A CONVEX-CONCAVE ELLIPTIC PROBLEM WITH A PARAMETER ON THE BOUNDARY CONDITION

In this paper we study existence and multiplicity of nonnegative solutions to  ∆u = u p + u in Ω, ∂u ∂ν = λu on ∂Ω. Here Ω is a smooth bounded domain of RN , ν stands for the outward unit normal and p, q are in the convex-concave case, that is 0 < q < 1 < p. We prove that there exists Λ∗ > 0 such that there are no nonnegative solutions for λ < Λ∗, and there is a maximal nonnegative solution for λ ≥ Λ∗. If λ is large enough, then there exist at least two nonnegative solutions. We also study the asymptotic behavior of solutions when λ → ∞ and the occurrence of dead cores. In the particular case where Ω is the unit ball of RN we show exact multiplicity of radial nonnegative solutions when λ is large enough, and also the existence of nonradial nonnegative solutions.


(Communicated by Juan Luis Vázquez )
Abstract. In this paper we study existence and multiplicity of nonnegative solutions to    ∆u = u p + u q in Ω, ∂u ∂ν = λu on ∂Ω.
Here Ω is a smooth bounded domain of R N , ν stands for the outward unit normal and p, q are in the convex-concave case, that is 0 < q < 1 < p. We prove that there exists Λ * > 0 such that there are no nonnegative solutions for λ < Λ * , and there is a maximal nonnegative solution for λ ≥ Λ * . If λ is large enough, then there exist at least two nonnegative solutions. We also study the asymptotic behavior of solutions when λ → ∞ and the occurrence of dead cores. In the particular case where Ω is the unit ball of R N we show exact multiplicity of radial nonnegative solutions when λ is large enough, and also the existence of nonradial nonnegative solutions.
1. Introduction. In the well-known paper [1], the following elliptic problem −∆u = u p + λu q in Ω, u = 0 on ∂Ω, (1.1) was considered. Here λ > 0 is a parameter and the main point is that the nonlinearity in the equation is a combination of a convex term and a concave term, that is, the exponents verify 0 < q < 1 < p. (1.2) Among other results, it was shown in [1] that there exists a value Λ > 0 such that problem (1.1) does not have positive solutions when λ > Λ, while it has at least a positive solution for λ = Λ and at least two positive solutions if 0 < λ < Λ (provided in addition that p is subcritical). These results have been subsequently generalized to deal with more general operators (cf. [11], [12] for the p-Laplacian or [8] for fully nonlinear operators) or boundary conditions (see [9] for mixed-type boundary conditions and [13] for a nonlinear boundary condition). The purpose of this paper is to consider a related problem where the presence of a convex and a concave term allows to have multiplicity of nonnegative nontrivial solutions. The problem we deal with is the following: where Ω is a smooth C 2,α domain or R N , ν stands for the outward pointing unit normal and p and q verify (1.2).
Observe that an important feature in (1.3) is the presence of the parameter λ in the boundary condition. Problems with bifurcation parameters in the boundary conditions of the form (1.3) appear in a natural way when one considers the Sobolev trace embedding H 1 (Ω) → L 2 (∂Ω) (see for instance [10] and the survey [27]), but in spite of this they are not very frequent in the literature. We quote the works [15], [16], [18] by the authors, which deal with different elliptic problems with the same boundary condition and [4], [5], [6], where the boundary conditions are more general.
We come next to the statement of our results. Let us begin with the questions of existence and nonexistence of nonnegative nontrivial solutions. (c) There exists Λ * * ≥ Λ * such that (1.3) admits a second nontrivial nonnegative solution u = v λ for λ > Λ * * . Moreover, v λ has nonnegative energy for large λ, i.e. E(v λ ) ≥ 0 for λ ≥ λ 0 .
In the literature on convex-concave problems, one actually has Λ * = Λ * * . We remark that in those cases the second solution can be obtained directly by means of the mountain pass theorem because the first solution can actually be obtained as a local minimum of the functional (1.4). To prove that this actually happens an important tool is the strong maximum principle: if u, u are respectively sub and supersolution with u ≤ u in Ω, then u < u in Ω unless u ≡ u and both are solutions. In our present situation, this property does not hold, due to the possible appearance of dead cores in the solutions (see below). Thus in general it is difficult to prove that Λ * = Λ * * . This can be shown in the particular case where the maximal solution u = u Λ * to (1.3) corresponding to λ = Λ * is strictly positive on ∂Ω, (this occurs, for instance, when Ω is a ball, cf. Theorem 1.4 below and Section 5).
