation (5) is satisfied almost everywhere in $Q.$ Thus, the existence of a regular solution to the boundary value problem (5), (2), (6₀) is proven.

The uniqueness of the solution to the boundary value problem (5), (2), (6₀) is proven by contradiction as usual. Suppose there are two solutions, $u_1(x,y)$ and $u_2(x,y)$, and let their difference be denoted as $u(x,y)=u_1(x,y)-u_2(x,y)$. Then, from the integral identity (7), we have the following:

$${\displaystyle \underset{Q}{\int }\left[\epsilon {\Delta }_y{u}_{xx}-u_{xx}-{\Delta }_{y}u-\left({\left|{\Delta }_y{u}_1\right|}^{p-2}{\Delta }_y{u}_1-{\left|{\Delta }_y{u}_2\right|}^{p-2}{\Delta }_y{u}_2\right)+c_{0}u\right]vdQ}=0,$$

(20)

for $v\in W_2^2\left(Q\right),\hspace{0.17em}\hspace{0.17em}$${\left.v\right|}_{\partial \Omega }=0, x\in [0,1],$$v(0,y)=0,{ \mathrm{v}}_x(1,y)=0, \hspace{0.17em}\mathrm{y}\in \Omega .$.

In the given identity, assuming $v(x,y)={\Delta }_{y}u(x,y),$ we obtain the following:

$$\begin{array}{l}\epsilon {‖{\Delta }_y{u}_x‖}_{2,Q}^2+{‖{\nabla }_y{u}_x‖}_{2,Q}^2+{‖{\Delta }_{y}u‖}_{2,Q}^2+\\ +{\displaystyle \underset{Q}{\int }\left({\left|{\Delta }_y{u}_1\right|}^{p-2}{\Delta }_y{u}_1-{\left|{\Delta }_y{u}_2\right|}^{p-2}{\Delta }_y{u}_2\right){\Delta }_{y}udQ}+c_0{‖{\nabla }_{y}u‖}_{2,Q}^2=0.\end{array}$$

(21)

Considering the monotonicity of the operator ${\left|{\Delta }_{y}u\right|}^{p-2}{\Delta }_{y}u$, we obtain the following:

$${\displaystyle \underset{Q}{\int }\left({\left|{\Delta }_y{u}_1\right|}^{p-2}{\Delta }_y{u}_1-{\left|{\Delta }_y{u}_2\right|}^{p-2}{\Delta }_y{u}_2\right){\Delta }_y(u_1-u_2)dQ}\ge 0.$$

(22)

From (22) and (21), the following is deduced:

$$\epsilon {‖{\Delta }_y{u}_x‖}_{2,Q}^2+{‖{\nabla }_y{u}_x‖}_{2,Q}^2+{‖{\Delta }_{y}u‖}_{2,Q}^2+c_0{‖{\nabla }_{y}u‖}_{2,Q}^2\le 0,$$

This implies $u(x,y)=0$. Therefore, the uniqueness of the solution to the boundary value problem (5), (2), $(6_0)$ is proven. □

Thus, Theorem 2 is completely proven.

The openness and closedness of set $\Lambda$ follow from a priori estimates. Let us derive these estimates.

Multiplying Equation (5) by the functions $-x{\Delta }_{y}u$ and $-x{\Delta }_y{u}_{xx}$, and then integrating over the cylinder $Q$, we obtain the following relations:

$$\begin{array}{l}\epsilon {\displaystyle \underset{Q}{\int }x{\left|{\Delta }_y{u}_x\right|}^{2}dQ}+{\displaystyle \frac{\epsilon }{2}}{\displaystyle \underset{\Omega }{\int }\left[{\left|{\Delta }_{y}u(1,y)\right|}^2-{\left|{\Delta }_{y}u(0,y)\right|}^2\right]}dy+{\displaystyle \underset{Q}{\int }x{\left|{\Delta }_y{u}_x\right|}^{2}dQ}+\\ +{\displaystyle \underset{Q}{\int }x{\left|{\Delta }_{y}u\right|}^{p}dQ}+c_0{\displaystyle \underset{Q}{\int }x|{\nabla }_{y}u{|}^{2}dQ}={\displaystyle \underset{Q}{\int }{\nabla }_y{u}_x{\nabla }_{y}udQ}-{\displaystyle \underset{Q}{\int }f}x{\Delta }_{y}udQ.\end{array}$$

