MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION


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1 MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe and subsolution method with Schaude s fixed point theoem to show the existence of multiple solutions of the pescibed mean cuvatue equation unde some special cicumstances. In this pape, we study the existence of multiple solutions of the following equation: { M(u) = nh(x, u) in Ω R n (0.) u Ω = ϕ, whee u M(u) := div ( + u ). / It is well known in [?], that equation (0.) has a unique solution if (0.) H 0 := sup (x,u) Ω R H(x, u) < n n inf x Ω H (x), whee H (x) stands fo the mean cuvatue of Ω at the point x, and if (0.3) H u (x, u) 0. Fo simplicity, we shall hencefoth conside the bounday condition u Ω = 0 instead. In ode to find multiple solutions of equation (0.), we must avoid assumption (0.3). Howeve, to ou knowledge, this condition has always played a cucial ole in the a pioi estimate fo the solution of equation (0.). It is quite difficult to study equation (0.) fo geneal Ω, so we shall only focus on the following special cases: () Fo n =, we set up the Mose theoy fo the functional I(u) := ( + u + H(x, u)) dx, with H(x, u) = u 0 Ω H(x, t) dt, and povide a multiple solution esult unde assumptions (H), (H), (H3). Because of the special featue of equation (0.), it is well known that, citical point theoy can not be set up on the W,p spaces fo p > using the above functional. But fo p = o othe function spaces, the lack of Palais Smale Condition is again a majo difficulty. Howeve, as the fist autho has noticed in [3], sometimes the heat flow, to which the Palais Smale Condition is ielevant, can be used as a eplacement 99 Mathematics Subject Classification. Pimay 53A0, 58E05; Seconday 35A5, 58C30. Key wods and phases. Mean cuvatue equation, Mose Theoy, multiple solutions.
2 K.C. CHANG AND TAN ZHANG of the pseudogadient flow in citical point theoy. We shall study the elated heat equation, and use the heat flow to set up the Mose theoy of isolated citical points fo the above functional. Citical goups fo isolated citical points ae counted, and Mose elation is applied. The main esult of the fist section is Theoem., which assets the existence of thee nontivial solutions. () Fo n >, we shall futhe assume H = H(, u) is otationally symmetic. Consequently, equation (0.) can be educed to the following O.D.E.: ü + n u ( + u ) = nh(, u)( + u ) 3/. We then use the sub and supe solution method to pove that thee ae at least two nontivial solutions if assumptions (H ), (H ) hold, and a degee agument to pove the existence of the thid nontivial solution unde the additional assumption (H3 ) and (H4). The main esult of the second section is Theoem.. This pape is divided into two pats. In pat, we deal with case (), using the heat flow method instead of the taditional pseudogadient flow method on some Hölde space. This appoach will theeby enable us to bypass the Palais Smale Condition. In pat, we deal with case (). We fist make some simple estimates, and then comes the cucial point in which, we constuct a positive small subsolution and a negative small supesolution of equation (0.). In the case n =, we have: M(u) = ü ( + u ) 3/. Let R > 0 and let = [ R, R]. Equation (0.) is now educed to the O.D.E.: (.) { ü = H(x, u)( + u ) 3/ fo x ( R, R) u(±r) = 0. We assume that H C ( R, R ) satisfies: (H) H 0 := sup (x,u) R H(x, u) < R. It is easy to veify that w ± λ (x) = ±( λ + R + λ x ), λ > 0, x R is a pai of supe and sub solutions of equation (.), povided λ < 3R. If we also assume: (H) H(x, 0) = 0 and H u (x, 0) > ( π R ), then x u ± (x) = ±ɛ cos( πx R ), x R ae positive sub and negative supe solutions of (.) espectively, povided ɛ > 0 is sufficiently small.
