PORTUGALIAE MATHEMATICA Vol. 56 Fasc. 3 1999 ON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT M. Guedda Abstract: In this paper we consider the problem u = λ u u + f in, u = u = 0 on, q c = N/(N 4), N > 4, is the limiting Sobolev exponent and is a smooth bounded domain in R N. Under some restrictions on f and λ, the existence of weak solution u is proved. Moreover u 0 for f 0 whenever λ 0. 1 Introduction In this article, we show that the problem (1.1) (P λ,f ) { ( u) = λ u q c u + f in, u = u = 0 on, is a smooth bounded domain in R N, N > 4, is the Laplacian operator and q c = N/(N 4), has weak solutions in H () = H () H0 1 () equipped with the norm ( ) 1/ u H = u. To this end we consider the functional F λ (u) = 1 u dx λ (1.) u dx q c fu dx, u H (), λ > 0. Under some suitable conditions, it is proved that (1.1) admits at least two solutions. Our arguments make use of the mountain pass theorem and of the Lions concentration-compactness principle. Received: November 10, 1997; Revised: January 15, 1998. AMS(MOS) Subject Classification: 35J65, 35J0, 49J45.
300 M. GUEDDA Recently, Van der Vorst [10] considered the following problem { } (1.3) S = inf u ; u H (), u = 1. He proved that the infimum in (1.3) is never achieved by a function u H () when is bounded. In contrast Hadiji, Picard and the author in [7] considered the problem (1.4) { S ϕ = inf u ; u H (), } u + ϕ = 1. They showed that the infimum in (1.4) is achieved whenever ϕ is continuous and non identically equal to zero. More precisely it is shown that, for any minimizing sequence (u m ) for (1.4), there exists a subsequence (u mk ) and a function u H () such that u mk u weakly in H () and u + ϕ = 1. On the other hand, Bernis et al. [1] considered a variant of (1.1) f is replaced by β u p u, 1 < p <. They proved the existence of at least two positive solutions for β sufficiently small. At this stage, we would like to mention that when = R N P.L. Lions [9] proved that S is achieved only by the function u ε defined by u ε (x) = N 4 [(N 4) (N ) N (N + ) ε] 8 (ε + x a ) N 4, x R N, for any a R N and any ε > 0. This note is organized as follows. In Section we verify that F λ satisfies the (PS) c condition. In Section 3 we prove the existence of a local minimizer u of F λ. Moreover, we show that u 0 whenever f 0 and λ 0. Section 4 is devoted to the existence of a second solution to (1.4). The results presented in this paper have been announced in [6]. Notice that if f 0, the result of Section 3 is valid and gives the trivial solution u = 0. The method we adopt is closely related to the one of [3]. Before the verification of the (PS) c condition, let us remark that if v is a solution to (1.1) then u = λ 1 v satisfies { ( u) = u q c u + g in, (1.5) g = λ 1 f. u = u = 0 on,
ON NONHOMOGENEOUS BIHARMONIC EQUATIONS 301 Verification of the (PS) c condition Let be a bounded domain in R N, N > 4, and f L (). We denote by F λ : H () R the functional defined by F λ (u) = 1 u dx λ (.1) u dx fu dx, q c is the Laplacian operator and λ is a real parameter. We first look for critical points of F def = F 1. We show that F satisfies the Palais Smale condition in a suitable sublevel strip. Let S be the best Sobolev embedding constant of H () into L (); that is { } (.) S = inf u ; u H (), u = 1 and (.3) K = N q q (4 q c ) q f q q, q = q c q c 1. Proposition.1. The functional F satisfies the (P S) c condition in the sublevel strip (, N S N 4 K); that is if {u m } is a sequence in H () such that (.4) F (u m ) c and df (u m ) 0 in H (), c < N S N 4 K, then {u m } contains a subsequence which converges strongly in H (). Proof: Let {u m } be a sequence in H () which satisfies (.4). From (.4) it is easy to see that {u m } is bounded in H (); thus there is a subsequence {u m k }, and an element u of H () such that (.5) u mk u weakly in H () and (.6) u mk u strongly in L p (), 1 p < q c and a.e. in. The concentration-compactness Lemma of Lions [9] asserts the existence of at most a countable index set J and positive constants {ν j }, j J such that (.7) u mk u + j J ν j δ xj,
30 M. GUEDDA weakly in the sense of measures, and (.8) u mk µ, for some positive bounded measure µ. Moreover, (.9) µ u + j J S ν N 4 N j δ xj, (.10) x j and ν j = 0 or ν j S N 4. We assert that ν j = 0 for each j. If not, assume that ν j0 0, for some j 0. From the hypothesis (.4), Using the Hölder inequality one has c = lim F (u m k ) 1 df (u mk ), u mk, k c u 1 fu + N N S N 4. c N S N 4 N q q (4 q c ) q f q q. This contradicts the hypothesis. Consequently ν j = 0 for each j and lim u mk = u, k which implies u mk u strongly in H (). The proof is complete. 3 Existence of a solution In this part we consider the problem of finding solutions to (P λ,f ). We show, under suitable conditions on f and λ, that F λ has an infimum on a small ball in H (). We suppose first that λ = 1, and denote by F the functional F 1. The proof is based on the following lemma.
