Full Text
UDC 517.9
MATHEMATICS
V. S. KLIMOV
ON SECOND SOLUTIONS OF BOUNDARY-VALUE PROBLEMS
(Presented by Academician I. G. Petrovskii, December 31, 1969)
In this paper, by methods of the theory of cones, boundary-value problems for nonlinear elliptic equations of second order are studied, as well as some boundary-value problems for ordinary differential equations of arbitrary order.
- Let \(K\) be a cone in a Banach space \(E\) \((^{1,2})\), and let \(K^*\) be the set of positive linear functionals. Let a nonlinear operator \(F\), acting and continuous from the Banach space \(E_1\) into \(E\) \((E_1 \subset E)\), and a linear operator \(A\), acting and completely continuous from \(E\) into \(E_1\), be positive: \(FE_1 \subset K,\ AK \subset K\).
Suppose that \(E_1\) is embedded in the Banach space \(E_0\). We shall call the linear operator \(A\) \(E_0\)-contracting if there exist a linear functional \(l \in K^*\), a vector \(u_0 \in K\), and positive numbers \(\lambda\) and \(k\) such that \(l(u_0)=1\) and, for every \(v \in K\), the inequalities
\[
l(Av) \geq \lambda l(v), \qquad l(Av) \geq k\|Av\|_{E_0}.
\tag{1}
\]
Theorem 1. Let the following conditions be satisfied:
1) \(A\) is an \(E_0\)-contracting operator;
2) \(F\theta=\theta\), where \(\theta\) is an isolated fixed point, completely continuous in \(E_1\), of the vector field \(\Phi u = u - AFu\), whose index is different from zero;
3) \(l(Fu) \geq a l(u) - b\) for every \(u \in K\) (here \(a,b\) are positive constants, with \(a\lambda>1\));
4) for solutions belonging to \(K \cap E_1\) of the family of operator equations
\[
u = (1-t)AFu + atAu \qquad (t \in [0,1])
\]
the relation
\[
\|u\|_{E_1} \leq \psi\bigl(\|u\|_{E_0}\bigr),
\]
holds, where \(\psi(s)\) is a continuous function.
Then the equation \(u=AFu\) has at least one nonzero solution.
Theorem 1 is a development of the well-known fixed-point principle of M. A. Krasnosel’skii for an operator stretching a cone. The greatest difficulties arise in verifying the first condition. The choice of the space \(E_0\) determines the character of the restrictions necessary for fulfilling the fourth condition, which is an abstract analogue of the classical \(L\)-condition of S. N. Bernstein.
- We give examples of the use of Theorem 1 for proving the existence of a solution of some boundary-value problems.
Let \(\Omega\) be a bounded domain in \(n\)-dimensional space with boundary \(\Gamma\) belonging to the class \(A^{2,\alpha}\) (see \((^{3})\)) for some \(\alpha \in (0,1)\).
Consider the differential equation
\[ \mathcal{L}u \equiv - \sum_{i,j=1}^{n} \frac{\partial}{\partial t_i} a_{ij}(t_1,\ldots,t_n)\frac{\partial u}{\partial t_j} + a(t_1,\ldots,t_n)u = f\left(t,u,\frac{\partial u}{\partial t_1},\ldots,\frac{\partial u}{\partial t_n}\right), \tag{2} \]
where \(t=\{t_1,\ldots,t_n\}\in \overline{\Omega}\), \(\overline{\Omega}=\Omega+\Gamma\). In what follows it is assumed that the operator \(\mathcal{L}\) is of elliptic type, that the coefficients \(a_{ij}(t), a(t)\) are sufficiently smooth functions \(\bigl(a_{ij}(t)\in C^{1+\alpha}, a(t)\in C^\alpha\bigr)\), with \(a(t)\geq 0\), and that the function \(f(t,u,p_1,\ldots,p_n)\) is defined and continuous for \(t\in \overline{\Omega}\) and arbitrary \(u,p_1,\ldots,p_n\).