(23)

$$\begin{array}{l}\epsilon {\displaystyle \underset{Q}{\int }x{\left|{\Delta }_y{u}_{xx}\right|}^{2}dQ}+{\displaystyle \underset{Q}{\int }x{\left|{\nabla }_y{u}_{xx}\right|}^{2}dQ}+{\displaystyle \frac{1}{p}}{\displaystyle \underset{\Omega }{\int }\left[{\left|{\Delta }_{y}u(1,y)\right|}^p-{\left|{\Delta }_{y}u(0,y)\right|}^p\right]}dy+\\ +(p-1){\displaystyle \underset{Q}{\int }x{\left|{\Delta }_{y}u\right|}^{p-2}{\left|{\Delta }_y{u}_x\right|}^{2}dQ}=-{\displaystyle \underset{Q}{\int }c(x,y)ux{\Delta }_y{u}_{xx}dQ+}{\displaystyle \underset{Q}{\int }fx{\Delta }_y{u}_{xx}dQ}\end{array}$$

(24)

Applying Hölder’s and Young’s inequalities to the right-hand side of relations (23) and (24), and taking into account the conditions theorem, we obtain the following estimates:

$$\begin{array}{l}\epsilon {\displaystyle \underset{Q}{\int }x{\left|{\Delta }_y{u}_x\right|}^{2}dQ}+\epsilon (1-\alpha ){\displaystyle \underset{\Omega }{\int }{\left|{\Delta }_{y}u(1,y)\right|}^2}dy+{\displaystyle \underset{Q}{\int }x{\left|{\Delta }_y{u}_x\right|}^{2}dQ}+\\ +{\displaystyle \underset{Q}{\int }x{\left|{\Delta }_{y}u\right|}^{p}dQ}+c_0{\displaystyle \underset{Q}{\int }x|{\nabla }_{y}u{|}^{2}dQ}\le C_1,\end{array}$$

(25)

$$\begin{array}{l}\epsilon {\displaystyle \underset{Q}{\int }x{\left|{\Delta }_y{u}_{xx}\right|}^{2}dQ}+{\displaystyle \underset{Q}{\int }x{\left|{\nabla }_y{u}_{xx}\right|}^{2}dQ}+{\displaystyle \frac{1-\alpha }{p}}{\displaystyle \underset{\Omega }{\int }{\left|{\Delta }_{y}u(1,y)\right|}^p}dy+\\ +{\displaystyle \underset{Q}{\int }x{\left|{\Delta }_{y}u\right|}^{p-2}{\left|{\Delta }_y{u}_x\right|}^{2}dQ}\le C_2.\end{array}$$

(26)

Similarly, multiplying in a scalar way Equation (5) by functions $-x{\Delta }_{y}u$ and$-x{\Delta }_y{u}_{xx}$ in $L_2(Q)$, and taking into account estimates (25) and (26), we obtain the following estimate for the function $u(x,y)$:

$$\epsilon {\displaystyle \underset{Q}{\int }{\left|{\Delta }_y{u}_x\right|}^{2}dQ}+{\displaystyle \underset{Q}{\int }{\left|{\Delta }_y{u}_x\right|}^{2}dQ}+{\displaystyle \underset{Q}{\int }{\left|{\Delta }_{y}u\right|}^{p}dQ}+c_0{\displaystyle \underset{Q}{\int }|{\nabla }_{y}u{|}^{2}dQ}\le C_3.$$

(27)

$$\epsilon {\displaystyle \underset{Q}{\int }{\left|{\Delta }_y{u}_{xx}\right|}^{2}dQ}+{\displaystyle \underset{Q}{\int }{\left|{\nabla }_y{u}_{xx}\right|}^{2}dQ}+{\displaystyle \underset{Q}{\int }{\left|{\Delta }_{y}u\right|}^{p-2}{\left|{\Delta }_y{u}_x\right|}^{2}dQ}\le C_4.$$

(28)

The obtained estimates will allow us to prove the openness and closedness of set $\Lambda$.

