On Strongly Nonlinear Elliptic Equations with Weak Coercitivity Condition László Simon

In this paper it will be proved existente and uniqueness of solutions of boundary value problems for the equation ON STRONGLY NONLINEAR ELLIPTIC EQUATIONS WITH WEAK COERCITIVITY CONDITION LÁSZLÓ SIMON We prove the existente and uniqueness of weak solutions ofboundary value problems in an unbounded domain 9 C IR" for strongly nonlinear 2m order elliptic differential equations . (-1)'D« [f« (x,D«u)]+ + E (1 ) I «ID« [g«(x, u, . . . , DQu, . . . )] = F in 9 «I 1, co > 0. Functions ,g« have some polynomial growth in Dpu, but on f« no growth restriction is imposed in D«u . Similar result has been proved in [1] f'or the equation 176 L. SIMON in a bounded 52 if the condition 9a(Sa)Sa ? col(.¡, cr, I al :~ m is fulfilled with some constante p > 1, cj > 0 . The proof of the existence theorem is based on a method called by F.E. Browder "elliptic superregularization" (see [1] [3]) . Our resulte can be extended to equations of the form lal-m, + Y-, (-1)I-IDa[9a(x,u, . . .,Dpu, . . .] =F lal~~-r where 191 < m (see [2] [5]) . It is to be mentioned that [6] is connected with our result where D. Fortunato has corrsidered equation Lu + f (x, u) = 0 ; by L is denoted a second order linear elliptic operator with weak coercitivity conditions in an unbounded domain . Similarly to our consideration, in [6] the solution u must satisf'y the "asymptotic condition" 1. Igrad u¡ 2 dx < +oo . 1 . The existence theorem Let 52 C Rn be an unbounded domain with bounded boundary 852, having the uniform C'-regularity property and 52,. = 52 f1 B, . where B,. = {x E Rn : Ix1 < r} (see [7]) . Denote by Wp(52) the usual Sobolev space of real valued functions u whose distributional derivatives belong to Lp(52) . The norm on W.'n(9) is B,y WP¡o"(S2) will be denoted the set of functions f such that wf E Wp (52) for all cp E Co (R'), Le . for all infinitely differentiable functions cp with corripact support . Denote by Wr;'o(52) the set of functions u E WP¡o"(0) satisfying the conditions : Dau E Lp(52) if la¡ = m and the trace of DQu on 852 equals to 0 if ~f 0 there is a function f,,, s such that .fas E L'(9,) for each r > 0 and 1f.(x,(a1 1 .f.,s(x) if j(',1 :5 s . Further, there exist constants cl, C2 > 0 and a function fá E L 1 (9) such that for a.e . x E 9 Lfa(x,(a)I ~ .fa(x')+C1I(aj'-1 ¡fl(al :5 C2 with some p > 1 . IV . There exists a constant c3 > 0 such that f'or all (C, E R, a.c . .r, E 12 fa(X,(a)I 1 C3l(a1'-1 . V. Functions ga : 52 x R' ,R (jal <_ m 1) satisfy the Carathéodory conditions . VI . There exists a bounded dornain S2' C S2 such that g,, (x, () = 0 for all ( E RN, a.e . x E S2\S2' ; furthcx, ga(x,()(a ? 0 . jal 0 ; epa , T. are continuous functions, cpa is monotone increasing, T a is strictly monotone increasing, ~Ga(0) = 0, xPa(0) = 0 and by c, c are denoted positive constante . Theorem 1. Assume that conditions I VII are fulfilled. Then for any G E V* (i .e . for linear continuous functional over V) with compact support there is u E V such that (1 .1) fa (x,Dau)Dau E L' (Q), fa (x, («) = X« (x )W.(Sa) + T-«-) CI(aIP-1 <= I a(Sa)I(Sa E R), 55 ¿IS.IP-1 lf IS .I < 1 I fa(x, DIXu)I :5 f.(11 + f(2) where f(1) E L1 (S2), .f«21 E Lq(9), p + 1 = 1, (1 .3) E J ,f,, (x,Da u)Davdx+ lala 9 for all v E Có (R') with viq E V. + 5 ga (x, u, . . . , Df3u, . . . )Dav dx = (G, v) lal <_m-1 ~, This theorem will be a simple consequence of Theorem 2 formulated below . Let V,. be the closure in WP (52,.) of { ,PIs1 . : W E Có (B,) n V} . Then V,. is a closed linear subspace of Wp(52,.) and -extending function u E V,. as 0 to 52\52,.the extensions belong to V . Let s > max{n, p} then by Sobolev's imbedding theorem Ws+1 (52, .) is continuously and also compactly imbedded into WP(9,) and CB (Q,) (see e.g . [7]) where CB (52,.) denotes the set of m times continuously differentiable functions WEAKLY COERCIVE NONLINEAR ELLIPTIC EQUATIONS 179 u with finite norm IIu¡I = supID«uI . Denote by Ws+ 1 (52,.) the n" closure in WS+ 1 (52,.) of 0 Then -extending u E Ws+ 1 (52,.) as 0 to 52\52, .the extension belongs to Ws+ 1 (52) . Further, let (Q, (u), v) {wlsl, : W E Có (Br)} . 0 Wr = W9+1(52T) n V,. with the norm of W9+1(52,.) . Then W,. is a closed linear subspace of W9+1(52, .) . Functions u E W,. will be extended to 52\52,. as 0 . For any u, v E W,. define (S,(u), v) = 5 IDaul s-2(D"u) (D' v) dx, Ial :5m+1' nr (T,(u), v) = 1: fa(x, Dau)D'vdx, lal<~, sZr By Hólder's inequality, Sobolev's imbedding theorem, assumptions I, III, V, VII S, ., T,., Q,. : W,. W; are bounded nonlinear operators Le . they map bounded sets of W,. onto bounded sets of W,* . Theorem 2 . Assume that conditions I VII are fulfzlled, G C V` has compact support and lim rI = +oo . Then for suiciently large l there 1-00 exists at least one soluton u` E Wr, of = ~, g .(x, u, , . . , DQu, . . . ) D vdx . lal<m 1 (1 .4) (S, ., (ut), v) + (T,., (u¡), v) + (Qr, (ul), v) = (G, v) for all v E W,., . Further, there is a subsequence (uí) of (ul) which is weakly converging in V to afunction u E V satisfying (1 .1) (1.3) . If (1.1) (1 .3) may have at most one solution then also (ut) converges weakly to u. Proof. Clearly, !S,, is a pseudomonotone operator . Since W,., is compactly imbedded into CB (52, .,) thus by use of assumptions I, III, V, VII