3 MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION 3 In fact, we have: M(u + ) = ( π R ) u + ( + ( u + ) ) 3/ < H u(x, 0)u +. It follows that M(u + ) H(x, u + ) fo ɛ > 0 small, i.e., u + is a subsolution. Hence, (u +, w + λ ) and (u, w λ ) ae two pais of sub and supe solutions of (.). Now we conside the heat equation elated to (.): (.) v t M(v) = H(x, v) in (0, T ) ( R, R), v(t, ±R) = 0, t [0, T ], v(0, x) = v 0, x, whee the initial data v 0 C +α () with v(±r) = 0 and α (0, ). We begin by intoducing the weighted paabolic Hölde space H ( α) +α (Ω) on Ω := [0, T ], fo 0 < α <, as follows: { v C(Ω) β+j d(x)max{β+j α,0} x β j t v(x) + d(x, Y )(Σ β+j= } β x j t v(x) β x j t v(y ) X Y + α xv(x) xv(y ) X Y ) < +, X, Y Ω, X Y +α whee X = (t, x), Y = (τ, y), X Y = ( x y + t τ ), d(x, Y ) = min{d(x), d(y )}, and d(x 0 ) = dist{x 0, ( Ω\{t = T }) {t < t 0 }}, fo X 0 = (t 0, x 0 ). The nom of H ( α) +α is defined by ( v = sup X,Y Ω,X Y β+j d(x)max{β+j α,0} x β j t v(x) ) +d(x, Y )(Σ β x j t v(x) β x j t v(y ) xv(x) xv(y ) β+j= X Y + α X Y ). +α Accoding to [?], the solution v H ( α) +α (Ω) exists fo any T > 0. Applying the Maximum Pinciple, we have: if Similaly, u + (x) v(t, x) w + (x), (t, x) Ω, u + (x) v 0 (x) w + (x), x. w (x) v(t, x) u (x), (t, x) Ω, if w (x) v 0 (x) u (x), x. We now view the solution v(t, x) as a flow, and conside the functional I(u) := [ + u + H(x, u)] dx,
4 4 K.C. CHANG AND TAN ZHANG with H(x, u) = equation (.). u Along the flow, we have: 0 H(x, t) dt, then the EuleLagange equation fo I is exactly d I(v(t, )) = dt = Hence, the functional I is noninceasing. v (t, x)[m(v(t, x)) H(x, v(t, x))] dx t ( ) v (t, x) dx 0 t We denote [u, v] = {w C +α () C 0 () u(x) w(x) v(x), x }, and notice that, if the initial data v 0 falls into the odeed inteval [u +, w + ] (o [w, u ] esp.), then v(t, x) is bounded, and I(v(t, )) is theefoe bounded fom below. Thus, c := lim t I(v(t, )) exists, and 0 ( ) v (t, x) dx dt = I(v 0 ) c t Thee must be a sequence t j + such that v j (x) := v t (t j, x) 0 in L (). Let u j (x) := v(t j, x) fo all j. Substituting these into equation (.), we obtain a sequence of equations: (.3) M(u j ) H(x, u j ) = v j, j We want to show that {u j } subconveges to a solution of (.). To this end, we shall pove that ü j is bounded as follows. Let z j = u j ( + u j )/. Since u j (±R) = 0, thee is ξ such that u j (ξ) = 0, i.e., z j (ξ) = 0. Accoding to equation (.3), we have: x z j (x) ( H(x, u j (x)) + v j (x) ) dx ξ H 0 R + (R) / ( v j dx) / It follows, fom v j 0 and (H), that thee is ɛ > 0, which depends only on H 0 and R, such that z j ɛ. Hence, u j ɛ. ɛ By setting M ɛ := ( + ( ɛ ) ) 3/, we have that ü j (H 0 + )M ɛ ; hence, u + W, () such that u j u + in W, (), and then u j u + in C ().
