Info

(3) We define the mixed boundary operator, B, by

[Suonri, where the operator B := dv + b with u G C1(ri,MJV) an outward pointing nowhere tangent vector-field and b G C1,Q(ri).

In this paper we study the following problems where a is a positive or negative regular function on Ti and 0 < q < 1 < p, r. We first study an elliptic equation with a logistic term on the boundary

Bu = fiu + a(x)ur on Ti, where /jgR will be regarded as bifurcation parameter. We do not know previous works in which (3) was analyzed. We characterize the existence, uniqueness and stability of positive solution in terms of the parameter fi (see Theorem 5.2).

Second, we study of the sublinear-superlinear equation

— = u on ail, on where n is the outward normal vector-field of ii. (The case —uT instead ur has been studied in Ref. [8].) Equation (4) has attracted a lot of attention in the last years with A = 0, see Refs. [6], [10], [18], [21], [22] and [26], among others, where basically the equation and its corresponding parabolic problem were analyzed in the particular case A = 0, and in Refs. [28], [29] where the local bifurcation was studied. We complete this study giving existence, non-existence and stability results in function of A (see Theorem 5.3).

Finally, we study the concave-convex equation

— = a(x)u in oil. on where m G C(fl) is nonnegative and non-trivial. Equation (5) was studied previously in Ref. [14] when Lu = -Au + u, and m = a = 1 by variational methods. When o < 0 we prove that there exists a positive solution of (5) if and only if A > 0. If a > 0 we complete and improve the results of Ref. [14] (see Theorems 5.4 and 5.5).

In order to study these equations we employ mainly sub-supersolution and bifurcation methods. We present in Sec. 2 results related with principal eigenvalues associated to these problems. In Sec. 3 we prove a general result of bifurcation from the trivial solution when the bifurcation parameter appears in both equation and boundary. As consequence, we can use it for equations (3) and (4). For the study of (5) we need a different result of bifurcation, where the parameter is in front of a non-linear term. In Sec. 4 we present results concerning to uniqueness, stability and a-priori bounds of positive solutions for general equations with nonlinear boundary conditions. Finally, in Sec. 5 we apply the results to the cited equations.

2. Some Preliminary Results: Eigenvalues Problems

Along this paper, we use the positive cone

P := (tíSC1^) : u > 0, u 0 in íí U Ti, Bu = 0 on dft}, and we say that u is positive if u e P and that u is strongly positive if u e int(P) := {u € P : u > 0 in ft U Ti, du/dn < 0 on r0}, where n is the outward normal vector-field of ft. On the other hand, the mixed operator B + m, m £ C(ri), means a similar operator to (2) with b + m instead of b in B. Finally, given two functions u, v we write (it, v) > 0 if u, v > 0 and some of the inequalities non-trivial. Consider the eigenvalue problem

H. Amann [2] proved the existence of a unique simple eigenvalue, the principal eigenvalue, whose associated eigenfunction can be chosen strongly positive in ft. We denote this eigenvalue by o\ [L, B], <J\[L, D] and a\ [L,J\i] stand for the principal eigenvalues under Dirichlet and Neumann homogeneous boundary conditions, respectively.

Some properties of o\ [L,B] have been studied in details by S. Cano-Casanova and J. López-Gómez [9] (see also Ref. [4]), we state some of them.

Proposition 2.1.

(1) a\ [L, B] > 0 if and only if there exists a positive supersolution of (£,£?, ft), i.e., a positive function ü such that Lñ > 0 ¿n ft and Bu > 0 on 9ft with some inequality strict.

(2) The map q £ L°°(ft) i—> &i[L + q, B] is increasing and continuous.

(3) The map m € C(rj) i—> o\ [L, B + m] is increasing and continuous.

(4) Suppose Ti 0 and consider a sequence bn € C(ri) such that

We suppose the following condition m € Ca(H), r e C1,Q(ri), E/x > 0 such that (c + (im, b + /zr) > 0. (7)

The following result provides us the existence of principal eigenvalue of (6). The second paragraph gives a characterization of the principal eigenvalue of (6) when m = 0, i.e., an eigenvalue problem at the boundary, the classical Steklov problem. In our acknowledge this result is new, although it nearly follows by the results on Ref. [9] (see Ref. [15] where a particular result is obtained.)

(1) Under condition (7), the eigenvalue problem (6) has a unique principal eigenvalue, ~fi[L, B], it is simple and its associated eigenfunction can be chosen strongly positive in fi.

