Weak solutions for some quasilinear elliptic equations by the sub-supersolution method

Nonlinear Analysis 4 (000) 995 100 www.elsevier.nl/locate/na Weak solutions for some quasilinear elliptic equations by the sub-supersolution method Manuel Delgado, Antonio Suarez Departamento de Ecuaciones Diferenciales y Analisis Numerico, C/Tara s/n, Universidad de Sevilla, 4101-Sevilla, Spain Received 6 March 1998; accepted 11 September 1998 Keywords: Quasilinear elliptic equation; Sub-supersolutions; Leray Schauder theorem; Gagliardo Nirenberg inequalities 1. Introduction Let be a bounded and regular domain in R N ;N, with a smooth boundary @. In this paper we study the solvability of the quasi-linear elliptic boundary value problem Lu = f(x; u; Du) in ; (1) u =0 on @ by the sub-supersolution method, where Du is the gradient of u and f : R R N R is a Caratheodory function satisfying f(x; s; ) g(x; s)+k with k R + 0 ;G(x):=sup s r g(x; s) Lp () for all r 0 and a positive constant with some restrictions that will be detailed below. In (1), L is an elliptic second-order operator of the form Lu := N i;j=1 @ @x i ( a ij (x) @u ) + @x j N i=1 b i (x) @u @x i and the precise assumptions on a ij and b i will also be explained below. Corresponding author. E-mail addresses: delgado@numer.us.es (M. Delgado), suarez@numer.us.es (A. Suarez) 036-546X/00/$ - see front matter? 000 Elsevier Science Ltd. All rights reserved. PII: S036-546X(99)00151-0

996 M. Delgado, A. Suarez / Nonlinear Analysis 4 (000) 995 100 The existence of a solution to a problem of this kind has been considered by several authors. Amann [] proved the existence of a minimal and a maximal classical solution (notions which appear already in papers of Satô [13], Mlak [1] and Akô [1]) for (1), assuming that a classical sub-supersolution couple exists. When p N, Amann and Crandall [3] proved a similar result in W ;p (), again with the help of a sub-supersolution couple for (1), with a more direct proof. Almost simultaneously, using the Leray Schauder xed point theorem, Kazdan and Kramer demonstrated in [9] that a classical solution exists. Here, the proof is considerably easier than in previous papers but the important information concerning the existence of minimal and maximal solutions is lost (see also [11], where the results are generalized including the case of some nonlinear boundary conditions). Several years later, Dancer and Sweers [5] remarked that, under the assumption of existence of a sub-supersolution couple in W 1; (), there exists a minimal and a maximal solution that belong to W 1; (). To this end, a nonconstructive argument which uses Zorn s lemma was employed by the authors. Moreover, Hess [8] proved the existence of a solution u W 1; () for (1), and so by a bootstrap argument u W ;m () where m = min{=; p}. In all these papers, the growth of the gradient in the nonlinear term is at most quadratic, i.e.. This is a reasonable assumption, in accordance with a well known result of Serrin [14] which states, roughly speaking, that if f grows faster than quadratically in Du and is smooth, there are smooth data for which (1) possesses no solution. More recently, several authors have used the Leray Schauder s theorem without any assumption concerning the existence of a sub-supersolution couple. Thus, Xavier [15] has deduced the existence of a solution in W ;p () in the case in which L is Laplace s operator and p N. It has been shown in [16] that the condition p N can be relaxed. In both papers, some conditions on the growth of f in terms of the rst eigenvalue 1 of Laplace s operator are imposed. In this paper, we will prove that, when p N, there exists at least one weak solution to (1) in W ;p (). We will use the sub-supersolution method for a general elliptic operator L.. The main result We consider problem (1), where N ( @ Lu := a ij (x) @u ) N + b i (x) @u : @x i @x j @x i i; j=1 i=1 We assume a ij W 1; (); a ij = a ji ;b i W 1; 0 () and div(b) 0, with b =(b 1 ;:::; b N ); we also assume there exists a positive constant such that N a ij (x) i j ; R N ; x : i; j=1 In (1), f : R R N R is a Caratheodory function, i.e. measurable in x and continuous in (s; ) R R N. In addition, we suppose that f satises f(x; s; ) g(x; s)+k ; ()