5 MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION 5 Finally, ϕ C0 (), fom [ ] ü j (x) ( + u H(x, u j(x)) ϕ(x) dx 0, j (x))3/ it follows that [M(u +) H(x, u +)]ϕ(x) dx = 0, i.e., M(u +(x)) = H(x, u +(x)) a.e., thus u + C 3 (). In summay, along the heat flow, thee is a subsequence {t j } + such that v(t j, ) u + in W, (), whee u + is a solution of (.). Let K be the set of all solutions of (.) in the ode inteval [u +, w + ],. Accoding to the pevious estimates, it is W, () bounded, so is compact in C (). Since inf{i(u) u [u +, w + ]} = inf{i(u), u K}, theefoe the functional I has a minimize, again denoted by u +, in the odeed inteval [u +, w + ]. A simila agument shows that we also have a solution u C 3 ( ) which minimizes the functional I in the odeed inteval [w, u ]. Since [w, u ] [u +, w + ] =, u + u ; they ae two distinct nontivial solutions of (.). In light of the above esult, we shall next seek a thid nontivial solution of (.). This is based on a Mosetheoetic appoach, we efe to [?] and [?] fo futhe details. When v 0 C +α () C 0 (), 0 < α <, it is known ([?]) that the solution v(t, x) H ( α) +α (Ω), which defines a defomation η : [0, + ) (C +α () C 0 ()) C +α () C 0 () by η(t, v 0 ) = v(t, ). It is easy to veify the continuity of the mapping η by standad aguments. In paticula, η : [0, + ) X X, whee X = [w, w + ] C +α () C 0 () is a closed convex set in the Banach space C +α () C 0 (). Now, a R, we denote I a := {u X I(u) a}. The following defomation lemma holds: Lemma.. (Defomation lemma) If thee exists no citical point of the functional I in the enegy inteval I [a, b], except pehaps some isolated citical points at the level a, then I a is a defomation etact of I b. Poof. It is sufficient to pove: If the obit O(v 0 ) = {η(t, v 0 ) t R +} I (a, b], and if the limiting set ω(v 0 ) is isolated; then η(t, v 0 ) has a weak W, limit w, (and then lim t + η(t, v 0 ) = w in C +α, 0 < α < ) on the level I (a). Indeed, by the pevious agument, w ω(v 0 ), we have a sequence t j + such that v(t j, ) w in W, (). Thus, v(t j, ) w C 0. Suppose the conclusion is not tue, then ɛ 0 > 0, t < t < t < t <, such that v(t j, ) w C = ɛ 0 v(t, ) w C v(t j, ) w C = ɛ 0, t I j := [t j, t j ], j =,,, and ω(v 0 ) B ɛ0 (w)\{w} =. The above inequalities imply the C boundedness of v(t, ) on I j, namely,
6 6 K.C. CHANG AND TAN ZHANG v(t, ) C w C + ɛ 0. Afte simple estimates, we have v(t, ) C C, t I j, whee C is a constant independent of j. But Since I j t v(t, ) dt = I(v(t j, )) I(v(t j, )) 0. ɛ 0 v(t j, ) v(t j, ) t j t t v(t, ) dt j (t j t j ) ( t j v(t, ) dt), t j it follows t j t j +, and then t j I j such that t v(t j, ) 0. Obviously, v(t j, ) C C, and v(t j, ) B ɛ 0 \B ɛ 0 (w). Again, by the pevious agument, z ω(v 0 ) such that v(t j, ) z. Then we have: z ω(v 0 ) B ɛ0 \B ɛ 0 (w). This is impossible, since ω(v 0 ) B ɛ0 (w)\{w} =. Fom lemma., the Mose elation holds fo I on X. In context, citical goups C q (I, u 0 ) = H q (U I c, (U \ {u 0 }) I c ) ae defined fo an isolated citical point u 0 of I, whee U is an isolated neighbohood of u 0, c = I(u 0 ), and H q (Y, Z) ae the gaded singula elative homology goups fo q = 0,,. In the estimation of numbe of solutions, we can assume, without loss of geneality, that thee ae only finitely many citical points {u, u,, u N } of I. Noticing that both u ± ae local minimizes of I, it follows that C q (I, u ±) = δ q0. On the othe hand, let β q be the qth Betti numbe of X, q = 0,,. Since the set X is contactible, β 0 =, and β q = 0, q. Let M q be the qth Mose type numbe of I: M q = Σ N j=ank C q (I, u j ), q. The Mose elation eads as Σ 0 (M q β q )t q = ( + t)p (t), whee P is a fomal powe seies with nonnegative coefficients. Theefoe, we must have at least one moe citical point u of I. If thee ae citical points othe than u and u ±, the conclusion follows, so we may assume u is the unique citical point othe than u ±. Then we have ank C 0 (I, u ) + ank C (I, u ) 0, accoding to the Mose elation. In ode to distinguish u fom θ (the tivial solution), we assume futhe (H3) H u (x, 0) > ( π R ).