(2) If m = 0 and r > 0, then, the principal eigenvalue exists for (6), denoted by X\[L,B\, if and only if <J\[L,D] > 0. Moreover, its associated eigenfunction can be chosen strongly positive in il.

Proof. The first paragraph follows with the same kind of arguments used in Theorem 2.2 of H. Amann [3] where To = 0.

It is clear that Ai is a principal eigenvalue of (6) with m = 0 if and only if fi(Ai) = 0 where n(X) := a\[L, B — Ar(z)].

We know by Proposition 2.1 that lrniA_>_oo mM = ci[L, D], /¿(A) is a decreasing and continuous function. So, it suffices to prove that limA-^+oo /¿(A) = — oo. Suppose the contrary, then lim.\->+oo MA) = —I-Take k G M large enough such that k + c(x) > 0 and k > I then, first part lim ai[L,B + bn]=ai[L,D], n—*+oo

(5) Suppose Fx ± 0, then a\[L,B] < a\[L,D], Consider now the eigenvalue problem

of the Theorem can be applied to the eigenvalue problem

Hence, there is a principal eigenvalue Ai that verifies 0 = /¿(Ai) This is a contradiction.

The following result will be very useful along this work.

Lemma 2.1. Assume (7) and (m,r) > 0. Then 71 [L,B] > 0 cti [L, B) > 0

Proof. We know by Theorem 2.1 that 71 [L, B] exists, and it is the unique zero of the application n{cr) = <j\ [L — am, B — ar].

Since ¡j, is a decreasing function, then /¿(0) > 0 implies n(ao) = 0 for (Jo = 71 [L, B] > 0 and the contrary. □

3. Bifurcation Results for Equations with Nonlinear Boundary

Consider the nonlinear equation

Bu = Xr(x)u + g(x,u) on Ti, where / e Ca(Tl x M), g € C1,Q!(ri x R), such that f(x,0) = 0Vx<Efi, s(x,0) = 0 VxeTi, (9)

(m, r) > 0 and satisfy condition (7) and A is a bifurcation parameter.

Remark 3.1. Due to the condition (7) we can assume, adding firn and ¡j,r to both sides of (8), that (c, b) > 0.

Now, we reduce the equation (8) to a suitable equation for compact operators. Define Cpo (£7) = {u € CQ(f2) : U|ro = 0} (analogously it can be defined C^Q(ii)) and the map K1 : Cpo(H) -» Cp;"(f2) by, given /, K\ (/) = u where u is the unique solution of the problem

We can extend this operator to Cr0 (ft)- Thanks to elliptic regularity results, this new operator, denoted again by K\, is compact as operator from Cr0 (ft) to Cr0(ft). We define now K2 : C{Vi) Cp'Q(ft) by, given g, K2(g) = u with u the unique solution of the problem

Again, it can be proved that the operator K2 : Cr0(dQ) —» Cr0(ft) is compact. Denote by 7 : C(ft) —> C(Ti) the trace operator. Following the same kind of arguments that in Ref. [3], Lemma 4.1, and denoting M, R, F and G by the Nemitski operators associated to m(x)u, r(x)u, f and g respectively, we have

Proposition 3.1. u satisfies u = K\ [XM(u) + F(u)} + iG[Ai?(7(ii)) + G(7(u))] if and, only if u is a classical solution of (8).

Since we are only interested in non-negative solutions of (8), we rewrite (8) as a problem with only non-negative solutions. Let u+ = maxju, 0}.

Lemma 3.1. If u is a solution of

Proof. Suppose that the problem (10) possesses solution u such that there exists a connected component fti C ft of the set ft' = {x € ft : u(x) < 0} such that u < 0 in fti. Observe that 9ft 1 n Ti / 0. Indeed, if fti C ft, then

Since c > 0, then by the maximum principle u = 0 in fti. Hence, ^ftiflTi ^ 0. Due to Lu > 0 in fti and c > 0 then, by the maximum principle, the minimum of u must be attained on dfti. As u < 0 in fti and u = 0 in ôfti PI To then, minimum must be attained on dili fl Ti, but in such points we have

= -b(x)u > 0, contradicting Hopf's Lemma (see Lemma 3.4 in Ref. [16]). □

Remark 3.2. Lemma 3.1 is still true if f(x, 0) > 0 and i(x,0) > 0. Consider the maps $a, $a : —> CrQ(ft) defined by

$A(«) = u - K1[XM(u+) + F(u+)} - K2[XR(-y(u+)) + G(7(u+))], ${(u) = « - tK1[XM{u+) + F{u+)} - «¡r2[Afl(7(u+)) + G(7(u+))],i > 0.

