M. P. SEMENOV

ON THE SOLVABILITY OF BOUNDARY-VALUE PROBLEMS FOR QUASILINEAR ELLIPTIC SYSTEMS

(Presented by Academician A. Yu. Ishlinskii, 15 V 1964)

Let (\Omega) be an (n)-dimensional domain with sufficiently smooth boundary (\Gamma). Consider the operator

[
Au(x)=\sum_{|\alpha|=|\beta|=m}(-1)^m D^\alpha\left[a_{\alpha\beta}(x)D^\beta u(x)\right],
]

where (a_{\alpha\beta}) are real, symmetric, sufficiently smooth matrices of order (N^2), satisfying the condition (a_{\alpha\beta}=a_{\beta\alpha}). Suppose that for any function (u(x)\in C_0^\infty(\Omega)), finite, infinitely differentiable, with compact support in (\Omega), the inequality

[
(Au,u){L_2}\geq \mu^2|u|,\quad \mu>0.}{W}{}^{(m)}_2}^{\,2
\tag{1}
]

is fulfilled.

Then the operator (A) can be extended in a unique way to the space (\overset{0}{W}{}^{(m)}_2(\Omega)) to a self-adjoint positive definite operator, and inequality (1) remains valid for the extension of the operator (A) and all (u(x)\in \overset{0}{W}{}^{(m)}_2(\Omega)) ({}^{1}). Assuming such an extension has been carried out, we consider the problem of the existence of solutions of the equation

[
Au(x)=f\left[x,u(x),D^\gamma u(x)\right],\quad |\gamma|\leq m,
\tag{2}
]

belonging to the space (\overset{0}{W}{}^{(m)}_2(\Omega)). In the present work it is proved that, under the fulfillment of certain conditions, the boundary-value problem has at least one solution. In the case when (f(x,0,0)\ne 0), i.e., when the function (u(x)\equiv 0) is not a solution of the boundary-value problem, conditions are given for the existence of at least one solution distinct from the identical zero.

1. Suppose that the function (f(x,u,\zeta_\gamma)) is continuous in the aggregate of the variables (x\in \overline{\Omega}), (|u|), (|\zeta_\gamma|<\infty), and satisfies the inequalities

[
|f(x,u,\zeta_\gamma)|\leq M(x,u,\zeta_\delta)\left[1+\sum_{k=k_0+1}^{m}\sum_{|\gamma|=1}|\zeta_\gamma|^{p_k}\right]^{1/p};
\tag{3}
]

[
uf(x,u,\zeta_\gamma)\leq a(x)+\sigma_0|u|^2+\sum_{k=1}^{m}\sigma_k\sum_{|\gamma|=k}|\zeta|^2,
\tag{4}
]

where (M(x,u,\zeta_\delta)) is continuous in the aggregate of the variables (x\in\overline{\Omega}), (|u|), (|\zeta_\delta|<\infty), (|\delta|\leq k_0=m-[n/2]), (p_k=2+\dfrac{4(2m-k)}{n-2(2m-k)}), (p>\dfrac{2n}{n+2m}), (a(x)\in L(\Omega)), and (\sigma_k) are nonnegative constants that will be determined below.

Theorem 1. If conditions (3) and (4) are fulfilled, then equation (2) has at least one solution (u(x)\in \overset{0}{W}{}^{(m)}_2(\Omega)).

Proof. Let (u(x)\in \overset{0}{W}{}^{(m)}_2(\Omega)) be a solution of equation (2). Then, by virtue of (1),

[
\mu^2|u|{\overset{0}{W}{}^{(m)}_2}^{\,2}(Au,u)
=\left(f\left[x,u(x),D^\gamma u(x)\right],u(x)\right)_{L_2}.
]

Hence, by virtue of (4), it follows that

[
\mu^2|u|{\overset{0}{W}{}^{(m)}_2}
\leq
a_0+\sum,}^{m} c_k\sigma_k|u|^2_{\overset{0}{W}{}^{(m)}_2
]

where (c_k) are the norms of the embedding operators of (\overset{0}{W}{}^{(m)}2(\Omega)) into (W^{(k)}_2(\Omega)), (0\leq k\leq m), (a_0=\int\Omega a(x)\,dx).

Assuming that (\sum_{k=0}^{m} c_k\sigma_k<\mu^2), we obtain from this that

[
|u(x)|{\overset{0}{W}{}^{(m)}_2}
\leq
\left[
a_0\Big/\left(\mu^2-\sum c_k\sigma_k\right)}^{m
\right]^{1/2}.
\tag{5}
]

The operator (A) establishes a one-to-one and continuous mapping of the space (\overset{0}{W}{}^{(m)}_2(\Omega)) onto the space (\overset{0}{W}{}^{(-m)}_2(\Omega)=(\overset{0}{W}{}^{(m)}_2)^); moreover the operator (A^{-1}) admits the factorization (A^{-1}=TT^), where (T) is a continuous operator acting from (L_2(\Omega)) to (\overset{0}{W}{}^{(m)}_2(\Omega)), and (T^*) is a continuous operator acting from (\overset{0}{W}{}^{(-m)}_2(\Omega)) to (L_2(\Omega)) ((^{2-4})).

Since (p>2n/(n+2m)), we have (L_p(\Omega)\subset \overset{0}{W}{}^{(-m)}_2(\Omega)). Consequently, the operator (T^*) is completely continuous as an operator acting from (L_p(\Omega)), (p>2n/(n+2m)), to (L_2(\Omega)).