7 MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION 7 Since d I(θ, ϕ) = ( ϕ + H u (x, 0)ϕ ) dx, unde the assumption (H3), we have C q (I, θ) = 0, fo q = 0 and. Again, this will be a contadiction, if besides u ±, I has only the citical point θ. Thus we have indeed established the following: Theoem.. Assume that H C ( R, R ) satisfies (H) and (H), then equation (.) has at least two distinct nontivial solutions, one positive and one negative. If in addition, (H3) is satisfied, then (.) will have at least thee distinct nontivial solutions. In the case n, we assume H = H(, u) is otationally symmetic. Let Ω = B R (0) R n, the ball centeed at the oigin of adius R. Equation (0.) is then educed to: (.) n du d ( d d n du d (+( du d ) ) / (0) = u(r) = 0, and assumption (0.) then becomes ) = nh(, u) fo (0, R) (.) H 0 := sup (,u) [0,R] R H(, u) < n n (.3) H 0 < R, if n = R, if n, and We shall use a fixed point agument in conjunction with the supe and sub solution method to tackle (.). We begin by deliveing the following a pioi estimate fo solutions of (.): Lemma.. Thee is a constant C, depending only on n, H 0, and R, such that all solutions u of (.) satisfy (.4) u C C. Poof. Fo n, it suffices to show () u () nh 0, and () u () n( n ) Fo then, u() is bounded by () and the bounday condition u(r) = 0; in the mean time, u is bounded by () and the altenative expession of (.): u + n u ( + u ) = nh(, u)( + u ) 3/. Fo simplicity, we let v = n u ( + u ) /.
8 8 K.C. CHANG AND TAN ZHANG It follows that v (.5) u = ± n v ( ( ) ), / n and (.6) v = n n H. It now emains to show v ( n )n. If so, fom (.5), we have: u ( n )[ ( n ) ] / n( n ), which poves (). Since u (0) = 0, v(0) = 0, by (.6), we have: hence, (.7) (.7) subsequently implies and v() 0 v (t) dt H 0 n, v() n H 0. v() n H 0R n, v ( ) u = n v ( ( ) ) nh / 0. n Fo n =, assumption (.) is eplaced by assumption (.3), and we have v H 0 H 0 R. Let ɛ := H 0 R, we then have: u ɛ. This completes the ɛ poof. Next, we constuct two pais of sub and supe solutions of (.) as in. Howeve, fo n >, the constuction of the second pai of such solutions is a bit moe complicated. Again, we let w λ () = ±( λ + R + λ ), λ > 0, 0 R. The following popeties hold: (w) w λ (R) = 0, ẇ λ (0) = 0, (w) M(w λ ) = M( w λ ) = (w3) ẇ λ (R) = R λ, (w4) w λ () R, ẇ λ λ λ. These popeties lead to the estimate: n (R + λ ) /,
9 MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION 9 povided n M(w λ ) = n(n (R + λ ) / n ) R nh(, w λ), n R, fo n >, λ λ 0 := n ɛr, fo n =, whee 0 < ɛ < (H 0R). This veifies ( w λ, w λ ) is a pai of sub and supe solutions of (.), when λ λ 0. Next, we conside the functions z n () := n n (), fo n =,,, whee n () denotes the ( n )ode Bessel function. Let µ n be the fist zeo of n (), and let we then have: v n () := z n ( µ n ), n =,,, R (v) v n (R) = 0, v n (0) = 0, (v) v n + n v n = ( µ n R ) v n, (v3) M n > 0 such that v n () M n. We intoduce assumption (H ) in place of (H) fo n > : (H ) H(, 0) = 0 and H u (, 0) > ( µ n R ), [0, R]. and define We compute to see that M(u + ) = u + := { δv n w λ, fo n > δv, fo n =, { δ( v ( + (u + ) ) 3/ n + n + n (δ 3 v n 3 3δ v nẇ λ + 3δ v n ẇλ) ( + ẇ λ = whee the emainde ( µn R ) ( + (u + ) ) 3/ δv n + + (u + ) v n ) [ẅ λ + n (ẇ λ + ẇ 3 λ)] } ) 3/ n Q(n, δ, λ)() + (R + λ ) / ( + (u + ) ), 3/ One has Q(n, δ, λ)() = n (δ 3 v n 3 3δ v nẇ λ + 3δ v n ẇλ).