Thanks to Proposition 3.1 and Lemma 3.1, u is a classical nonnegative solution of (8) if and only if <&\(u) — 0 in Cr0(ft). Assume that lim = o unif. in H, lim ^^ = 0 unif. on IV (11)

Finally, denote by 71 := 71 [L, B], and £1 its strongly positive eigenfunction associated.

Lemma 3.2. Let A C R be a compact interval such that X < 71 for all X G A. Then, there exists S > 0 such that $a(u) / 0 Vu £ Cr0 (CI) with IMIe(H) = IMI e (0,5), VA G A andMt G [0,1].

Proof. Suppose the contrary, that there exist A n,tn eR and un G Cp0 (CI) such that An -> A, tn —» t, ||u„|| —> 0 and (un) = 0. By Lemma 3.1, un > 0 and dividing by ||un|| we obtain

„ * K {KM(un) + F(un)\ , , /rAnfl(7(Un)) + g(7(«n))

where vn = p^j- Thanks to (11) we have that the terms inside K\ and K2 are uniformly bounded in Cl and on Ti, respectively. Since K\ and K2 are compact operators, then the sequence vn is a relatively compact in C(Cl). Therefore, we can suppose that vn —> v in C(Cl). By (11), we have

Passing to the limit in (12), we conclude that v = t\XKi(M(v)) + XK^R^iv)))}.

Thanks to un > 0, ||wn|| = 1 and by the maximum principle, v is a strongly positive function in CI. Due this fact At = 71 but this is not possible because At < 71 by the choice of the set A. □

We are going to use the following notation: for R > 0, let BR = {« G Cr0(ft) : IMI < R}. Then, deg($a, BR, 0) stands for the degree of $a on

Br with respect to 0, and i($\,uo,0) denotes the index of the solution uq of the equation = 0.

Proof. If X > 0 consider the interval A = [0, A] in the contrary case consider A = [A, 0]. Thanks to the Lemma 3.2, we know that 3(5 > 0 such that Vu G Cro(Ô) with IMI G (0,6) we have 0, Vi G [0,1]. Therefore by homotopy invariance of the degree we obtain t(SA, 0,0) = deg($\ = $A, Bs, 0) = deg($° = B5,0) = 1. □

Lemma 3.3. Let A > 71. Then, there exists 8 > 0 such that Vu G Cr0(H) with ||u|| G (0,5), $A(u) + rfi, Vr > 0.

Proof. Assume that there exist sequences rn > 0, un G Cr0(^) such that ||un|| —> 0 and <&\(un) = rn£i. Thanks to Proposition 3.1 and similar arguments that we have employed in Lemma 3.1, we have that u„ > 0 is a classical solution of the problem

Lun = Xm(x)un + f(x, un) + 7iTnm(x)£i in ÎÎ, un = 0 on To,

Since by Remark 3.1 we can assume that (b, c) > 0, positive constants are supersolutions of (L,B,Q), and so by Proposition 2.1 it follows that <j\\L,B] > 0, and so that by Lemma 2.1, 71 > 0. Thanks to conditions (11), we obtain

Hence, un is a strict positive supersolution of (L — A771(2) + e, B — Xr(x) + e, ÎÎ), and then

SE(X) =<r1[L- Am(x) + e,B- Xr(x) + s] > 0. (13)

On the other hand, we know that 71 is the unique zero of the continuous and decreasing function ¿(A) = a\[L — Xm(x),B — At-(x)]. Since A > 71 then 5(A) < 0. Moreover, by Proposition 2.1, we infer that exists £ > 0 such that 5e(A) < 0, contradicting (13). □

Proof. Let e 6 (0,8) where 8 is given in Lemma 3.3. Since $a is bounded on Be, then by Lemma 3.3, there exists a > 0 such that $\(u) ^ tat;i, Vu e BE, Vt e [0,1]. Hence,

0,0) = deg($A, Be, 0) = deg($A - oft, Be, 0) = 0. □

Let CcMx Cr0 (fl) be the closure of the set of positive solutions of (8). Then,

Theorem 3.1. Assume that (m,r) > 0, (7), (9) and (11). Then 71 is a bifurcation point from the trivial solution, and it is the only one for positive solutions. Moreover, there exists an unbounded continuum Co C C of positive solutions emanating from (71,0).

Proof. The result follows by Corollaries 3.1 and 3.2 and Ref. [5], Proposition 3.5. We only remark that the uniqueness of 71 follows with the same kind of arguments as in the proof of Lemma 3.2. □

Remark 3.3.