ON NONHOMOGENEOUS BIHARMONIC EQUATIONS 303 Lemma 3.1. There exist constants r and R > 0 such that if f R, then (3.1) F (u) 0 for all u H () = r. Proof: Thanks to the Sobolev and Hölder inequalities we have F (u) 1 u 1 S ( u ) 1 ( 1 S 1 f q c u ) 1/ (3.). Inequality (3.) can be written ) (3.3) F (u) h ( u H, h(x) = 1 x λ 0 x λ 1 x, λ 0 = 1 q c S and λ 1 = f 1 1 S 1. Let g(x) = 1 x λ 0 x 1 λ 1 for x 0. There exists λ > 0 such that, if 0 < λ 1 λ, g attains its positive maximum and we get (3.1), with thanks to (3.3). ( S r = ) 1 1 and R = 1 + 1 S λ, Remark 3.1. Arguing as above we can see that there exists a constant α > 0 such that F (u) α, for all u H = r. Proposition 3.1. Let R and r be given by Lemma 3.1. Suppose that f 0 and ) (3.4) max ( f, f q < min(r, R), R = N 4 q c S N 4q ( (q c 1) ) 1 q Then there exists a function u 1 H () such that. (3.5) F (u 1 ) = min B r F (v) < 0,
304 M. GUEDDA } B r = {v H, v H () < r, and u 1 is a solution to (P 1,f ). Moreover, u 1 0 whenever f 0. Proof: a. Let Without loss of generality, we can suppose that f(a) > 0 for some u ɛ (x) = ε N 4 4 φ(x) (ε + x a ) N 4, ε > 0, φ C0 () is a fixed function such that 0 φ 1 and φ 1 in some neighbourhood of a. Since fu ɛ dx > 0, for a small ε, we can choose t > 0 sufficiently small such that Hence (3.6) F (t u ɛ ) < 0. inf B r F (v) < 0. Let {u m } be a minimizing sequence of (3.6). From (3.4) and Lemma 3.1 we may assume that (3.7) u m H < r 0 < r. According to the Ekeland variational principle [5] we may assume (3.8) u m u m f 0 in H (). On the other hand, from (.3) and (3.4), we get (3.9) 1 N S N 4 K > 0. We deduce, from (3.8) (3.9) and Proposition.1, that {u m } has a subsequence converging to u 1 H (), and u 1 is a weak solution to (P 1,f ). Now we suppose that f 0. Let v H () be a solution to the following problem v = u 1.
ON NONHOMOGENEOUS BIHARMONIC EQUATIONS 305 As in [10, 11] we get v > 0, v u 1 in, v = u 1 and It then follows that v u 1. F (v) F (u 1 ) and v H r. Consequently F is minimized by a positive function. This method allows us under suitable conditions on f and λ, to prove the existence of solutions to (P λ,f ). Theorem 3.1. Suppose that f 0, then there exists λ f > 0 such that if the following condition is satisfied ( ) 0 < λ f < λ 1 1 1 (3.10) < min, min(r, R), f f q Problem (P ) λ,f has at least one solution u λ. Moreover u λ 0 whenever f 0. Proof: For the proof we consider Problem (P 1,g ) g = f λ 1. Condition (3.10) implies that g satisfies (3.4). So the existence follows immediately from Proposition 3.1. Now suppose, on the contrary, that u λ exists for any λ such that ( ) 0 < λ 1 1 1 < min, min(r, R). f f q Note that, since λ 1 u λ is the solution to (P 1,g ) obtained by (3.5), we have u λ H () r λ 1. It follows from this that u λ H () 0 as λ 0. Passing to the limit in (P λ,f ) we deduce that f 0, which yields to a contradiction. 4 Existence of a second solution In this section we shall show, under additional conditions that (P λ,f ) possesses a second solution. Here we use the mountain pass theorem without the Palais Smale condition [, 8]. As in the preceding section, we first deal with the case λ=1.