Let$\left\{{\lambda }_k\right\}$ be a sequence of elements from set $\Lambda$, converging to the number ${\lambda }_0$. We will show that ${\lambda }_0$ is also an element of $\Lambda$.

For every ${\lambda }_k$, there corresponds a function $u_k(x,y),$ belonging to the space $V_p(Q)$, and representing the solution to the boundary value problem (5), (2), $(6_{{\lambda }_k})$. For the family of functions $\{u_k(x,y)\}$, the uniform estimates (25)–(28) with respect to k will hold. From these estimates, it follows that there exists a function $u(x,y).$ in the space $V_p(Q)$ and a subsequence $\{u_k(x,y)\}$ such that for $k\to \infty$ the following convergences hold:

$$u_k(x,y)\to u(x,y) \mathrm{weakly} \mathrm{in} V_p(Q).$$

From these convergences, it follows that the limiting function $u(x,y).$ will be a solution to equation (5), and that it will satisfy conditions (2) and $(6_{{\lambda }_0})$. The estimates (25)–(28) for the solutions to problem (5), (2), $(6_{{\lambda }_0})$ further imply that the function $u(x,t)$ will belong to the space $V_p(Q)$. From this, it follows that the number ${\lambda }_0$ will be an element of set $\Lambda$.

Membership of a limit point for set $\Lambda$ to the set itself implies its closedness.

Let us now prove the openness of set $\Lambda$.

Suppose the number ${\lambda }_0$ is an element of set $\Lambda$. Set $\Lambda$ will be open if the numbers $\lambda ={\lambda }_0+\tilde{\lambda }$ for small values of $\left|\tilde{\lambda }\right|$ also belong to this set.

Let us take an arbitrary function $u(x,y)$ from the space $V_p(Q)$ and consider the boundary value problem (5), (2) and

$$u(0,y)={\lambda }_0\alpha u(1,y)+\tilde{\lambda }\alpha w(1,y), u_x(1,y)=0, \hspace{0.17em}\mathrm{y}\in \Omega .$$

(29)

According to the definition of set $\Lambda$, this problem has a solution $u(x,y)$, belonging to the space $V_p(Q)$. Thus, the boundary value problem (5), (2), (29) generates an operator $K,$ translating space $V_p(Q)$ into itself $K(w)=u$. We will show that for small values of $\left|\tilde{\lambda }\right|$, this operator will be contractive.

Let $w_1(x,y)$ and $w_2(x,y)$ be functions from the space $V_p(Q)$; $u_1(x,y)$ and $u_2(x,y)$ are corresponding images under acting operator $K.$ Let us denote$w(x,y)=w_2(x,y)-w_1(x,y),$$u(x,y)=u_2(x,y)-u_1(x,y)$. For the function $u(x,y)$, the equation

$$\epsilon {\Delta }_y{u}_{xx}-u_{xx}-{\Delta }_{y}u-\left({\left|{\Delta }_y{u}_2\right|}^{p-2}{\Delta }_y{u}_2-{\left|{\Delta }_y{u}_1\right|}^{p-2}{\Delta }_y{u}_1\right)+c_{0}u=0,$$

and boundary conditions are satisfied:

$${\left.u\right|}_{\partial \Omega }=0, x\in [0,1],$$

$$u(0,y)={\lambda }_0\alpha u(1,y)+\tilde{\lambda }\alpha w(1,y), u_x(1,y)=0, \hspace{0.17em}y\in \Omega .$$

Estimates (26)–(29) give the inequality for the function $u(x,y)$:

$${‖u(x,y)‖}_{V_p(Q)}\le C\left|\tilde{\lambda }\right|\left({‖w‖}_{L_2(Q)}+{‖w(1,y)‖}_{W_2^1(\Omega )}\right).$$

