MATHEMATICS

S. I. POKHOZHAEV

ON A BOUNDARY-VALUE PROBLEM FOR THE EQUATION (\Delta U = U^2)

(Presented by Academician M. A. Lavrent'ev, January 5, 1961)

In the present note we consider the boundary-value problem

[
\Delta U = U^2, \qquad U\big|_{\Gamma} = \varphi(s)
\tag{1}
]

for twice continuously differentiable real functions (U(x,y)) in a bounded domain (G) with boundary (\Gamma), under continuous boundary values (\varphi(s)).

First, the existence of a nontrivial solution of problem (1) is established for the case of zero boundary values. Then a theorem is proved on the existence of two distinct solutions of the boundary-value problem (1) under certain conditions on the function (\varphi(s)). Further, the boundary-value problem (1) is investigated for the case of nonpositive values of the function (\varphi(s)). In this case some sufficient conditions for the existence or nonexistence of solutions of the boundary-value problem (1) are indicated.

Let (K(P,Q)) be the Green’s function of the operator (L(U)=\Delta U-qU), where the function (q(x,y)) has continuous first-order derivatives; (P) and (Q) are points of the domain (G), and suppose that the equation

[
\psi(P)=\lambda \int_G K(P,Q)\psi(Q)\,dQ
]

has only positive eigenvalues.

Theorem 1. There exists a (\mu) such that the integral equation

[
\mu V(P)=\int_G K(P,Q)V^2(Q)\,dQ
\tag{2}
]

has a nontrivial solution.

Proof. Consider the auxiliary variational problem: to find the greatest value of the functional

[
F(x_1,x_2,\ldots,x_n,\ldots)
=
\frac{1}{3}\int_G
\left(\sum_1^\infty x_k\varphi_k(Q)\right)^3 dQ
]

under the condition

[
H(x_1,x_2,\ldots,x_n,\ldots)
=
\frac{1}{2}\sum_1^\infty \lambda_k x_k^2
=
c^2,
]

where (\varphi_k(Q)) and (\lambda_k) are, respectively, the orthonormal eigenfunctions and eigenvalues of the kernel (K(P,Q)). From the theorems of M. Vainberg ((^2)) on the properties of quadratic integral forms in the spaces (L_q) ((q \le 2)), it follows that the functional (F) is defined on the manifold (H=c^2) and is bounded in absolute value.

To prove the existence of a solution of this variational problem we apply the Ritz method.

The maximizing sequence of elements (X^{(n)}=(x_1^{(n)},x_2^{(n)},\ldots,x_n^{(n)})) is found from the condition that the functional (F_n), defined by the equality

[
F_n(x_1,x_2,\ldots,x_n)=F(x_1x_2,\ldots,x_n,0,\ldots)
]

attains at the point (X^{(n)}) its maximum (d_n) under the condition

[
H_n(x_1,x_2,\ldots,x_n)=H(x_1,x_2,\ldots,x_n,0,\ldots)=c^2.
]

The coordinates of the point (X^{(n)}) satisfy the system of equations

[
\mu_n\lambda_i x_i^{(n)}
=
\int_G
\left(\sum_1^n x_k^{(n)}\varphi_k(Q)\right)^2
\varphi_i(Q)\,dQ,
\tag{3}
]

where

[
\mu_n=\frac{3}{2c^2}\,d_n.
]

By a known device, from the sequence of elements

[
\widetilde X^{(n)}
=
\left(\sqrt{\lambda_1}x_1^{(n)},\sqrt{\lambda_2}x_2^{(n)},\ldots,\sqrt{\lambda_n}x_n^{(n)}\right)
]

one extracts a subsequence ({\widetilde X^{(n_k)}}) weakly convergent in (l_2), such that for every (i)

[
\sqrt{\lambda_i}\,x_i^{(n_k)}
\to
\sqrt{\lambda_i}\,x_i^{(0)}
\quad \text{as } k\to\infty.
]

Applying the theorems of Vainberg ((^2)), from system (3) we obtain the system of equations

[
\mu\lambda_i x_i^{(0)}
=
\int_G
\left(\sum_1^\infty x_k^{(0)}\varphi_k(Q)\right)^2
\varphi_i(Q)\,dQ;
]

the element (X^{(0)}=(x_1^{(0)},x_2^{(0)},\ldots,x_n^{(0)},\ldots)) also solves the original variational problem. For the quantity (\mu) the inequality

[
\frac{\sqrt{2}\,c}{\lambda_1^{3/2}}\,
|\varphi_1|{L_s^3}
\le
\mu
\le
\frac{(\sqrt{2}c)^{3/2}}{3}\,
|K|.}(G\times G)
]

In obtaining the lower bound for (\mu) it was assumed that (\varphi_1(P)\geq 0). This is ensured if (q(x,y)\geq 0), or if (|q(x,y)|) is sufficiently small.

The sequence of functions

[
V_m(P)=\sum_1^m x_k^{(0)}\varphi_k(P)
]

converges uniformly in the domain (G) to the function (V(P)). The function (V(P)) is not identically zero and satisfies equation (2).

Remark. The function (U(P)=\dfrac{1}{\mu}V(P)) is not identically zero and satisfies the equation

[
U(P)=\int_G K(P,Q)U^2(Q)\,dQ.
]

Theorem 2. The boundary-value problem (1) with (\varphi\equiv 0) has a nontrivial solution.