Consider in (L_2(\Omega)) the operator equation

[
v(x)=T^* f[x,Tv(x),D^\gamma Tv(x)].
\tag{6}
]

It is easy to see that, if the function (v(x)\in L_2(\Omega)) is a solution of equation (6), then the function (u(x)=Tv(x)) belongs to (\overset{0}{W}{}^{(m)}_2(\Omega)) and satisfies equation (2).

To prove the solvability of equation (6), we use the theorem (see ((^5))):

If a completely continuous operator (B), acting in a Hilbert space, satisfies the inequality

[
(Bu,u)<(u,u),\qquad |u|=r,
]

then the equation (u=Bu) has at least one solution (u_0), and (|u_0|<r).

Let (|v(x)|{L_2}=r), (r>\left[a_0/\left(\mu^2-\sum). Then, putting}^{m} c_k\sigma_k\right)\right]^{1/2

[
Bv(x)=T^* f[x,Tv(x),D^\gamma Tv(x)],
]

we shall have, by virtue of (4),

[
\begin{aligned}
(Bv,v){L_2}
&=
\bigl(T^* f[x,Tv(x),D^\gamma Tv(x)],v(x)\bigr) \
&=
\int_\Omega f[x,Tv(x),D^\gamma Tv(x)]\,Tv(x)\,dx \
&\leq
|T|^2
\left(a_0+\sum_{k=0}^{m} c_k\sigma_k|v|^2_{L_2}\right),
\end{aligned}
]

where (|T|) is the norm of the operator (T), acting from (L_2(\Omega)) to (\overset{0}{W}{}^{(m)}_2(\Omega)). It is easy to see that (|T|<1/\mu). Consequently,

[
(Bv,v){L_2}<(v,v)=r.},\qquad |v|_{L_2
]

Thus, equation (6), and consequently also the boundary-value problem, have at least one solution.

We note that all the arguments remain valid if, instead of the space (\overset{\circ}{W}{}^{(m)}_2(\Omega)), one takes any closed subspace (V\subset \overset{\circ}{W}{}^{(m)}_2(\Omega)) containing (C^\infty_0(\Omega)). It is easy to show that the solutions of the boundary-value problems are continuously differentiable (2m-1) times.

1. Let now (f(x,0,0)\equiv 0). Then equation (2) has in (\overset{\circ}{W}{}^{(m)}_2(\Omega)) the trivial solution (u(x)\equiv 0). The following theorem on the existence of solutions of the boundary-value problem different from the trivial one holds.

Theorem 2. Let the conditions of Theorem 1 be satisfied. Suppose, in addition, that:

a) (f(x,0,0)\equiv 0);

b) the function (f(x,u,\xi_\gamma)) has partial derivatives with respect to the variables (u=(u_1,\ldots,u_N)) and all variables (\xi_\gamma), (|\gamma|\leqslant m), continuous in some neighborhood of the point (u=0,\ \xi_\gamma=0);

c) the matrix (f_u(x,0,0)=(f_{iu_j}(x,0,0))) does not depend on (x), and the matrices (f_{\xi_\gamma}(x,0,0)\equiv 0);

d) there exists an independent, positive definite, completely continuous operator (Q), acting in (L_2(\Omega)), such that

[
\mu_1 Qv(x)\leqslant T^* f_u(x,0,0)Tv(x)\leqslant \mu_2 Qv(x),\qquad v(x)\in L_2(\Omega),
\tag{7}
]

where

[
\sqrt{\mu_2^2-\mu_1^2}<\min_i\left|\mu_2-\frac{1}{\lambda_i}\right|,
]

where (\lambda) are the eigenvalues of the operator (Q);

e) the number (\mu_2) belongs to one of the intervals

[
\left(\frac{1}{\lambda_{2i-1}},\frac{1}{\lambda_{2i}}\right),\qquad i=1,2,\ldots .
]

Then equation (2) has at least one solution (u(x)\in\overset{\circ}{W}{}^{(m)}_2(\Omega)) different from the trivial one.

We give the scheme of the proof of Theorem 2.

Under the conditions of Theorem 1, the rotation of the vector field (I-B) on the sphere (|v|=r),

[
r>\left[a_0/\left(\mu^2-\sum_{k=0}^m c_k\sigma_k\right)\right]^{1/2},
]

is equal to 1. The operator (B_1v(x)=T^*f_u(x,0,0)Tv(x)) is the Fréchet derivative of the operator

[
Bv(x)=Tf[x,Tv(x),D^\gamma Tv(x)]
]

at the point (v(x)\equiv 0).

From inequalities (7) and (8) it is easy to derive the homotopy of the vector fields (I-B_1) and (I-\mu Q) ((\mu_1\leqslant\mu\leqslant\mu_2)) on spheres with center at the origin. From the homotopy of vector fields and the theorem of Leray and Schauder ({}^{(6)}), it follows that the index of the zero solution is equal to ((-1)^\beta), where (\beta) is the sum of the multiplicities of the characteristic numbers of the operator (Q) lying in the interval ((0,\mu)).

Since, by d), (\beta) is odd, the index of the zero solution is equal to (-1), while the rotation of the vector field (I-B) on the sphere (|v(x)|=r) is equal to 1; hence there exists at least one solution (v(x)\ne 0) of equation (6), with (|v(x)|_{L_2}