Inequality obviously follows from this inequality:

$${‖u(x,y)‖}_{V_p(Q)}\le C\left|\tilde{\lambda }\right|{‖w‖}_{V_p(Q)}.$$

If now the number $\tilde{\lambda }$ is such that $C\left|\tilde{\lambda }\right|<1$, then the operator $K$ will become a contraction operator. A contraction operator has a fixed point—a function $u(x,y)$ such that $K(u)=u$. For this function, the equation will hold as follows;

$$\epsilon {\Delta }_y{u}_{xx}-u_{xx}-{\Delta }_{y}u-\left({\left|{\Delta }_y{u}_2\right|}^{p-2}{\Delta }_y{u}_2-{\left|{\Delta }_y{u}_1\right|}^{p-2}{\Delta }_y{u}_1\right)+c_{0}u=f,$$

and with conditions as follows:

$${\left.u\right|}_{\partial \Omega }=0, x\in [0,1],$$

$$u(0,y)={\lambda }_0\alpha u(1,y)+\tilde{\lambda }\alpha u(1,y), u_x(1,y)=0, \hspace{0.17em}y\in \Omega .$$

This means that the number${\lambda }_0+\tilde{\lambda }$is an element of set $\Lambda$, and set $\Lambda$ is open.

From the proven result, it follows that the boundary value problem (5), (2), $(6_1)$ when the function $u(x,y)$ belongs to the space $V_p(Q)$ will have a solution $u\left(x,y\right),$ which also belongs to the space $V_p(Q)$. Thus, this boundary value problem generates an operator ${\rm T}$, which maps the space $V_p(Q)$ into itself ${\rm T}(u)=u$. We will show that the operator ${\rm T}$ has fixed points in $V_p(Q)$.

For the solutions of the boundary value problem (5), (2), the estimates (25)–(28) hold. It is not difficult to obtain the following estimate:

$${‖u(x,y)‖}_{V_p(Q)}\le C^{\prime }\left(M+{‖f‖}_{L_2(Q)}\right).$$

Consequently, the operator ${\rm T}$ will map any ball in the space $V_p(Q)$ with radius $r\ge C^{\prime }\left(M+{‖f‖}_{L_2(Q)}\right)$ into itself.

Let us now show that the operator ${\rm T}$ is completely continuous.

Let the sequence of functions $\{u_m(x,y)\}$ be a bounded sequence of functions in the space $V_p(Q)$. From the estimates (25)–(28) due to uniform boundedness, it follows that the sequence $\{u_m(x,y)\}$ will be bounded in the space $V_p(Q)$. From the boundedness of the sequences$\{u_m(x,y)\},\hspace{0.17em}\hspace{0.17em}\{{\Delta }_y{u}_m(x,y)\},\hspace{0.17em}\hspace{0.17em}\{{\Delta }_y{u}_{mxx}(x,y)\}$ and the compactness of embedding $W_2^1(Q)$ into$C(Q)$, it follows that there exists subsequences $\{u_{m_k}(x,y)\},\hspace{0.17em}\hspace{0.17em}\{{\Delta }_y{u}_{m_k}(x,y)\},\hspace{0.17em}\hspace{0.17em}\{{\Delta }_y{u}_{m_{k}xx}(x,y)\}$ of the corresponding sequences, and functions $u(x,y),\hspace{0.17em}\hspace{0.17em}{\Delta }_{y}u(x,y),\hspace{0.17em}\hspace{0.17em}{\Delta }_y{u}_{xx}(x,y)$ such that for $k\to \infty$ the following convergencies hold:

$$u_{m_k}(x,y)\to u(x,y)\mathrm{weakly} \mathrm{in} L_2(Q),$$

$${\Delta }_y{u}_{m_k}(x,y)\to {\Delta }_{y}u(x,y)\mathrm{weakly} \mathrm{in} L_2(Q),$$

$${\Delta }_y{u}_{m_{k}xx}(x,y)\to {\Delta }_y{u}_{xx}(x,y)\mathrm{weakly} \mathrm{in} L_2(Q),$$