(1) Assume that there exist constants ci,c2 £l such that lim S^ = c\ unif. in fi, lim ^ = C2 unif. on IV

Then, we can apply the above result to the problem L\u = Xm(x)u+ fi(x,u) in fi, u = 0 on To and B2u = Xr(x)u + g2(x,u) on Ti, where L\ = L — c\, B2 = B — c2, f\{x,u) = f(x,u) — c\u and g2(x,u) = g{x,u) — c2u, and so f\ and g2 satisfy (11).

(2) The case that to > 0, r = 0 (i.e., the bifurcation parameter only in the equation) can be included in the Theorem 3.1. Indeed, if b > 0 then (7) is verified. If b < 0 or changes sign we can perform a change u = eM^v where ip is the function that appears on Ref. [20], Proposition 3.4, and the original problem is transformed into a similar new problem where the new 6, say b > 0.

(3) It is also possible to cover the case m = 0, r > 0 (i.e., the bifurcation parameter only at the boundary). Indeed, if o\ [L, D] < 0 then it can be proved that bifurcation from the trivial solution does not occur. Now, assume cr\\L,D] > 0. By Proposition 2.1 there exists \ir with ¡i enough big such that a\ [L, B + fir] > 0. Then, there exists a unique solution h > 0 in ft of the problem

Now, we perform the change u = hv, which transforms the original problem into a new problem where the new c, c > 0.

(4) A similar result can be obtained for bifurcation from infinity.

(5) We have not found in the literature a general result similar to Theorem 3.1. In Ref. [27] the author studied bifurcation form infinity for a similar equation with nonlinearities asymptotically linear. In Ref. [7] the bifurcation method is studied but with nonlinearities only at the boundary. In both papers, Lu = — Au + u and b(x) > 0.

In the rest of the section we consider the problem Lu = Xf(x, u) in ft, u = 0 on To, (14)

Bu = g(x,u) onTi, where / G Ca(ft x R), g G C1,a(ri x R). Throughout the rest of the section we assume the following conditions (c, b) > 0, (9) and fix s) —

We have that u is a classical nonnegative solution of (14) if and only if = 0 in Cr0(ft), where : CTo(ft) Cr0(ft) is defined as

Lemma 3.4. If X < 0 then there exists S > 0 such that Vu G Cr0(ft) with IMIc(n) = IMI € (M) we have t^ 0, Vi G [0,1].

Proof. Suppose the contrary, then there exist S6QU6I1C6S tfi G M, VLjx G Cr„(ft) such that tn t, ||wn|| 0 with ^"(un) = 0. Dividing by ||u„||, we obtain w K fF(Un)\,, K (G{ 7K)) vn = XtnKi -J—r + tnK2 -

where vn — jj^- Since A < 0, the fact that j|un|| > ||«n||ri and using (15) and (16) we get that vn —> 0 in C(fl), a contradiction because ||un|| = 1. □

Lemma 3.5. If A > 0 then there exists 5 > 0 such that Vu € Cr0(fi) ivith ||u|| S (0,5) and Vr > 0 we have SS!\(u) rcpi, where ipi is a positive eigenfunction associated to <j\[L,B}.

Proof. Let us assume that for some sequence un € Cr0(^) with ||itnj| —> 0 and numbers rn > 0, ty\(un) = Tnipi. It is clear, by the maximum principle, that un> 0 and it is a classical solution of the problem

Lun = Af(x, un) + <ji [L, B]Tnipi in fi, un = 0 on r0,

Take e > 0, and M > cr\ [L, B + e}. Since a\ [L, B] > 0 and due to un —> 0 in C(fi) we have, using (15) and (16), that there exists no such that Vn > no

Lun = Af(x,un) + o"i [L,B]Tn(pi > Mun in fi, un= 0 on r0, (17)

Therefore, un is a positive strict supersolution of (L — M,B + e,il), then <Ti [L — M, B + e] > 0, and so M < a\ [L, B + e], a contradiction. □

Theorem 3.2. Under conditions (c,b) > 0, (9), (15) and (16), A = 0 is a bifurcation point from, trivial solution and it is the only one for positive solutions. Moreover, there exists an unbounded continuum Co of positive solutions emanating from (0,0).

