Hilbert space methods for nonlinear elliptic equations

JOURNAL OF DIFFERENTIAL Hilbert 32, 2X%257 (1979) EQUATIONS Space Methods for Nonlinear Elliptic Equations PETER WT. BATES**+ Department of Mathematics, Pan American Univewity, Ed&burg, Texas 78539 Received December 16, 1977 1. INTRODUCTION In [ 121 Mawhin gives an abstract theorem which he uses to obtain the existence and uniqueness results of Lazer and Sanchez [IO] and Lazer [8] for the periodically perturbed conservative system Y” + grad G(Y) =.f(t) (l-1) y’(0) = y’(24. y(O) = Y&4 It is assumed thatf is continuous and 2rr-periodic and G E Cz is such that there exists and integer m and real numbers 4 and p such that m21 < 41 < (PG(a)/& ax?) < pI < (m $- 1)21, for all a E W. Using similar techniques, Kannan and Locker in [5] obtained the same result and, in addition, they extend the result of Lazer and Leach [9] which gives the existence of solutions to nonlinear boundary value problems of the form y” + Yt, y, Y’)Y =f(c y(r) Y(O) = a, Y, Y'>* (1.2) = 6. Lazer and Leach assume that h and f arc continuous, that f is bounded, and that there exists a positive integer flz and 6 > 0 such that (m + 8)2 $ h(t, y, JX)< (m + 1 - S)2 for all t, y, Z. Kannan and Locker’s theorem in [5] gives the existence of at least one solution of Ty - h(t, y ,..., y’“-l’)y = f(t, y ,..., y(“-lb), By = 0, where T is an nth order linear, symmetric differential a<t<b (1.3) operator with Coo coeffi- * This research was partially supported by a grant from the Faculty Research Council Pan American University. College of Science, Texas A & M + Current address: Department of Mathematics, University, College Station, Tex. 77843. at 250 0022-0396/79/0.50250-08$02.00/O Copyright All rights 0 1979 by AcademicPress, of reproduction in any form Inc. reserved. NONLINEAR ELLIPTIC EQUilTIONS 251 cients, the leading coefficient non-vanishing in [a, b]. With the boundary condition By = 0, T defines a self-adjoint operator, L, in L, . The assumptions on h andf are (i) h and f are continuous, (ii) f is bounded, (iii) there exist real numbers 4 < h(t, y1 ,...T Yn) d P q < p so that for t E [a, b], yi E R, l<i<?Z, and none of the eigenvalues of L lie in [q, p]. In this paper, Mawhin’s theorem (stated as Theorem 2.1) is used with topological methods to dispense with condition (ii) and weaken condition (iii) of Kannan and Locker’s theorem. The abstract results are applied to elliptic equations, thus, also extending some of the results of Landesman and Lazer in [7]. In [3] the author uses the Mawhin theorem to obtain existence and uniqueness of weak solutions to other nonlinear systems of P.D.E’s, including a nonlinear perturbation of the wave equation. Recently, Ward has shown in [ 131 that solutions to (1.1) exist under weaker hypotheses on G and f. 2. ABSTRACT RESULTS Let H be a (real or complex) Hilbert space with inner product (., .> and norm 1 . 1. Let L : domL C H -+ H be a linear, self-adjoint operator with spectrum (T and N: H + H be a mapping having a GPteaux derivative N’(u) E B(H), the bounded linear operators on H, and such that X’(u) is symmetric for all ti E H. Suppose that there exist real numbers q < p so that q1 < N’(U) < pI for all u E H, I being the identity in B(H) and the partial ordering being defined by A > 0 if and only if (A u, U) 3 0 for all u E H. The main result in [12] may be stated as follows: THEOREM 2.1 (Mawhin). Let L and IV be as above. Suppose that l-4, PI n fJ = 4 (2.1) Lu - Nu = y (2.2) then has a unique solution for each y E H. Furthermore, l(L - WY, where C = l/dist([q, - (L - W’YZ I < c I Y1- y2 I, p], 0). Now, let HI and Hz be Hilbert spaces with inner products (., .)r and (*, -)a and 252 PETER W. BATES norms 1 . II and / . Ia, respectively, such that HI is continuously imbedded in H, and H, is continuously and compactly imbedded in HI , with 1 * II < 1 * I2 . Suppose that L satisfies a coercivity condition (4 dom L C H, and there exists M such that ]ula <M([uI + ILuI)foralluEdomL. Let R > 0 and let h be a mapping satisfying: h: B,(O, R) + B(H) @I) is continuous where B,(O, R) is the open ball in HI which has center 0 and radius R, VkJ w h(u) is symmetric for each zu E &(O, R), and there exist real numbers q < p such that q1 < h(w) < p1 for each w E Z&(0, Ii). THEOREM 2.2. Let L and h be as abooe and, in addition, satisfy