M. Delgado, A. Suarez / Nonlinear Analysis 4 (000) 995 100 997 where k; R + 0 ;gis also a Caratheodory function and satisfy: (H1) For all r 0, sup g( ;s) L p (); s r (H) (a) If N =, then 1 p and (b) If N 3, then N N + p N or p +1 : and p +1 N N p N p N and p +1 : (3) Denition 1. A couple {u 0 ;u 0 }, where the functions u 0 and u 0 belong to W ;p () L (), is a sub-supersolution couple for (1) (in W ;p ()) if: (1) u 0 0 u 0 on @, () u 0 u 0 in, (3) Lu 0 f(x; u 0 ;Du 0 ) 0 Lu 0 f(x; u 0 ;Du 0 ). Remark 1. Notice that the last inequalities have a meaning a.e. in. Indeed; whenever v W ;p () L (); we have f( ;v;dv) L q () for some q 1; since f(x; v; Dv) g(x; v) + k Dv a:e: in : In this paper, the main result is the following: Theorem 1. Assume 1 p if N = and N=(N +) p N if N 3. Also; assume (H1) and (H). If there exists a sub-supersolution couple for (1) in W ;p (); then (1) possesses at least one solution in W ;p (). Proof From (H), we know that p = Np=(N p). We also know that the interval [p; p ) is nonempty. For any q [p; p ), one has W ;p (), W 1;q () with a compact imbedding. Fixed q 0 [p; p ), let T : W 1;q0 () W 1;q0 () L () be the truncation operator associated to the sub-supersolution couple {u 0 ;u 0 } u 0 (x) if u 0 (x) u(x); Tu(x)= u(x) if u 0 (x) u(x) u 0 (x); u 0 (x) if u(x) u 0 (x): Observe that Tu(x) max{ u 0 (x) ; u 0 (x) } = m a:e: in ; whence Tu L () for any u W 1;q0 ().

998 M. Delgado, A. Suarez / Nonlinear Analysis 4 (000) 995 100 On the other hand, if u W 1;q0 () then Tu W 1;q0 (). Indeed, Tu(x)= u(x)+u0 (x)+u 0 (x) u(x) u 0 (x) 4 + u(x)+u0 (x) u 0 (x) u(x) u 0 (x) 4 and, by Corollary A:6. in [10] for example, it follows that Tu W 1;q0 (). Let us introduce U 0 =u 0 +K and U 0 =u 0 K, where K is a positive constant chosen in order to have U 0 1 and U 0 1: (4) Observe that TU 0 = u 0 and TU 0 = u 0. Dene now a(x) = max{ LU 0 (x); LU 0 (x); 1}: Then a L p () and a 1. For each 0 t 1, let us consider the boundary value problem Lu +(1 t)a(x)u = tf(x; Tu; D(Tu)) in ; u =0 on @: (5) Then, {U 0 ;U 0 } is a sub-supersolution couple for (5). Indeed, LU 0 +(1 t)a(x)u 0 LU 0 +(1 t)a(x) LU 0 (1 t)lu 0 = tlu 0 tf(x; u 0 ;Du 0 )=tf(x; TU 0 ;D(TU 0 )): Similarly, we can prove that U 0 is a sub-solution of (5). Furthermore, if u W ;p () is a solution of (5) for some t [0; 1], then U 0 u U 0 : (6) Indeed, let us just check that u U 0. Let us set w = u U 0. Then w W ;p () and we want to prove that w + = 0, where w + (x) = max{0;w(x)}. We know that Lu +(1 t)a(x)u = tf(x; Tu; D(Tu)) LU 0 +(1 t)a(x)u 0 tf(x; TU 0 ;D(TU 0 )) = tf(x; u 0 ;Du 0 ): Subtracting, we nd Lw +(1 t)a(x)w t[f(x; Tu; D(Tu)) f(x; u 0 ;Du 0 )]: (7) Multiplying this inequality by w + and integrating over, we obtain: (Lw)w + +(1 t) a(x)(w + ) t [f(x; Tu; D(Tu)) f(x; u 0 ;Du 0 )]w + t (g(x; Tu)+g(x; u 0 ))w + + k ( D(Tu) + Du 0 )w + : (8)