306 M. GUEDDA Assume that condition (3.4) is satisfied and that f > 0 in some neighbourhood of a. Set v ε = u ε. u ε The main result of this section is the following. Theorem 4.1. There exists t 0 > 0 such that if f satisfies f q q < t 0 (4.1) fv ε dx, for small enough ε > 0, K 1 K 1 = N q q (4 q c ) q then (P 1,f ) has at least two distinct solutions. Proof: The proof relies on a variant of the mountain pass theorem without the (PS) condition. We have, for ε sufficiently small (see [4]),, (4.) Set v ε = S + O(ε N 4 ). h(t) = F (t v ε ) = 1 t X ε 1 t t fv ε dx for t 0, q c X ε = v ε. Since h(t) goes to as t goes to +, sup t 0 h(t) is achieved at some t ε 0. Remark 3.1 asserts that t ε > 0, and we deduce (4.3) h (t ε ) = t ε (X ε tε ) fv ε dx = 0 and h (t ε ) 0, thus (4.4) ( 1 ) 1 q c 1 X 1 ε t ε X 1 ε. Let t 0 = 1 ( 1 q ) 1 c 1 S 1. We deduce from (4.) and (4.4) that, for ε 0 small, (4.5) Thus t 0 < t ε for ε (0, ε 0 ). sup h(t) = sup h(t). t 0 t t 0
ON NONHOMOGENEOUS BIHARMONIC EQUATIONS 307 On the other hand, since the function t 1 t X ε 1 q c t interval [0, X 1 ε ], we get thanks to (4.). Hence (4.6) Consequently if we let (4.7) we deduce that (4.8) h(t ε ) N S N 4 tε h(t ε ) N S N 4 t0 t 0 fv ε dx + O(ε N 4 ), fv ε dx + O(ε N 4 ). fv ε dx > K 1 f q q, sup F (tv ε ) < t 0 N S N 4 K. Note that there exists t 1 large enough such that is increasing on the (4.9) F (t 1 v ε ) < 0 and t 1 v ε H > r, r is given by Lemma 3.1. Hence α c = inf γ Γ max s [0,1] F (γ(s)) < N S N 4 K, Γ = { ( } γ C [0, 1], H () ): γ(0) = 0, γ(1) = t 1 v ε, provided ε is small enough. Then, according to the mountain pass theorem without the (PS) condition, there exists a sequence {u m } in H () such that F (u m ) c and df (u m ) 0 in H (). Since c < N S N 4 K, we deduce from Proposition.1 that there exists u such that c = F (u ) and u is a weak solution to (P 1,f ). This solution is distinct from u 1 since c 1 < 0 < c. So the proof is complete. Finally, by using Theorem 4.1, we deduce the Corollary 4.1. Assume (3.10). If λ q 1 < t 0 K 1 f q q fv ε dx, for ε small enough, then problem (P λ,f ) has at least two solutions.
308 M. GUEDDA ACKNOWLEDGEMENTS The author thanks C. Picard and M. Kirane for their useful discussions and for informing him of Ref. [1] and the referee for careful examination of the paper and valuable remarks. This work was partially supported by C.C.I. Kénitra (Maroc). REFERENCES [1] Bernis, F., Garcia-Azorero, J. and Peral, I. Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Advances in Dif. Equat., 1() (1996), 19 40. [] Brezis, H. and Nirenberg, L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437 477. [3] Chabrowski, J. On multiple solutions for the nonhomogeneous p-laplacian with a critical Sobolev exponent, Diff. and Integ. Equat., 8(4), (1995), 705 716. [4] Edmuns, E., Fortunato, D. and Jannelli, E. Critical exponents, critical dimensions and the biharmonic operator, Arch. Rational Mech. Anal., 11 (1990), 69 89. [5] Ekeland, I. On the variational principle, J. Math. Anal. Appl., 47 (1974), 34 353. [6] Guedda, M. A Note on Nonhomogeneous Biharmonic Equations Involving Critical Sobolev Exponent, report LAMIFA, Univ. de Picardie, Fac. de Maths et d Info, Amiens. [7] Guedda, M., Hadiji, R. and Picard, C. in preparation. [8] Guedda, M. and Veron, L. Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. The. Meth. and Appl., 13(8) (1989), 879 90. [9] Lions, P.L. The concentration-compactness principle in the calculus of variations, the limit case, Parts 1 and, Riv. Mat. Iberoamericana, 1 (1985), 45 11, 145 01. [10] Van Der Vorst, R.C.A.M. Best constant for the embedding of the space H H0 () into L N N 4 (), Diff. and Integ. Equat., 6() (1993), 59 76. [11] Van Der Vorst, R.C.A.M. Fourth order elliptic equations with critical growth, C.R.A.S., 30(I) (1995), 95 99. M. Guedda, LAMFA, Faculté de Mathématiques et d Informatique, Université de Picardie Jules Verne, 33, rue Saint-Leu, 80039 Amiens FRANCE