An eigenvalue problem for non-bounded quasi-linear operator

Proceedings of the Edinburgh Mathematical Society Submitted Paper Paper 30 March 2004 AN EIGENVALUE PROBLEM FOR NON-BOUNDED QUASI-LINEAR OPERATOR Jos´e Carmona1and Antonio Su´arez2 1Dpto. de ´ Algebra y An´alisis Matem´atico, Facultad de Ciencias, Ca˜nada de San Urbano, Almer´ıa, Spain e-mail: jc[email protected] 2Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Sevilla, Spain e-mail: suar[email protected] (Received ) Abstract In this paper we study the eigenvalues associated with a positive eigenfunction of a quasilinear elliptic problem with a not necessarily bounded operator. For that, we use the bifurcation theory and obtain the existence of positive solution for a range of values of the bifurcation parameter. AMS 2000 Mathematics subject classification: Primary 35J60, 35J25 Secondary 35D05 1. Introduction Let Ω be a bounded open subset of RNwith sufficiently smooth boundary ∂Ω and let A(x, s) be a real symmetric matrix which coefficients, aij : Ω×R+ 0→R, are Carath´eodory functions. We assume that there exists a positive constant αsatisfying for every (x, s, ξ)∈Ω× R+×RN, A(x, s)ξ·ξ≥α|ξ|2.(A1) In this paper we analyze the nonlinear eigenvalue problem (−div(A(x, u)∇u) = λu, x ∈Ω, u= 0, x ∈∂Ω,(Pλ) where, we say that λis an eigenvalue for this problem if (Pλ) admits a positive and nontrivial solution, that is, if there exists u∈H1 0(Ω), u≥0, u6≡ 0, such that A(x, u)∇u∈ (L2(Ω))Nand ZΩ A(x, u)∇u· ∇v=λZΩ uv, ∀v∈H1 0(Ω). 1 2J. Carmona and A. Su´arez In addition to the interest itself in the study of (Pλ), this kind of equation has been used to model a species inhabiting in Ω where its diffusion depends on the density of the species, which arises in more realistic models, see [3] and references therein. Problem (Pλ) is well known when Adoes not depend on s, i.e., when A(x, s) = B(x) with B= (bij) and bij ∈L∞(Ω), bij ≥b0>0 in Ω. In this case, there exists the principal eigenvalue, denoted by λ1(B), for the problem: (−div(B(x)∇u) = λu, x ∈Ω, u= 0, x ∈∂Ω,(1.1) being the unique eigenvalue with a positive eigenfunction, see for instance [5]. In [2], assuming that Asatisfies (A1) and |A(x, s)| ≤ β, for each (x, s)∈Ω×R, (A2) the author proved that for each r > 0, there exists λr>0 and a positive solution ur∈H1 0(Ω), of (Pλr) such that kurk2=r. Moreover, denoting by λ0:= λ1(A(x, 0)), he showed that if r→0, then λr→λ0and ur rconverges to a positive eigenfunction associated to λ0in H1 0(Ω). Finally, if Aalso verifies lim s→∞ A(x, s) = A∞(x),uniformly in x∈Ω,(A3) then λr→λ∞and ur rgoes to a positive eigenfunction associated to λ∞in H1 0(Ω) as r→ ∞, where λ∞:= λ1(A∞(x)). In [4], a slightly modification of (Pλ) is analyzed. Under conditions (A1−3), λu +h(x) for some 0 ≤h∈L2(Ω) is considered instead of λu. But the arguments used to prove the existence of solution leads to the trivial one in the case h≡0. In [1], assuming in addition the existence of an Osgood function ω:R+ 0→Rsuch that |A(x, s1)−A(x, s2)| ≤ ω(|s1−s2|),(A4) for every (x, s1),(x, s2)∈Ω×R, using a bifurcation analysis, the authors study a more general problem (−div(A(x, u)∇u) = f(λ, x, s), x ∈Ω, u= 0, x ∈∂Ω, for f:R×Ω×R7→ Rand Asatisfying (A1−4). In the particular case