the nonresonance condition [q, p] n a = +. Suppose that the mapping f: &(O, R) -+ H is continuous, bounded, and satisfies (F> I f (w)l < RK for all w E HI such that j w II = R, where K = &P1((max{] Then the equation q 1, 1p I} + 1)C + 1)-l. Lu - h(u)24 = f(u) (2.3) has at least one solution in H, n B,(O, 2;)). Remark 2.3. In the event that (hl) - (hJ hold for all positive R and thatf is defined on all of HI, then condition (F) will be satisfied if f has sublinear growth outside some bounded set in HI , that is, in case or if f has superlinear growth inside some neighborhood of 0 in HI , that is, if ,$J, 1 If bw w II = 03 however, in this case zero is a solution. This is avoided if superlinear function. (2.5) f is a slightly perturbed Proof of Theorem 2.2. Fix w E &(O, R) and define N(u) = h(w)u, y = f(w), then Theorem 2.1 applies and there exists a unique solution, II, of Lu - h(w)u = f(w). (2.6) NONLINEAR ELLIPTIC 253 EQUATIONS Thus, the mapping T: &(O, R) -+ dom L C H, C HI defined by Tw = u, where ZI is the unique solution of (2.6), is well defined. The operator N = h(w) is linear for each fixed w E &(O, R) so Theorem 2.1 shows that Thus, writing (2.6) as Tw = a = (L - h(w))-If(w), G3 we have Now, by condition (L) I aI2 <J,f(lu! + IW) < M-(1 21j + 1Lu - h(zu)u ! + /I R(w)lj ! 2l i) = AU 24I + If@)1 + IIh(w)li I u I> d IV If(fu)l (C + 1 + C =$I 4 1,I P I>), that is, 12-zI2 < K-l Suppose satisfies that wr , .w2E B,(O, R) and (2.9) I f(w)l* ur = Tzq , uz = Tzo, , then zli - al, L(2.8,- u2j - ~(w,)(u, - 2~~)= @a(~,)- ~(w,))u, +f(wl) -f(~uJ which is of the form (2.6). Thus, by (2.9) and T is continuous from &(O, R) into H2 . Since the imbedding Hz C HI is continuous and compact, the mapping T: B,(O, R) + HI is completely tontinuous. We now show that the Leray-Schauder degree, (see [Ill), d(1T, B,(O, R), 0) is defined and nonzero. Consider the map defined by S(t, w) = zu - t Tw for (t, .ZU)E (0, l] X B,(O, R). S satisfies the usual continuity and compactness conditions and does not vanish on the surface of B,(O, R) for each t E [0, 11. Indeed, if S(t, W) = 0 for some t E [O! 11, ! .Wjl = R, then letting u = Tw, (2.9) and (F) give R=Iw/,=tjTwl/l< Iu/,<K-l[f(W)j <R, 254 PETER W. BATES a contradiction. The invariance of Leray-Schauder degree under homotopy implies that d(S(t, .), B,(O, R), 0) is defined and constant for 0 < t < 1. Thus d(l- T, B,(O, R), 0) = d(L B,(O, R)> 0) = 1. It follows that there exists a w E B,(O, J2) such that Tw = w, and so w E Hz and by (2.8) Lw - h(zu)w = f(w). This completes the proof. 3. ELLIPTIC SYSTEMS Let G be a bounded domain in R”, rrz 3 1, having a smooth boundary. j > 0 let Hj denote the completion of P(G) with respect to the norm where a is a multiindex For a = (ur ,..., a,). For K > 1 let k H = fi Ho, Hl = n H”“+, i=l i=l and H, = fi H”i, i=l where IQ >, I, 1 < i < K. Clearly, the imbeddings H, C HI C H are continuous and Rellich’s theorem (see, e.g., [l]) shows that the imbedding Hz C HI is compact. Also, 1 . /i < 1 . je . Let T = diag(T,), where Ti is a linear, formally self-adjoint, n,th order, uniformly elliptic differential operator. We shall impose boundary conditions by means of a boundary operator of the form B = diag(B,), where for i = l,..., K, ( Ti , BJ defines a self-adjoint operator in L, with domain in H’“i. For instance, Bi could be a generalized Dirichlet or Neumann boundary operator (see, e.g. [3], [4], [14]). It follows that T, l3 defines a self-adjoint elliptic operator L in H with domain in Hz and I~I,~w~~+lL~I), for u E dom L, for some constant M, (see Agmon [l], p. 266). Thus, the operator L satisfies condition (L) of section 2, this being the only reason to restrict our attention to elliptic operators. Let u denote the spectrum of L. As for the mapping h, we could consider several interesting cases, however, for simplicity we shall take h to be the evaluation map associated with a symmetric NONLINEAR ELLIPTIC 255 EQUATIOXS k x k matrix function (IQ). It is assumed that this matrix is measurable with respect to its m space variables and continuous with respect to those I (say) arguments accomodating the components of the vector zl and those derivatives which are admissible for u E Hr. Further, assume that h,,(., q(.),..., q(.)) lies in Ho, 1 < i, j < K, whenever c‘s E Ho, 1 < s < r, thus assuring the continuity of