Proof. It is possible, thanks to Lemmas 3.4 and 3.5, reasoning as Theorem 3.1 to prove that there exists an unbounded continuum Co- We only need to prove uniqueness of bifurcation point. By Lemma 3.4 we can prove that bifurcation from the trivial solution does not occur for points of the form (Ao,0), Ao < 0. Let us assume that there exists a sequence (A„,uaJ € M x Cr0(ft) verifying (A„,maJ (Ao,0) in K x Cr0(Ti) with A0 > 0. Then

Lu\n = Kf{x,u\n) > XnMu\n, BuXn = g(x,uXn) > -eu\n.

At this point we only need to follow the reasoning of Lemma 3.5 to obtain a contradiction. □

Remark 3.4.

(1) A similar result is obtained under the condition f(x,s) = m(x)f(s), with m € Ca(U), m{x) > 0, and non-trivial, / £ Ca(R) and lims_0+ = +0°) instead of (15).

(2) Similar results still are true for equations of the form

Bu = Ai(x,u) on Tj, where h and i play the same role as g and /, respectively. 4. Stability, Uniqueness and a-priori Bounds

In this section we present (without proofs) some results concerning to the stability, uniqueness and a-priori bounds of the solutions of the problem

Bu = g(x,u) on Ti, where / and g are regular functions.

Let u a non-negative solution of (18). For the study of the stability of u, we linearize (18) around u and consider the eigenvalue problem:

Thanks to Theorem 2.1, we know that the eigenvalue problem has a unique principal eigenvalue 71 (u) =71 [L — fu(x,u),B — gu{x,u)}.

Theorem 4.1. Let u a nonnegative solution of (18).

(1) If 71 (u) > 0, then u is linearly asymptotically stable (I. a. s.).

In general, determinating the sign of 71 (u) is not easy. Due this fact, we give the following characterization using the following related problem:

Using Lemma 2.1 and Proposition 2.1, we get

Theorem 4.2. -fi(u) > 0 (resp., 71 (u) < 0) if and only if the problem (20) admits a positive strict supersolution (resp., subsolution).

With respect to the uniqueness, we have:

Theorem 4.3. Assume f € CJ(Q x [0,+oo)) and g £ C1^! x [0,+oo)) and that f(x,t) g(x,t) . . . , „

11—► ———-, 11—> —1—- are nonmcreasing functions in t > v, and at least one of them is a decreasing function. Then, problem (18) admits at most one positive solution.

We assume / e C(fi x [0, +00)), g 6 C1,a(Ti x [0, +00)) and there exists P e (X> 9 e (1- A^) fchat verifies lim (21)

uniformly in CI with h £ C(fl) a positive function and hm (22)

uniformly on Ti with i £ C1,Q(f2) a positive function.

Theorem 4.4. Let u £ C2(f2)nC1(f2) a nonnegative solution of the problem (18). Suppose that one of the following conditions is satisfied:

(2) The maximum ofu is attained on 80,, (22), (21) is satisfied for any function h and p < 2q — 1.

Then, there exists C(p,q,Cl) a positive constant depending on p,q and CI such that for all x £ CI

Remark 4.1.

(1) The condition p ^ 2q — 1 appears in other papers, see Ref. [12] and it is necessary to apply a Gidas-Spruck argument.

(2) The proofs of Theorems 4.1, 4.3 and 4.4 can be found in Ref. [23]. Theorem 4.1 complements and improves Theorem 5.6.2 of Ref. [25] and Theorem 3.1 of Ref. [28]. Theorem 4.3 is proved in Ref. [24] where other uniqueness results can be found. Finally, a-priori results have been shown in Ref. [30] with nonlinearities only at the boundary (see also Ref. [12] for systems) and Ref. [14] for particular nonlinearities in the equation and on the boundary.

5. Some Applications

In this section we are going to study some equations with nonlinear boundary. The first equation is where r > 1 and a G C1>a(ri). Theorem 5.1.

(1) Assume that a < 0. Then, (23) has a positive solution if and only if ai [L, B] < A < <j\[L,D]. Moreover, if the solution exists, it is unique and I. a. s.

(2) Assume that a > 0. If u is a positive solution of (23) then, A < ai [L, B], 7/1 < r < NN__2 then there exists at least a positive solution of (23) for all A < o\ [L, B\. Moreover, all positive solutions of (23) are unstable.

Proof. (1) Assume a < 0. Suppose u is a nonnegative solution of (23), then, by the strong maximum principle, u is strongly positive, and so, A = &i[L, B — a(x)ur~1]. Applying Proposition 2.1, we obtain

Now, we construct a sub-supersolution for the problem (23). Fix Ao G (<ji[L, B], <ji[L, D]). By Proposition 2.1 there exist fci and k2 such that <7i[L, B + k2] > Ao > ci[L, B + fci]. Now, the pair u = eipi and u = Mip2 with e small and M large enough, and </?* a strongly positive eigenfunction associated to o\[L, B + fci], is a sub-supersolution of (23).