Let \(\mu_1\) denote the first eigenvalue of the operator \(\mathcal{L}\) under the boundary condition
\[ u_\Gamma=0, \tag{3} \]
and let \(\alpha_n,\beta_n\) be numbers satisfying the inequalities
\[ \alpha_n(n-1)<n+1,\qquad \beta_n n<n+1,\qquad \alpha_n>0,\qquad \beta_n>0. \]
Theorem 2. Let the function \(f(t,u,p)\) be nonnegative, have continuous partial derivatives with respect to each of its arguments, let
\[
f(t,0,0,\ldots,0)=0
\]
and let the inequalities
\[
f(t,u,p)\geq a_1|u|-b_1 \qquad (b_1>0,\ a_1>\mu_1),
\]
\[
f(t,u,p)\leq C(u)(1+p^2)\quad (n=1),
\]
\[
f(t,u,p)\leq C(1+|u|^{\alpha_n}+|p|^{\beta_n})\quad (n>1);
\]
hold. Here \(C(u)\) is a continuous function, \(C\) is a positive constant, and
\[
|p|=\sqrt{p_1^2+p_2^2+\cdots+p_n^2}.
\]
Then the boundary-value problem (2)—(3) has a nonzero solution belonging to the class \(C^{2+\alpha}\).
An important role in the proof of Theorem 2 is played by
Lemma 1. Let \(\varphi(t)\) be a nonnegative eigenfunction, and let \(G(t,s)\) be the Green’s function of the operator \(\mathcal{L}\) under the boundary condition (2). Then
\[ \varphi(t)\geq k_1\|G(t,s)\|_{L_p(\Omega)}, \]
where \(k_1,p\) are positive constants, with \(p(n-1)<n\).
The proof of Lemma 2 is based on estimates of the type of Giraud—Oleinik (3).
- In this section we study the question of the existence of a solution of equation (2) satisfying, for \(t\in \Gamma\), the boundary condition
\[ \partial u/\partial \nu=\beta u. \tag{4} \]
Here \(\partial u/\partial \nu\) is the derivative along the inward normal, and the function \(\beta=\beta(t)\in C^{1+\alpha}\) is positive. Let \(\mu_2\) denote the first eigenvalue of the operator \(\mathcal{L}\) under the boundary condition (4).
Theorem 3. Let the function \(f(t_1,\ldots,t_n,u,p_1,\ldots,p_n)\) be nonnegative, have continuous partial derivatives with respect to each of its arguments, let
\[
f(t,0,\ldots,0)=0
\]
and satisfy the inequalities
\[ f(t,u,p)\geq a_2|u|-b_2 \qquad (b_2>0,\ a_2>\mu_2), \]
\[ f(t,u,p)\leq C\bigl(\exp |u|^{\gamma_2}+|p|^{\delta_2}\bigr), \quad \text{for } n=2, \]
\[ f(t,u,p)\leq C(1+|u|^{\gamma_n}+|p|^{\delta_n}), \quad \text{for } n>2. \]
Here \(C,\gamma_n,\delta_n\) are positive numbers, with
\[ \gamma_2<1,\qquad \delta_2<2,\qquad \gamma_n(n-2)<n,\qquad \delta_n(n-1)<n\quad (n>2). \]
Then the differential equation (2) has a positive solution satisfying the boundary condition (4) and belonging to \(C^{2+\alpha}\).
4. Let us consider the problem of finding a solution of the ordinary differential equation
\[ Mu \equiv \sum_{i=0}^{m} p_i(t)u^{(i)} = g\bigl(t,u,u',\ldots,u^{(m-1)}\bigr), \tag{5} \]
satisfying the boundary condition
\[ u(a_1)=u(a_2)=\ldots=u(a_m)=0. \tag{6} \]
Here \(p_i(t)\) are functions defined and continuous on the interval \([0,T]\) \(\bigl(|p_m(t)|>0\bigr)\), \(g(t,u_1,\ldots,u_m)\) is a function defined for \(t\in[0,T]\), \(-\infty<u_i<\infty\) \((i=1,\ldots,m)\); \(a_1,a_2,\ldots,a_m\) are certain numbers from the interval \([0,T]\), with \(0=a_1<a_2<\ldots<a_m=T\). We shall assume that every solution of the equation \(Mu=0\) has on the interval \([0,T]\) no more than \(n-1\) zeros (counting multiplicities), and that the function \(g(t,u_1,\ldots,u_m)\) is nonnegative, continuous jointly in all variables together with the partial derivatives \(g_{u_i}\) \((i=1,\ldots,m)\).