f(λ, x, s) = λs, from their results it can be deduced the existence of an unbounded continuum (closed and connected subset) of positive solutions bifurcating from the trivial solution at λ=λ0and Non-bounded quasi-linear operator 3 meeting with infinity at the value λ=λ∞. Thus, as a consequence, there exists positive solution of (Pλ) for λ∈(λ0, λ∞) or (λ∞, λ0). In the following section we complete this study for Asatisfying (A1−4) by giving sufficient conditions for the uniqueness of positive solution. The main goal of this work (see Section 3) is to analyze (Pλ) when Ais not necessarily bounded and/or does not satisfy (A3). In this case, we show that there exists an unbounded continuum of positive solutions bifurcating from the trivial one at λ=λ0. If, in addition there exists a continuous function g:R+ 0→R, with lim s→+∞g(s)=+∞, satisfying for every (x, s, ξ)∈Ω×R+×RN, A(x, s)ξ·ξ≥g(s)|ξ|2≥α|ξ|2.(A∞) then, the bifurcation from infinity at λ=λ∞(which exists in the bounded case) “disappears”. Specifically, there exists at least a positive solution uλfor λ∈(λ0,∞) and kuλk→∞as λ→ ∞. However, if Ais bounded in a subset of Ω, then again a bifurcation to infinity exists. Along the work we will use the following notation: •H1 0(Ω) and E=C0(Ω) are the usual Sobolev space and the space of the continuous functions in Ω vanishing on ∂Ω endowed with the norms kuk=k∇uk2and kuk0= supΩ|u|, respectively. •cl(D) denotes the closure of the set D. • S denotes the set S= cl{(λ, u)∈R×E:uis solution for (Pλ), u ≥0, u 6≡ 0}. Any continuum subset of Swill be called a continuum of positive solutions of (Pλ), although it may contain the trivial solution (λ, 0) for some value of λ > 0. •Iwill denote both the identity matrix and the identity operator. •Given square matrices B1, B2we say that B1>0 (respect. B1≥0) if the quadratic form induced by B1is definite positive (respect. semidefinite positive). We say that B1< B2(respect. B1≤B2) if B2−B1>0 (respect. B2−B1≥0). •The map ProjR:R×E7→ Rstands for the projection of the product space R×E onto R. 2. The case of bounded matrices A In order to study problem (Pλ), let us recall that, for matrices Asatisfying (A1,2), if u∈H1 0(Ω) is solution of (Pλ) then using the De Giorgi-Stampacchia Theorem ([8, Th´eor`eme 7.3] and [6, Theorem I] or [7, Theorem 8.29]), u∈C0,γ(Ω) for some 0 < γ < 1. Moreover, if the coefficients of the matrix Asatisfy aij ∈C1,γ0(Ω ×R),for some 0 < γ0<1,(2.1) 4J. Carmona and A. Su´arez then by Theorem 15.17 in [7] we have that u∈C2,γγ0 0(Ω). We also recall that for every (λ, u)∈ S with u∈ C1(Ω) and u6≡ 0, using the Hopf maximum principle, we have that u > 0 in Ω and the normal exterior derivative ∂u ∂neis negative in ∂Ω. The following lemma provides us necessary conditions in λ∈Rfor which (Pλ) admits solution in some special cases. Lemma 2.1. Assume (A1,3)and that (Pλ)admits a positive solution. Then 1. λ0≤λ(respect. <, ≥, >) if for every s∈R+,A(x, 0) ≤A(x, s)(respect. <, ≥, >). 