Uniqueness follows by Theorem 4.3. For the stability we use Theorem 4.2. Choose u = u with u solution of (23) then, (L — A)u = 0 in f2,

and u = 0 on To, (B — a(x)rur~1)u > Bu - a(x)ur = 0 on Ti, i.e., u is a positive strict supersolution of the linearized problem around u, (L~ \,B — ra(:r)ur_1, fl).

(2) Assume now that a > 0. If u is a nonnegative solution of (23) then A = <J\[L, B — a(x)ur~1] < a\[L,B\. By Theorem 3.1, there exists a unbounded continuum Co emanating from (<Ji[L, JB],0). Its direction is subcritical by the limitation of A and, under condition 1 < r < jfz^, Theorem 4.4 proves us that the projection of Co on A-axis V\(Co) = (—oo, ai[L, B]). Positive solutions of (23) are unstable because if u is a positive solution then,

<j\[L — \,B — ra(x)ur~l] < ax\L - A, B - a(x)ur_1] =0. □

Remark 5.1. In the case a < 0 and thanks to the subsolution that we have built, it could be proved that for K. a compact subset of SI \ To, lim minuA = +oo. A—<7i [L,D]- K

5.1. Elliptic equation with a logistic term, at the boundary

From the results obtained of the equation (23), we can deduce results for the equation (3).

Theorem 5.2.

(1) If a\[L,D] < 0, then (3) does not have positive solutions.

(a) If a < 0, then (3) has positive solutions if and only if p>X1[L,B}.

Moreover, if the positive solution exists, is unique and I. a. s.

(b) If a > 0 and there exists a positive solution of (3), then p, < Ai[L, B], Moreover, under condition 1 < r < 7735, there exists at least one positive solution if p. < Ai [L, B]. Furthermore, positive solutions of (3) are unstable.

Proof. We only need to put A = 0 and b(x) = b(x) — p in Theorem 5.1. If a < 0, (3) possesses a positive solution if and only if a\\L,B — < 0 < <7i [L,D], By the definition of Ai [L, B\, the result follows. Analogously the case a > 0. □

5.2. A sublinear-superlinear equation

Now, we study the equation (4).

Theorem 5.3. (0,0) is the unique point of bifurcation from the trivial solution, and there exists an unbounded continuum Co of positive solutions emanating from (0,0). Moreover,

(1) Respect bifurcation direction:

(a) If p < r (resp., p > r) then the bifurcation direction is supercritical (resp., subcritical).

(b) If p = r then the bifurcation direction is supercritical (resp., subcritical) for |fi| > \dCl\ (resp., |fi| < |<9f2|j.

(2) If p — r and |fi| < (4) does not have positive solutions for X >0.

(3) Ifp < 2r — l, (4) does not have positive solutions for X large enough.

(4) If p <r and X < 0 every positive solution is unstable.

(5) If p < 2r — 1 and r < jf^ then every positive solution is bounded in L°° norm.

(6) If p > 2r — 1, there exists solution for all X > 0.

Proof. Due to Theorem 3.1, we have a unbounded continuum Co of positive (4) emanating from (crif-A,^ = 0,0). We study the bifurcation direction. Consider Xn —> o"i[—A, A/] and its solutions associated un. Then, multiplying the equation by <pi = c > 0, the eigenfunction associated to the eigenvalue a\{—A,TV], we obtain

(ci[—A,A/] — Xn) / un(pidx= / u^tpxda - / u^dx. (24) Jo. Jen Jn

Assume for example that p < r, multiply (24) by and taking into account that i^ji" —> fi in C(Q) (see the proof of Lemma 3.2), it follows

5g(<Ji[-A,A/] - Xn) = Sg(-Ja tf+'dx^ , hence, <ti[-A,A/] < A„. All results related to local bifurcation can be proved by the same way.

Let u a positive solution of (4) with p — r. Then, if we multiply the equation (4) by l/ur, and integrating by parts, we get

-r f «~r-1|V«|2 — |ail| + |il| = A f u!-r. Jti Jn

Then, paragraph (2) follows.