$$u_{m_k}(0,y)\to u(0,y)\mathrm{strongly} \mathrm{in} C(\Omega ),$$

$$u_{m_k}(1,y)\to u(1,y)\mathrm{strongly} \mathrm{in} C(\Omega ),$$

$$u_{m_{k}x}(1,y)\to u_x(1,y)\mathrm{strongly} \mathrm{in} C(\Omega )$$

From these convergences, it follows that the function $u(x,y)$ will be associated with the problem (5), (2), (6₁). Let us denote$w_k(x,y)=u_{m_k}(x,y)-u(x,y)$. The following equation holds:

$$\epsilon {\Delta }_y{w}_{\mathrm{kxx}}-w_{\mathrm{kxx}}-{\Delta }_y{w}_k-\left({\left|{\Delta }_y{u}_{m_k}\right|}^{p-2}{\Delta }_y{u}_{m_k}-{\left|{\Delta }_y{u}_m\right|}^{p-2}{\Delta }_y{u}_m\right)+c_0{w}_k=0,$$

This equation, along with the estimates (25)–(28) for the functions $w_k(x,y)$, the boundedness of $V_p(Q)$ in the space $V_p(Q)$, and the aforementioned convergences of the subsequence $\{u_{m_k}(x,y)\}$, leads to the following:

$${‖w_k(x,y)‖}_{V_p(Q)}\to 0 \mathrm{for} k\to \infty .$$

From any bounded sequence $\{u_m(x,y)\}$ in the space $V_p(Q)$, it is possible to extract a convergent subsequence from the sequence $\{{\rm T}(u_m)\}$. This means that the operator T is completely continuous in the space $V_p(Q).$

The complete continuity of the operator ${\rm T}$, mapping a ball in the space $V_p(Q)$ of sufficiently large radius into itself, and Schauder’s theorem, give us that the operator ${\rm T}$ has a fixed point in the space $V_p(Q)$. This point, i.e., the function $u(x,y)$ from the space $V_p(Q)$, will be the desired solution of the boundary value problem (5), (2), $(6_1)$.

The limiting transition for $\epsilon$ in Equation (5) is established based on the a priori estimates obtained earlier. We select a sequence of numbers $\left\{{\epsilon }_m\right\}$ converging to zero. From the sequences $\{u_m(x,y)\}$ of solutions to the boundary value problems (5), (2), $(6_{\lambda })$ in the case $\lambda =1,\hspace{0.17em}\hspace{0.17em}\epsilon ={\epsilon }_m$, we extract a subsequence $\{u_{m_k}(x,y)\}$$k\to \infty$ such that the following holds:

$$u_{m_k}(x,y)\to u(x,y)\mathrm{weakly} \mathrm{in} L_2(Q),$$

$${\Delta }_y{u}_{m_k}(x,y)\to {\Delta }_{y}u(x,y)\mathrm{weakly} \mathrm{in} L_2(Q),$$

$${\epsilon }_{m_k}{\Delta }_y{u}_{m_{k}xx}(x,y)\to 0\mathrm{weakly} \mathrm{in} L_2(Q),$$

$$u_{m_k}(0,y)\to u(0,y)\mathrm{strongly} \mathrm{in} C(\Omega ),$$

$$u_{m_k}(1,y)\to u(1,y)\mathrm{strongly} \mathrm{in} C(\Omega ),$$

$$u_{m_{k}x}(1,y)\to u_x(1,y)\mathrm{strongly} \mathrm{in} C(\Omega ).$$

It follows that the limit function $u(x,y)$ will be the desired solution to the boundary value problem (1)–(3).

Thus, we have proved Theorem 1. □

3. Conclusions

In this paper, we investigated a nonlocal boundary value problem for a quasilinear elliptic equation. In conclusion, we note that similar results can be obtained for more general equations, for example, as follows:

$$-u_{xx}-{\Delta }_{y}u+c_{0}u+A(u,\hspace{0.17em}\nabla u,\hspace{0.17em}\Delta u)=f(x,y),$$

where the operator $A(u,\hspace{0.17em}\nabla u,\hspace{0.17em}\Delta u)$ can take the form of $|u{|}^{p-2}u,\hspace{0.17em}\hspace{0.17em}|u{|}^{p-2}\Delta u,\hspace{0.17em}\hspace{0.17em}\left|u{|}^p\right|\Delta u{|}^{q-2}\Delta u$ among others.