2. λ∞≥λ(respect. >, ≤, <) if for every s∈R+,A∞(x)≥A(x, s)(respect. >, ≤, <). Proof. The result follows from the fact that for given symmetric matrices B1(x), B2(x) for which there exist λ1(B1) and λ1(B2), with 0 < B1≤B2then λ1(B1) = inf ½ZΩ B1(x)∇u· ∇u, u ∈H1 0(Ω),kuk2= 1¾≤λ1(B2). Thus, if u∈H1 0(Ω) is a solution of (Pλ), we conclude by taking into account that λ=λ1(A(x, u)). ut The main result of this section is the following: Theorem 2.2. Assume (A1−4). We have that λ0and λ∞are the only bifurcation points from the trivial solution and from infinity, respectively, and there exists a continuum Σ⊂ S of positive solutions meeting (λ0,0) and (λ∞,∞), in particular, (Pλ) possesses a positive solution for every λ∈(λ0, λ∞)or λ∈(λ∞, λ0). Moreover, •the bifurcation from λ0is subcritical (resp. supercritical) if there exists s0>0such that A(x, s)< A(x, 0),(respect. A(x, s)> A(x, 0)),∀s∈(0, s0), •the bifurcation from λ∞is subcritical (resp. supercritical) if A(x, s)< A∞(x),(resp. A(x, s)> A∞(x)),∀s∈R+. Furthermore, •if A(x, 0) < A(x, s)< A∞(x)for every s∈R+, then there exists nontrivial solution for (Pλ)if, and only if, λ∈(λ0, λ∞), in particular ProjRΣ = [λ0, λ∞). If, in addition, A(x, s)is increasing in sand it verifies (2.1), the solution is unique. •If A(x, 0) > A(x, s)> A∞(x)for every s∈R+, then there exists nontrivial solution for (Pλ)if, and only if, λ∈(λ∞, λ0), in particular ProjRΣ = (λ∞, λ0]. Non-bounded quasi-linear operator 5 Proof. The existence of the continuum Σ of positive solutions follows by Theorem 5.1 in [1], and so the existence of positive solutions for every λin (λ0, λ∞) or in (λ∞, λ0). The description ProjRΣ, in the cases A(x, 0) < A(x, s)< A∞(x) or A(x, 0) < A(x, s)< A∞(x) for every s∈R+, follows directly from Lemma 2.1. Moreover, arguing as in that lemma we get the laterality of the bifurcations. Now, assume that A(x, s) is increasing in sand (2.1) is satisfied. In order to prove the uniqueness of solution for (Pλ), let us suppose that there exist λ∈(λ0, λ∞) and u1, u2∈E, solutions of (Pλ) with u16≡ u2. We claim that u1, u2can be chosen such that u1≤u2. Indeed, this is a consequence of the existence of a sequence (λn, un) with λn→λ0and un→0 in E. In fact, by regularity results, un→0 in C1(Ω). Thus, for λn< λ,unis a subsolution for (Pλ) and for large n,un≤min{u1, u2}. Then, by the sub and supersolution method, there exits w∈Esolution of (Pλ) with un≤w≤u1, un≤w≤u2. This implies that w6≡ u1or w6≡ u2, and the claim is proved by taking u1=wand u2=uifor some i= 1,2. Now we take v=u2 2 u1as test function in the equation satisfied by u1and v=u2in that satisfied by u2. Thus, subtracting both equalities we have that: 0 = ZΩ A(x, u1)∇u1· ∇ µu2 2 u1¶−ZΩ A(x, u2)∇u2· ∇u2 =−ZΩ A(x, u1)µu2 u1 ∇u1− ∇u2¶·µu2 u1 ∇u1− ∇u2¶ −ZΩ (A(x, u2)−A(x, u1)) ∇u2· ∇u2<0. This contradiction gives the uniqueness. ut 3. The case of unbounded matrices A In this section, we study (Pλ) when Ais not necessarily bounded and does not satisfy (A3). We prove firstly that every solution of (Pλ) is bounded. More precisely we have Lemma 3.1. Let A(x, s)satisfy (A1)and u∈H1 0(Ω) be a solution of (Pλ), then u∈E. Moreover, there exist positive constants c1, c2, γ1, γ2such that kukγ1 0≤c1+c2kukγ2.