Assume that the problem (4) has a positive solution for every A > 0. Consider the parabolic problem wt-Aw = -wp in ÎÎ x (0, T), on dCl x (0,T), (25)

We know by Ref. [6], Theorem 2.3, that if p < 2r — 1 then all positive solutions of (25) blow-up in finite time for wo with large L°° norm. Take u\ a solution of (4), if we prove that u\ is supersolution of (25) for large A, then u\(x) > w(x,t) for all t e (0,T) which is a contradiction. In order to prove this, we only need that u\ > wo- It is clear that for A > 0 u\ is supersolution of the problem

As solutions of (26) are, for A > 0, A1^-1) then ux>X1^p~1\ (27)

Now, there exists A > 0 large enough such that ||i«o||oo < A1/^-^ < u\, this concludes paragraph (3).

Let u a positive solution, we are going to prove that under condition p < r this solution is unstable. For that, thanks to Theorem 4.2, we have to show that

<7i[—A — A + pup_1,A/' — rur~1] < 0. (28)

For this fact we choose as subsolution, u = uq, where q will be fixed later. We have that

(—A - A + pup~1)u = q{ 1 - q)uq~2\Vu\2 + Xu"(q - 1) + up+9_1(p - q).

Choosing q such that p < q < r, it follows (28), so that paragraph (4).

By (27), u attains its maximum on dCl. So, paragraph (5) follows from Theorem 4.4.

For the last paragraph we only need to find a sub-supersolution of (4) for every A > 0. We choose as subsolutiona u = ee~5<t>1

where e,5 > 0 can be chosen later and <j>i is the positive eigenfunction associated to ox = <ti[-A,D] with ||^i||oo = 1/2. After some calculations we obtain

Vu = -eSe'^Wcj) i, A u = —sSe-5^1 (—<5| V0i )2 + A<£i). Thanks to Hopf's Lemma, it follows that max =CU Ci >0. (29)

Since, = 0 on 9ft, we only need to verify on the boundary that

In the equation we must check that

-52\V4>! I2 - 6ai4>! + eP-ie-^iiP-D < (31)

Observe that if A > 0, we only need to choose e and S positive and small enough for that (30) and (31) hold. So that, we are going to study the case A = 0. From (30) we choose 6 = so that (31) transforms into

Now, due to </>i = 0 on dil, but on the boundary d<f>i/dn < 0, there exist some constants C2, C3 >0 such that

|V0i| > C2 en fix := {a; e f2: 0i(x) < C3}. (33)

In this way, in fii for that the condition (32) must be fulfilled we need that p - 1 > 2(r - 1) thus p + 1 > 2r.

On the other hand, in ft \ Cli we need that p — 1 > r — 1. The supersolution follows by Ref. [22], it was used also in Ref. [18], in both cases for the particular case A = 0. We choose u:= Ma[2-( 1-<M

aThis subsolution appears on Ref. [18] for the particular case A = 0.

where M > 0 will be chosen large and A, B and C will be fixed later. Observe that

Vu = BCMa{2 - (1 - 0i)B]C-1(l - <MB-1V</>i, Au = BCMa[2 - (1 - 4>i)B]c~2{l - <t>i)B~2-

{(c - i)(i - <j>i)B\y<t>i\2 + [2 - (1 - <MB]AM 1 - h)-

[2 — (1 — <I>i)b](B — l)|V0i|2} . Taking into account (29), on the boundary it must be verified (observe that (¡>i=0 and so that [2 - (1 - <t>i)B] = 1):

For the equation, we need that

+BC(B - 1)[2 - (1 - ti)3]-1^ ~ 0i)B"2|V0i|2 ^ J +Ma(p-V[2 - (1 - ^)b]cCp-I).

Take A > 0, C = -1/Ci and B = Mb, with b to be fixed later. With this choice, for condition (34) it will be needed b + A(l-r)>0. (=> b > 0). (36)

Now, we study the term (35). First term in the right hand tends to —oo or zero (the term (1 — </>i)2(B_1) can tend to zero). Second term is similar, we remind that — A<pi — a\[—A,£>]0i. Third term tends to — oo with order M26 and the last one to +oo with order MA(p~l\ so we have to impose A(p— 1) > 2b. This last inequality and (36) are possible because p+1 > 2d

Remark 5.2. Except paragraphs (2) and (3), Theorem 5.3 is true for more general operators L and B.

5.3. The concave-convex equation

Finally, we consider (5). Assume the following conditions c > c0 > 0 in H, with Co e R. (37)

We distinguish two different cases: a negative and positive.