Denote by \(P(t,s)\) the Green’s function of the operator \(M\) under the boundary condition (6). It follows from the results of [4] that the integral equation
\[ \mu \int_{0}^{T} |P(t,s)|z(t)\,dt = z(s) \]
has a nonnegative solution \(y(s)\) for some \(\mu=\mu_3>0\).
Lemma 2. There exists a positive constant \(k_2\) such that, for all \(s\in[0,T]\),
\[ y(s)\geq k_2 \max_{t\in[0,T]} |P(t,s)|. \]
In the proof of Lemma 2, estimates of the Green’s function \(P(t,s)\) established in [4] are essentially used.
Theorem 4. Suppose that \(g(t,0,\ldots,0)=0\) and that, for any \(t\in[0,T]\), \(-\infty<u_i<\infty\) \((i=1,\ldots,m)\), the inequalities
\[ c_3|u_1|-d_3 \leq g(t,u_1,\ldots,u_m)\leq D(u_1)\left(1+\sum_{i=1}^{m-1}|u_{i+1}|^{n/i-\varepsilon}\right), \tag{7} \]
hold, where \(D(u_1)\) is a continuous function; \(c_3,d_3\) are positive constants, with \(c_3>\mu_3\).
Then the boundary-value problem (5)—(6) has a nonzero solution.
5. We shall give conditions for the existence of a solution of the differential equation (5) satisfying the linear homogeneous boundary conditions
\[ l_1(u)=l_2(u)=\ldots=l_m(u)=0 \tag{8} \]
We shall assume that under the boundary conditions (8) there exists a positive Green’s function \(Q(t,s)\) of the operator \(M\). Denote by \(\mu_4\) the first eigenvalue of the operator \(M\) under the boundary condition (8), and by \(x(s)\) the corresponding first eigenfunction. The existence of an eigenfunction follows from the positivity of \(Q(t,s)\).
Lemma 3. There exists a positive constant \(k_3\) such that, for all \(s\in[0,T]\),
\[ x(s)\geq k_3\left\{\max_t\left|Q_{tn-2}^{(n-2)}(t,s)\right|+\sup_{\tau_1,\tau_2}\frac{\left|Q_{tn-2}^{(n-2)}(\tau_1,s)-Q_{tn-2}^{(n-2)}(\tau_2,s)\right|}{|\tau_1-\tau_2|}\right\}. \]
An immediate consequence of Lemma 3 is
Theorem 5. Let \(g(t,0,\ldots,0)=0\) and let the inequality
\[
g(t,u_1,\ldots,u_m)\geq c_4|u_1|-\bar d_4
\quad (c_4>\mu_4,\ \bar d_4>0)
\]
hold.
Then the differential equation (5) has a positive solution satisfying the boundary conditions (8).
- As was shown in \((^6)\), the differential equation
\[ \Delta u+|u|^r=0 \]
for \(r\geq (n+2)/(n-2)\) \((n>2)\) has no nonzero solutions satisfying the boundary condition (3). This example shows that the requirement of sublinear growth of the function \(f(t,u,p)\) with respect to the variables \(u,p_1,\ldots,p_n\) is essential for the validity of the assertion of Theorem 2 and, in the general case, cannot be discarded.
Theorem 4 is an immediate generalization of the results of \((^5)\). The scheme developed in the first section is applicable to systems of differential equations.
The author expresses his gratitude to M. A. Krasnosel’skii and Yu. S. Kolesov, who drew the author’s attention to the works of S. I. Pokhozhaev \((^6,^7)\), in which related questions are studied by other methods.
Voronezh State University
named after the Lenin Komsomol
Received
25 XI 1969
REFERENCES
\(^1\) M. G. Krein, M. A. Rutman, Uspekhi Mat. Nauk, 3, no. 1 (23), 3 (1948).
\(^2\) M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Moscow, 1962.
\(^3\) K. Miranda, Partial Differential Equations of Elliptic Type, IL, 1957.
\(^4\) Yu. V. Pokornyi, Mat. Zametki, 4, no. 5, 533 (1968).
\(^5\) M. A. Krasnosel’skii, Yu. V. Pokornyi, ibid., 5, no. 2, 253 (1969).
\(^6\) S. I. Pokhozhaev, Dokl. Akad. Nauk SSSR, 165, no. 1, 36 (1965).
\(^7\) S. I. Pokhozhaev, Mat. Sb., 78 (120), no. 2, 237 (1969).