h: HI + B(H), ( see, e.g., [6] p. 22). Suppose that there exist two real numbers, q <p, such that the matrix h(.zu) is bounded below by 41 and above by ~1, for all w E B,(O, R), for some R > 0. Thus, the mapping h satisfies (A,) - (h,) of section 2. Suppose [q, p] n 0 = 2. Let have components which obey the same continuity assumptions as the components of the matrix h, thus, the evaluation map F associated with f is continuous from HI into H. We requireF to map bounded sets in HI into bounded sets in H. We also assume that F satisfies condition (F). For instance, it is routine to check that this condition is satisfied if we assume that R = cy;!and for x E G, (vi) E BP, where ~((vi))/~(vi)~ -+ 0 as [(vi)1 -+ co, i(q)1 being the Euclidean norm of the vector (vi) E [WV. With the above conditions satisfied, Theorem 2.2 guarantees the existence of a weak solution to the elliptic system diag(Ti) U(X)- (i&(x, h(x)))kXk u(x) = f(.? Du(x)), diag(&)n (3.1) = 0, where Du represents that vector whose components of ui of order less than lzi , 1 < i < k. EXAMPLE 3.1. XEG are the partial derivatives Let G = (0, rr) x (0, r) and consider the 2 x 2 system Ll*u, - 2422 + sin2(D(1~0)u,/[ x 1)) - ug cos(D”**~z4,) = (D(“,l)u, f lj1j3 (3.2) -Au, - u1 cos(D(1%,) - ~~(23 - u2*/(u2* + exp(D(s%r))) with the boundary = (De%+ + l)a!a conditions u&x) = d,(x) = 0, XEaG, (3.3) (au,zlz)(x) = 0, XEaG, where R is an outward normal to 6G at x. 256 PETER W. BATES It is easy to check that the spectrum of the self-adjoint (“,”?A) and (3.3) is (7 Furthermore, 2OI< the matrix 21I< operator associated with + k2 : j, K are integers) inequality, 22 + sin2(a) cos(b) cos(b) 23 - c”/(c’ + exp(d)) ’ 241 < 25’* holds for all real a, 6, c, d. Note that lf(~, vr ,..., v,)l = ((vi + 1)2/3 + (vj + 1)4/3)1/8 for some i,j and so I~(x, vr ,..., v,)l/l(v, ,..., v,.)[ -+ 0 as [(vi ,..., v,r)/ + co, thus, condition(F) holds. Consequently, (3.2)-(3.3) has at least one weak solution u ). It can be shown that the solution is actually a classical solution and using (Ul? 2 Theorem 2.2 a bound for this solution in H3 x H1 can be obtained. Remark equations a classical regularity dimension conditions, 3.2. In the event that (3.1) is a system of ordinary differential and ($j), f are continuous functions, then the weak solution is actually solution. When (3.1) is a system of partial differential equations, of slutions depends on the relationship which exists between the of the space, the order of the elliptic operator, and the boundary (see [l], [2], [4]). REFERENCES 1. S. AGMON, “Lectures on Elliptic Boundary Value Problems,” Van Nostrand, N. J., 1965. S. AGMON, A. DOUGLIS, AND L. NIRENBERG, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions: I, Comm. Pure Appl. Math. 12 (1959), 623-727. P. W. BATES, Hilbert space methods for nonlinear systems of partial differential equations, Pan American University. Equations, ” Holt, Rinehart & Winston, New York, A. FRIEDMAN, “Partial Differential 1969. R. K~~NNAN AND J. LOCKER, On a class of nonlinear boundary value problems, J. Differential Equations 26 (1977), 1-8. “Topological Methods in the Theory of Nonlinear Integral M. A. KRASNOSEL’SKII, Equations,” Macmillan Co., New York, 1964. E. M. LJINDESMAN AND A. C. LAZER, Linear eigenvalues and a nonlinear boundary value problem, Pacific J. Math. 33 (1970), 311-328. A. C. LAZER, Application of a lemma on bilinear forms to a problem in nonlinear oscillations, Proc. Amer. Math. Sot. 33 (1972), 89-94. A. C. LAZER AND D. E. LEACH, On a nonlinear two-point boundary value problem, J. Math. Anal. Appl. 26 (1969), 20-27. A. C. LAZER AND D. A. S.&CHEZ, On periodically perturbed conservative systems, Michigan Math. J. 16 (1969), 193-200. Princeton, 2. 3. 4. 5. 6. 7. 8. 9. 10. NONLINEAR 11. J. LERAY ELLIPTIC EQUATIONS 257 mm J. SCHAUDER, Topologie et equations fonctionnelles, Ann. .?&le NOYNP. Sup. 51, No. 3 (1934), 45-78. 12. J. MAWHIN, Corm-active mappings and periodically perturbed conservative systems, Arch. Math. 2, Scripta Fat. Sci. ‘Nat. UJEP Brztnensis 12 (1976), 67-74. Periodic solutions of perturbed conservative systems, Proc. Anzer. Math. 13. J. R. \vARD, Sac. 72 (1978), 281-285. value problems for linear hyperbolic partial dif14. C. H. WILCOX, Initial-boundary ferential equations of the second order, Arch. Rat. Mech. Anal. 10 (19621, 361-400.