(3.1) Proof. Once we know that u∈L∞(Ω), and kukγ1 ∞≤c1+c2kukγ2for some positive constants c1, c2, γ1, γ2, then the result follows directly from the De Giorgi-Stampacchia Theorem. Let us prove the L∞(Ω)-estimate. We consider for every k∈R+the function Gk:R+ 0→R+ 0given by Gk(s) = (0 0 ≤s≤k, s−k s > k. 6J. Carmona and A. Su´arez Thus, we can take v=Gk(u) as test function in the weak equation satisfied by uand using (A1) we have αk∇Gk(u)k2 2≤ZΩ A(x, u)∇u∇Gk(u)≤λZ Ωk uGk(u),(3.2) where Ωk≡ {x∈Ω : u(x)> k}. Using the Sobolev and H¨older inequalities, in the case N > 2, by (3.2) we yield, for u∈Lr(Ω) with r > 2∗ 2∗−1, and some positive constant c, kGk(u)k2 2∗≤ckukrkGk(u)k2∗(meas Ωk)(1−1/r−1/2∗).(3.3) Taking into account that, for every h > k,Gk(u)≥h−kin Ωh, (3.3) implies that (h−k)(meas Ωh)1/2∗≤ckukr(meas Ωk)(1−1/r−1/2∗), or equivalently meas Ωh≤ckuk2∗ r(meas Ωk)2∗−1−2∗/r (h−k)2∗.(3.4) We can now apply the Stampacchia Lemma ([8, Lemma 4.1]) to deduce that: i) if u∈Lr(Ω) with r > N 2, then u∈L∞(Ω) and kuk∞≤ckukr, ii) if u∈Lr(Ω) with r=N 2, then u∈Lt(Ω) for t∈[1,∞) and kukt t≤c+c0kukt r, iii) if u∈Lr(Ω) with r < N 2, then u∈Lt(Ω) for t=2∗r (2−2∗)r+2∗−δand δ > 0 arbitrarily small. Moreover, kukt t≤c+c0kukt+δ r. Since u∈L2∗(Ω) and 2∗>2∗ 2∗−1, we can argue as before for r0= 2∗. Thus, if 2∗>N 2 we conclude by item i). In the case 2∗=N 2we use item ii) in order to take r1>N 2and conclude again by item i). Finally, in the case 2∗<N 2we can take r1=2∗r0 (2 −2∗)r0+ 2∗−δ1> r0. As before, if r1≥N 2we easily conclude. In other case we take r2=2∗r1 (2 −2∗)r1+ 2∗−δ2. By an iterative argument we conclude after a finite number of steps. Indeed, in other case, we have that rnis bounded, where rnis defined recurrently by    r0= 2∗ rn+1 =2∗rn (2 −2∗)rn+ 2∗−δn+1. Non-bounded quasi-linear operator 7 where limn→∞ δn= 0. Moreover, rnis non decreasing and so it converges to r∈(2∗,N 2] that satisfies r=2∗r (2 −2∗)r+ 2∗, that is, 2∗= (2 −2∗)r+ 2∗, which implies that r= 0 and this is a contradiction. Observe that the estimate (3.1) follows, after this finite number of steps, from estimates in items i)-iii), and the Sobolev embedding. Finally, in the case N= 2 we can choose r > q q−2for any q > 2 and argue as before with 2∗replaced by q. In this case we finish by item i). ut Along this section, we assume, instead of (A2), that for each s0∈R+there exists β(s0) such that |A(x, s)| ≤ β(s0),(˜ A2) for (x, s)∈Ω×[0, s0]. We consider the truncated problems (−div(A(x, Tn(u))∇u) = λu, x ∈Ω, u= 0, x ∈∂Ω,(Pλ,n) being Tn(s) the map defined, for each n∈N, by Tn(s) = (s0≤s≤n, n s > n. By Theorem 2.2, there exist Σnunbounded maximal continua of positive solutions such that (λ0,0) ∈Σnfor each n∈N. Now, we can prove Theorem 3.2. Suppose that Asatisfies (A1,4)and (˜ A2). Then, there exists an unbounded continuum Σ⊂ S such that (λ0,0) ∈Σ. Proof. Firstly, we denote by Σn kthe connected component of Σk∩(R×Bn(0)) containing (λ0,0). We claim that Σn k= Σn nfor k≥n. (3.5) Indeed, if k≥nand (λ, u)∈Σn kthen uis solution of (Pλ,n). Thus, Σn kis a closed and connected subset of cl{(λ, u)∈R×E:uis solution non-trivial of (Pλ,n)} containing (λ0,0). So, Σn k⊂Σn, whence we deduce that Σn k⊂Σn n. We can reason similarly and obtain that Σn n⊂Σk∩(R×Bn(0)), and so it follows (3.5). So, we get Σn n= lim kΣn k. Therefore, for each n∈Nwe have a continuum Σn n⊂cl{(λ, u)∈R×E:uis a non-trivial solution of (Pλ)} 8J. Carmona and A. Su´arez containing (λ0,0) and if (λ, u)∈Σn nthen kuk0≤n. Now, we are going to prove that Σn n⊂Σn+1 n+1 for each n∈N.