10 0 K.C. CHANG AND TAN ZHANG M(u + ) = ( µn R ) Fom (w4) and (v3), we obtain: ( µn R ) ( + (u + ) ) u 3/ + + ( + (u + ) ) w 3/ λ ( ) + ẇ 3/ λ n Q(n, δ, λ)() ( + (u + ) ) (R + λ ) / ( + (u + ) ). 3/ = + o() as δ 0, λ +, ( + (u + ) 3/ ) Q(n, δ, λ)() = O(δ 3 + δ λ + δλ ), and ( µ n R ) w λ ( + ẇλ) 3/ n (R + λ ) µ n / λ Let ɛ := H u (, 0) ( µn R ), we have: M(u + ) < H u (, 0)u + ( + (u + ) ] ) 3/ + O(δ 3 ), with λ = δ as δ 0. n (R + λ ) / If one can choose δ > 0 so small that (.8) ( ( R ) ) n (R + λ ) / [ ɛ u + µ n ( ( R ) λ ) + [ ( µ n R ) + ɛ ] (R )δ n δɛv n () + (R + δ 4 ), / then u + is a subsolution of (.) fo n >. This is possible, since v n () > 0 in [0, R), and since the last tem in the ight hand side of (.8) is of ode δ, futhemoe, it is a positive constant on [0, R]. So, (.8) holds fo small δ > 0. As to n =, it has aleady been shown in section. In summay, we obtained, as in the pevious section, two pais of sub and supesolutions of (.), (u +, w + ) and (w, u ) whee w ± = ±w λ0 and u = u +. In the next step, a fixed point agument is applied to obtain a thid nontivial solution of (.). Let X := {u C ([0, R]) u (0) = u(r) = 0}. We define an opeato T : X X C ([0, R]) via v := T u, u X, and if v is a solution of the linea O.D.E.: { v + n v = ( + u ) 3/ (nh(, u) n u 3 ) v(0) = v(r) = 0 It is clea that u is a fixed point of T if and only if u is a solution of (.). Accoding to the Maximum Pinciple, T maps the odeed inteval [w, u ] X and [u +, w + ] X into themselves. By the Schaude s Fixed Point Theoem, thee is a fixed point u ± in each of these odeed intevals. We may assume, without loss of geneality, that u ± is the only fixed point of T in O ±, whee O + = int ([u +, w + ] X) and O = int ([w, u ] X), the inteio
11 MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION of each of the coesponding set. Fo othewise, thee must be a thid nontivial solution. Applying the LeaySchaude s degee theoy, we have ind (I T, u ±) =, whee ind (I T, u ±) denotes the LeaySchaude index of I T at u ±, espectively. We can then use Amann s Thee Solutions Theoem [?] to confim the existence of a thid solution u. One may again assume, without loss of geneality, that u is the only fixed point in [w, w + ] X othe than u ±. By a degee computation, ind (I T, u ) =, povided deg (I T, int ([w, w + ] X), θ) =. As befoe, we should distinguish u fom θ, the tivial solution. Fo this eason, stonge assumptions ae imposed. Let µ j n be the jth zeo of the Bessel function n, and let m j n be the multiplicity of µ j n. We assume that j 0 satisfying (H3 ) and µ j0 n < H u (, 0)R < µ j0+, [0, R], n (H4) Thus, This implies u θ. m n = Σ j0 j= mj n is odd. ind (I T, θ) = ( ) mn =. Lastly, we conclude ou pape by stating the following: Theoem.. Assume that H C ([0, R] R, R ) satisfies (.) and (H ), then equation (.) has at least two distinct nontivial solutions, one positive and one negative. If in addition, (H3 ) and (H4) ae satisfied, then (.) will have at least thee distinct nontivial solutions. Remak In the case n =, by combining methods used in sections and, we may obtain a esult slightly diffeent fom both Theoem. and Theoem.. Namely, if H C ( R, R ) satisfies H 0 < R, and H(x, u) = H( x, u), x [ R, R], u R, H(x, 0) = 0 and H u (x, 0) > ( 3π R ) x [ R, R], then thee exist at least thee nontivial symmetic solutions of equation (.).
12 K.C. CHANG AND TAN ZHANG The poof is the same as that of Theoem., with an impovement of the a pioi estimates of the solutions obtained in the fist few paagaphs of, unde the assumption that u is symmetic. Refeences. Amann, H. On the numbe of solutions of nonlinea equations in odeed Banach spaces,. Funct. Anal. 4 (973), Chang, K. C. Infinite dimensional Mose theoy and multiple solution poblems, Bikhause, (993). 3. Chang, K. C. Heat method in nonlinea elliptic equations, Topological methods, Vaiational methods, and thei applications, (ed. by Bezis, H., Chang, K. C., Li, S.., Rabinowitz, P.) Wold Sci. (003), Gilbag, D., Tudinge, N., Elliptic patial diffeential equations of second ode, Gundlehen de Mathematischen Wissenschaften 4, (983). 5. Liebemann, G. M., Second ode paabolic diffeential equations. Wold Sci. (996). LMAM, School of Math. Sci., Peking Univ., Beijing 0087, China, Dept. Math & Stats., Muay State Univ., Muay, KY 407, USA, eseach suppoted by NSFC, MCME
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