M. Delgado, A. Suarez / Nonlinear Analysis 4 (000) 995 100 999 Since w + =0 on @, one has (Lw)w + Dw + 1 div(b)(w + ) : (9) Here, we have used that p, i.e. that p N=(N + ). Observe that every integral in (8) is well posed. Certainly, g( ;Tu)+g( ;u 0 ) L p () by (H1), while w + L p () with p = p=(p 1) (this follows in a standard manner from Sobolev s imbedding and (H)). On the other hand, from the fact that D(Tu) L p (), we have D(Tu) + Du 0 L p = (): But denoting by s the conjugate of p =, we also have w + L s () (here, we use that (Np N +p)=(n p), which is also implied by (H)). Let us introduce A = {x :u(x) U 0 (x)}; B = {x :u(x) U 0 (x)}: We can write the last two integrals in (8) as the sum of integrals over A and B. Since w + =0 in A and Tu = Tu 0 = u 0 in B, we nd from (8) and (9) that w + = 0. The proof that U 0 u is similar. Let us introduce the mapping S :[0; 1] W 1;q0 () W 1;q0 (), where S(t; u)=v and v is the solution of Lv +(1 t)a(x)v = tf(x; Tu; D(Tu)) in ; v =0 on @: From the assumptions (H1) and (H), it follows that f( ; Tu; D(Tu)) L p () for any u W 1;q0 (). According to the L p theory of elliptic equations (see [7,9]), S(t; u) W ;p () for all t [0; 1] and for all u W 1;q0 (). Furthermore, from the Sobolev s imbedding theorem, it is not dicult to prove that S is continuous and compact. Let us show that u 1;q0 C; (10) whenever u W 1;q0 () and satises S(t; u) =u for some t [0; 1]. Then, from the Leray Schauder s xed point theorem (see for example [7], Theorem 11:6) we will be able to arm there exists u W 1;q0 () such that S(1;u)=u, i.e. satisfying Lu = f(x; Tu; D(Tu)) in ; (11) u =0 on @: Since this u belongs to W 1;q0 (), we will have f( ; Tu; D(Tu)) L p () and u W ;p (). Arguing as we did in the proof of (6), we will also have u 0 u u 0 ; whence Tu = u, that is to say u solves (1).

1000 M. Delgado, A. Suarez / Nonlinear Analysis 4 (000) 995 100 Hence, the proof of Theorem 1 will be achieved if we are able to establish (10). First, we observe that u 1;q0 C u ;p C( a p u + g( ;Tu) p + D(Tu) p ): Thus, one also has u ;p C( a p u + g( ;Tu) p + D(Tu) p ): (1) We know that u W ;p () L r (); r 1. From the Gagliardo Nirenberg inequalities (see [6], Theorem 10:1), we have Du p C u ;p u r 1 (13) with Nr pr pn = (Nr pr pn ) ( [1=; 1]): (14) This gives u ;p C(1 + g( ;Tu) p + u ;p u r (1 ) ): (15) Notice, from (6) that u r C where C is independent of t and r 1. From (H), we see that r can be chosen such that r(n p) pn and r 1; thus [1=; 1] r p + r N N p : (16) On the other hand, we have 1 N +r N + r : (17) Thus, for any and r satisfying r +r 1 r; r(n p) pn and N p + r N + r ; (18) we have 1 and, consequently, (10). We can consider the functions h 1 and h, with h 1 (r) r p + r ; h (r) N +r N + r : They both are increasing and satisfy lim h i(r)=: r + Moreover, h 1 (1) h (1) and their graphs only intersect when r(n p)=pn and, at this value, h i (r)=n=(n p). Thus, if p N and for any with =(p +1) N=(N p), we can nd one value of r which satises (18). If p N, the functions can only intersect at a nonpositive value, so that for any [=(p +1); ) we can nd a value for r which satises (18). In the case p = N the functions never intersect, so we have the same conclusion as the case p N. The proof is thus nished.