Proof. It suffices to set the function (q(x,y)) equal to zero and use the remark to Theorem 1.

Let (K_0(P,Q)) be the Green’s function of the Laplace operator for the domain (G). Denote

[
\max_G \int_G K_0(P,Q)\,dQ=K_1,\qquad
\max_\Gamma |\varphi(s)|=B.
]

Theorem 3. If the function (\varphi(s)) satisfies one of the conditions:
a) (\varphi(s)\geqslant 0); b) (4BK_1<1), then the boundary-value problem (1) has two distinct solutions.

Proof. The existence of the first solution (U_0(P)), nonpositive in case a) and small in the sense that

[
|U_0(P)| \leqslant \frac{1-\sqrt{1-4BK_1}}{2K_1},
]

in case b), was proved in papers ((^3,^4)). Put now (U(P)=U_0(P)+W(P)). Then the function (W(P)) satisfies the following conditions:

[
\Delta W-2U_0W=W^2,\qquad W|_{\Gamma}=0.
\tag{4}
]

Let us use Theorem 1. Take as the function (q(x,y))

[
q(x,y)=2U_0(x,y).
]

In case a) the function (q(x,y)) is nonpositive, and in case b) the function (q(x,y)) is sufficiently small in absolute value, so that all eigenvalues of the kernel (K(P,Q)) are positive and the first eigenfunction (\varphi_1(P)\geqslant 0).

By virtue of the remark to Theorem 1, there exists a nontrivial function (W(P)) satisfying conditions (4).

Let us pass to the study of the boundary-value problem (1) in the case of nonpositive values of the function (\varphi(s)).

Lemma. There exists a constant (C_*) such that the boundary-value problem for the disk (r\leqslant R)

[
\Delta V=V^2,\qquad V(R)=C
\tag{5}
]

has a solution if (C\geqslant C_/R^2), and has no solution if (C<C_/R^2).

This fact follows from the study of the family of solutions of the boundary-value problem (5) depending only on (r). A numerical solution of the corresponding ordinary differential equation gives for the value (C_) the approximate value (C_=-1.40).

From Theorem 2 it follows that in the closed domain (\overline{G}) there exists a solution (V(P)) of the equation (\Delta V=V^2), taking on the boundary (\Gamma) negative values (\psi(s)).

Theorem 4. If the function (\varphi(s)) is such that

[
\psi(s)\leqslant \varphi(s)\leqslant 0,
]

then a solution of the boundary-value problem (1) exists.

Proof. Consider the following method of successive approximations:

[
\Delta U_k=U_{k-1}^2,\qquad U_k|_{\Gamma}=\varphi(s)\quad (k=1,2,\ldots),
]

where (U_0(x,y)) is the harmonic function corresponding to the boundary values (\varphi(s)).

We shall show that the sequence of functions (U_k) is a monotone sequence. Indeed, for the functions (U_1(x,y)) and (U_0(x,y)) the inequality

[
U_1(x,y)\leqslant U_0(x,y),
]

holds, since

[
\Delta(U_1-U_0)=U_0^2\geqslant 0,\qquad U_1-U_0|_{\Gamma}=0.
]

Similarly, by induction we obtain the inequalities

[
U_k(x,y)\leq U_{k-1}(x,y)\qquad (k=1,2,\ldots)
]

The sequence ({U_k}) is bounded below. Indeed, for the function (U_1(x,y)) we have

[
\Delta(V-U_1)=V^2-U_0^2\geq 0,\qquad V-U_1\big|_\Gamma=\psi(s)-\varphi(s)\leq 0;
]

therefore, everywhere in the domain (G) the inequality

[
V(x,y)\leq U_1(x,y)
]

holds. It is also not difficult to see that

[
V(x,y)\leq U_k(x,y)\qquad (k=1,2,\ldots).
]

Writing the integral equation for the limiting function (U(x,y)),

[
U(x,y)=\lim_{k\to\infty} U_k(x,y),
]

we obtain that the function (U(x,y)) solves the boundary-value problem (1).

Corollary. The boundary-value problem (1) has a solution if

[
\frac{C_*}{R^2}\leq \varphi(s)\leq 0,
]

where (R) is the radius of a disk containing the domain (G), and (C_*) is the quantity defined in the lemma.

Proof. By the lemma, for a disk of radius (R) there exists a solution of the boundary-value problem

[
\Delta V=V^2,\qquad V(R)=\frac{C_*}{R^2}.
]

This solution (V(P)) assumes on the boundary (\Gamma) of the domain (G) values (\psi(s)), which, by the subharmonicity of the function (V(P)), satisfy the inequality

[
\psi(s)\leq \frac{C_*}{R^2}.
]

Theorem 5. The boundary-value problem (1) has no solution if

[
\varphi(s)<\frac{C_*}{R_0^2},
]

where (R_0) is the radius of a disk contained in the domain (G), and (C_) is the quantity defined in the lemma.*

Proof. Suppose the contrary. Then in the domain (G) there exists a solution of the boundary-value problem (1), which on the boundary (\Gamma_0) of the disk of radius (R_0) assumes values (\psi(s)\leq \varphi(s)). Put (\max_{\Gamma_0}\psi(s)=A) and consider the following boundary-value problem:

[
\Delta V=V^2,\qquad V(R_0)=A.
]

Since (\psi(s)\leq A\leq 0), a solution of this boundary-value problem depending only on (r) exists by Theorem 4. But this contradicts the lemma, since (A