(3.6) Indeed, observe that Σn n= Σn n+1 ⊂Σn+1 ∩(R×Bn(0)) ⊂Σn+1 ∩(R×Bn+1(0)), so, since Σn+1 n+1 is the connected component of Σn+1 ∩(R×Bn+1(0)) containing (λ0,0) and Σn nis a connected of such subset containing it, (3.6) follows. Finally, we show that the set Σ = ∞ [ n=1 Σn n satisfies the theorem. Firstly, observe that since Σnis unbounded, Σ is also unbounded. Indeed, since ProjRΣnis bounded, so there exists a connected subset of Σn∩(R×Bn(0)) containing (λ0,0) and intersecting with R×∂Bn(0) for each n∈N; i.e., for each n∈N there exists (λn, un)∈Σn n, with kunk0=n. On the other hand, since Σn nis connected and (λ0,0) ∈Σn nfor each n∈N, it follows that Σ is connected. Finally, we will prove that Σ is closed. Let (λ, u)∈Σ. Since Σ is connected, there exists a connected and bounded set Σ0⊂Σ containing (λ0,0) and (λ, u). Thus, there exists n∈Nsuch that Σ0⊂cl{(λ, u)∈R×E:kuk0≤n, u is non-trivial solution of (Pλ,n)}. In particular, Σ0⊂Σn∩(R×Bn(0)) whence Σ0⊂Σn nand so, (λ, u)∈Σn n⊂Σ. ut Remark 3.3. 1. We would like to point out that the above result is true even in the case that the limit of A(x, s) does not exist as s→ ∞. 2. In the case Abounded in some subset of Ω, then we can conclude that ProjRΣ is bounded. Indeed, assume that |A(x, s)| ≤ γif x∈B, where Bis a ball such that B⊂Ω, then using the monotony of the principal eigenvalue with respect to the domain, we obtain λ=λ1(A(x, u)) ≤λB 1(A(x, u)) ≤λB 1(γI) = γλB 1(I). 3. In this case we can obtain a similar result to the main one in [2]. Indeed, for each r > 0 there exists λr>0 and ur∈H1 0(Ω) solution of (Pλ) with kuk0=r. In the next result we show that when A(x, s) tends to infinity as s→ ∞ in the sense of (A∞), then the bifurcation at infinity disappears, in some sense λ∞→+∞when A(x, s) tends to infinity. Non-bounded quasi-linear operator 9 Theorem 3.4. Assume that Asatisfies (A4),(˜ A2)and (A∞). Then, there exists a continuum Σ⊂ S such that (λ0,0) ∈Σ. Moreover, the interval (λ0,+∞)⊂ProjRΣand lim λ→+∞ (λ, uλ)∈Σ kuλk0= +∞. Proof. The existence of the continuum unbounded Σ bifurcating from (λ0,0) follows by Theorem 3.2. Since λ=λ1(A(x, u)) ≥λ1(αI) = αλ1(I), there do not exist positive solutions for λsmall. So, it suffices to prove that it is not possible bifurcation from infinity. In order to do that we observe that problem (Pλ) can be written as (−div(B(x, u)g(u)∇u) = λu, x ∈Ω, u= 0, x ∈∂Ω,(Pλ) where gis given by hypothesis (A∞) and B(x, u) := A(x, u) g(u). Moreover, if we perform the change of variable w= ˜g(u) = Zu 0 g(t)dt, problem (Pλ) is equivalent to (−div(C(x, w)∇w) = λf(w), x ∈Ω, w= 0, x ∈∂Ω,(Qλ) where C(x, w) := B(x, ˜g−1(w)) and f(w) := ˜g−1(w). Now we argue by contradiction, and assume that there exists a sequence of solutions (λn, un) of (Pλn) such that λn→λ > 0 and kunk0→ ∞. Then, by (3.1) we have that kunk→∞and taking wn= ˜g(un), it is clear that kwnk0→ ∞. In addition, since (A∞) implies that α2kunk2≤ kwnk2, we also have that kwnk → ∞. For the normalized sequence zn:= wn kwnkwe know the existence of z∈H1 0(Ω), such that zn→zstrongly in L2(Ω), and a.e. in Ω. and so, taking wn/kwnk2as a test function in (Qλn), we obtain that α≤ZΩ C(x, wn)∇zn· ∇zn=λnZΩ f(wn) kwnkzn.(3.7) Now, taking into account that f(s) s→0 as s→ ∞,