M. Delgado, A. Suarez / Nonlinear Analysis 4 (000) 995 100 1001 Remark. Notice that our solution satises u W ;p () and also u L () (remember (6)). Remark 3. We can use a similar argument when p=n. We have W ;p (), W 1;q () for any q [1; ), so every integral of (8) is well posed. We can dene S :[0; 1] W 1;q () W 1;q () for any q [1; ) and we need u 1;q C with C being a constant independent of t and u. From the Sobolev imbedding theorem, we have again u 1;q C u ;p : The Gagliardo Nirenberg inequalities can be also applied. In this situation, we obtain the restrictions N +1 : Moreover, the solution belongs to W 1;q () for any q [1; ). Remark 4. In the case N=(N +) p N=, we obtain in Theorem 1 that the upper bound is N=(N p). This function is increasing in p, so its inmum is achieved for p =N=(N + ). For this value, N=(N p)=(n +)=N which is just the bound in [15,16]. Remark 5. As we say in the Introduction, in [9] the author proved that there exists a solution u W ;m () for (1) where m = min{=; p}, and so, if =p u W ;p (). However, if =p then u belongs to W ;= (), W 1;q () where q = N N : If (N +)=N then q, so following [8] it cannot go further to obtain more regularity of the solution. In particular, it cannot prove the solution belongs to W ;p () by this way. 3. An example In this section, we analyze a simple (academic) example to which Theorem 1 can be applied. Let us consider the particular case N = 3, and the quasilinear problem u = n(x)+u( m(x)u)+(b(x) u) in ; (19) u =0 on @; where R; 3 and (H) m L (); n L (); b (L ()) 3 ; n 0; m m 0 0: In the context of population dynamics, any positive solution of (19) can be viewed as a steady-state population density. The coecient m = m(x) is associated to the limiting

100 M. Delgado, A. Suarez / Nonlinear Analysis 4 (000) 995 100 eect of crowding in the population, while the transport eect and the inuence of the surronding media are responsible for the presence of b=b(x) and n=n(x), respectively. With these assumptions, it is easy to prove that {u 0 ;u 0 } = {0;K} is a sub-supersolution couple for (19), provided K 0 is suciently large. So, we can deduce from theorem 1 that there exists a positive solution of (19). When p N, the existence of a sub-supersolution couple for (1) is ensured by others conditions that can be found in [15,4] Acknowledgements The authors would like to ackowledge support under grant DGICYT PB95-14. They also thank the referee for helpful suggestions. References [1] K. Akô, On the Dirichlet problem for quasi-linear elliptic dierential equations of the second order, J. Math. Soc. Japan 13 (1961) 45 6. [] H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z. 150 (1976) 81 95. [3] H. Amann, M. Crandall, On some existence theorems for semilinear equations, Indiana Univ. Math. J. 7 (1978) 779 790. [4] N.P. Cac, Some remarks on a quasilinear elliptic boundary value problem, Nonlinear Anal. 8 (1984) 697 709. [5] E.N. Dancer, G. Sweers, On the existence of a maximal weak solution for a semilinear elliptic equation, Di. Int. Equa. (1989) 533 540. [6] A. Friedman, Partial Dierential Equations, Holt Rinehart and Winston. New York, 1969. [7] D. Gilbarg, N.S. Trudinger, Elliptic Partial Dierential Equations of Second Order, Springer, Berlin, 1977. [8] P. Hess, On a second-order nonlinear elliptic boundary value problem, Nonlinear Anal, in: L. Cesari, R. Kannan, H. Weinberger (Eds.), A Collection of Papers in Honor of E. Rothe, Academic Press, New York (1978), pp. 99 107. [9] J. Kazdan, R. Kramer, Invariant criteria for existence of solutions to second-order quasilinear elliptic equations, Commun. Pure Appl. Math. 31 (1978) 619 645. [10] D. Kinderlehrer, G. Stampacchia, An Introduction to Varitional Inequalities and Their Applications, Academic Press, New York 1980. [11] J. Mawhin, K. Schmitt, Upper and lower solutions and semilinear second order elliptic equations with non-linear boundary conditions, Proc. Roy. Soc. Edinburgh 97A (1984) 199 07. [1] W. Mlak, Parabolic dierential inequalities and Chaplighin s method, Ann. Polon. Math. 8 (1960) 139 15. [13] T. Satô T, Sur l equation aux derivees partielles z = f(x; y; z; p; q), Composition Math. 1 (1954) 157 177. [14] J. Serrin, The problem of Dirichlet for quasilinear elliptic dierential equations with many independent variables, Phil. Trans. Roy. Soc. London 64 (1969) 413 496. [15] J.B.M. Xavier, Some existence theorems for equations of the form u = f(x; u; Du), Nonlinear Anal. 15 (1990) 59 67. [16] Z. Yan, A note on the solvability in W ;p () for the equation u = f(x; u; Du), Nonlinear Anal. 4 (1995) 1413 1416.