Theorem 5.4. Assume that a < 0. The problem (5) has a positive solution v\ if and only if X > 0. For X > 0, it is the unique positive solution, it is I. a. s. and lim |Moo=0. (38)

Proof. Thanks to the maximum principle (5) does not posses nonnegative solutions for A < 0. By Theorem 3.2, there exists a continuum Co emanating from (0,0) supercritically. On the other hand, v = Mipi is, for M large enough, supersolution of (5) where ipi is the positive eigenfunction associated to ai[L,N]. This is true because o\[L,N] > 0, which follows by (37). Since M can be chosen large enough that Mtp\ > v\ for A > 0 small, where v\ denotes the solution of the problem founded by bifurcation. Then, we have a family of supersolutions such that a solution belonging to the continuum is smaller than the supersolution. Now, adapting the proof of main theorem of Ref. [13], we have that there exists at least a positive solution for all A > 0. Finally, for A > 0

so the stability follows. Uniqueness follows by Theorem 4.3. □

(1) From (0,0) emanates supercritically an unbounded continuum Co of positive solutions. Moreover, it is the unique bifurcation point from the trivial solution.

(2) There exists A* > 0 such that for A > A* problem (5) does not have positive solutions.

(3) There exists S > 0 such that there exists at most a positive solution u\ of (5) such that ||ua||oo <

(4) Moreover, if L is self-adjoint and r < jjz^ then:

(b) There exists at least two positive solutions in (0, A).

(c) There exists a unique positive solution in (0, A) I. a. s.

Proof. Since the proof follows the same lines that Theorem 6.9 in Ref. [11], we only sketch it.

The existence of the continuum Co follows by Theorem 3.2. We prove now that the bifurcation direction is supercritical. Assume that there exist An < 0 and uXn € C(ft), uXn > 0 such that (A„,uaJ -> (0,0) in R x C(H). Then, for n > no we get

LuXn <0 in SI, = a(x)u\n < eaMuXn on dfi, where a« — maxan a,. On the other hand, since o\ [L, N] >0 then, for e > 0 small, cti [L, N — com] > 0, and applying the strong maximum principle we obtain that u\n = 0, a contradiction.

Now, we are going to prove paragraph (2). Suppose that there exists positive solution u\ of (5) for all A, in particular for A > 1. Let v\ be the unique positive solution of

Since u\/X is supersolution of (39) for A > 1, then u\ > Ai>i for A > 1. On the other hand, since u\ is a positive solution of (5), we get

0 = ai[L - Am(x)u®-1, A/" — a(x)u^-1] < ax\L,M - A1"-1^-1^-1], where ao = mingn a(x). This is an absurdum. Indeed, since r > 1, we have lim a\ [L,N — Ar-1ag-1?;[_1] = -oo.

Now, define

We have proved that 0 < A < +oo. Moreover, it is not difficult to prove the existence of a minimal solution u\ for all A G (0, A).

The following result shows properties of the principal eigenvalue, denoted by 71(A), of the linearized around the minimal solution u\, i.e., dt . > mf flo (40)

L£ — Aqm(x)uqx = 71 (A)^ in Q, ae dn or equivalently, the unique zero of the map

(3(a) = a\[L — A qm(x)u^'1 — a, M — ra(x)u^_1 — a]. Lemma 5.1.

(1) If u\ is the minimal solution of (5), then 71(A) > 0.

(2) If 7i(Ao) > 0, for some Ao > 0, then the set of positive solutions of (5) can be parametrized in a neighborhood of (Ao, wo) by a regular and increasing function on A.

(3) If 71 (Ao) = 0, for some Ao > 0, then the set of positive solutions of (5) can be parametrized by a new parameter s G (—e,e), such that (A(s),u(s)) is a positive solution of (5) and

A(s) = Ao + s2A2 + o(s3), u(s) = uAo+s$o + s2,i'o + o(s3), (41)

where $o iS the positive eigenfunction associated to 71 (Ao) and fn $0^0 =0. Moreover,

Finally, if L is self-adjoint,

Remark 5.3. Except the first paragraph, the result is true for any positive solution u not necessarily being the minimal.

Proof. (1) Assume that 71 < 0 and denote by (j>\ the positive eigenfunction associated to 71. It is not difficult to show (see Ref. [11]) that u\ — ce(f> 1 is supersolution of (5), for a > 0 small. Since u\ > v\, where v\ is the unique positive solution obtained in Case 1 (a(x) < 0), and v\ is subsolution of (5), it follows the existence of a solution u < u\ of (5), an absurdum because u\ is the minimal solution.

(2)-(3) Except (43), these two paragraphs follow by el Propositions 20.6, 20.7 and 20.8 of Ref. [1]. Using (41) and the definition of we get

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