Our next concern is the asymptotic behavior of solutions to (1.3) as λ → +∞. We will show that there are essentially two behaviors: the maximal solution increases and converges to a boundary blow-up solution, while all bounded solutions tend to zero at a precise rate. Theorem 1.2. Under the assumptions of Theorem 1.1, the following properties on the asymptotic behavior of (1.3) as λ → ∞ hold true.
(a) The maximal solution is a family of solutions to (1.3) such that sup Ω v λ ≤ C for λ ≥ λ 0 then there exist positive constants C 1 , C 2 , such that Remarks 1. a) Theorem 1.2 elucidates the behavior for large λ of two specific kinds of solutions. Namely, those which became uniformly unbounded on ∂Ω as λ → ∞ (the family of maximal solutions u λ ) and those solutions to (1.3) that remain uniformly bounded on ∂Ω for large λ (notice that solutions to (1.3) are subharmonic). Indeed, in the case where Ω is a ball and solutions are radially symmetric the full asymptotic behavior of (1.3) is the one described in Theorem 1.2. However, in the general case, asymptotic responses that are a combination of the ones in cases a) and b) are possible (see Section 6). b) An optimum version of estimate (1.6) on the asymptotic amplitude of nonnegative solutions that stay bounded as λ → ∞ can be obtained in the case of radially symmetric solutions (Theorem 1.4 (b) below).
As we have already mentioned, one of the difficulties that appear when dealing with nonnegative solutions u to problem (1.3) is the possible appearance of dead cores, that is, the set O = {x ∈ Ω : u(x) = 0} could be nonempty. We next state some conditions which ensure that nonnegative solutions do or do not have dead cores. Theorem 1.3. Assume that the hypotheses of Theorem 1.1 hold. Then, (a) If {v λ } λ≥λ0 is a family of nonnegative solutions with nonnegative energy, that is E(v λ ) ≥ 0, then v λ exhibits a dead core for large enough λ. Moreover, v λ (x) = 0 for all x ∈ Ω satisfying with d(x) = dist (x, ∂Ω) and C > 0 a constant that does not depend on λ. In particular, the family {v λ } obtained in Theorem 1.1 (c) exhibits a dead core for large λ.
has a dead core for large enough λ. In this case, v λ (x) = 0 when If Ω lies between parallel hyperplanes π 1 , π 2 with dist(π 1 , π 2 ) < 2R 0 , R 0 as in (c), then the maximal solution u λ is strictly positive when λ is large enough. Finally, we will concentrate in the particular case where Ω is a ball in R N . In this case, we have more precise information, particularly when dealing with radial solutions. We will show exact multiplicity of radial nonnegative solutions to (1.3) when λ is large enough, and also the existence of nonradial nonnegative solutions, in the spirit of [16]. Theorem 1.4. Under the same conditions as in Theorem 1.1, assume in addition that Ω = B R , the open ball in R N with radius R and center x = 0. Then, (a) There exists Λ * > 0 such that problem (1.3) has no nonnegative nontrivial solutions for λ < Λ * , while it has at least a nonnegative nontrivial radial solution when λ = Λ * and at least two nonnegative nontrivial radial solutions if λ > Λ * , one of them being the maximal solution u λ . (b) There exists λ 0 > 0 such that problem (1.3) has exactly two nonnegative nontrivial radial solutions u λ and z λ when λ > λ 0 , whose asymptotic behavior is given by parts (a) and (b) in Theorem 1.2, respectively. Moreover, as λ → ∞ where ρ(λ) stands for the distance from the dead core {z λ = 0} to the boundary |x| = R of B R . (c) There exists λ 0 > 0 such that problem (1.3) admits a nonradial nonnegative solution v λ when λ > λ 0 .
The rest of the paper is organized as follows: in Section 2 we prove Theorem 1.1. In Section 3 we study the asymptotic behavior of solutions when λ → ∞, while dead core formation is analyzed in Section 4. Section 5 is devoted to the problem in a ball of R N . Finally, Section 6 collects some results on multiplicity of solutions.
2. Existence and nonexistence of solutions. In this section we prove Theorem 1.1, which will be split into a series of lemmas. We begin by part (a), that is, nonnegative solutions do not exist when λ is small. We are always assuming that Ω is a smooth bounded domain and that p, q verify (1.2). Proof. Assume that u ∈ H 1 (Ω) is a nonnegative nontrivial weak solution, then we have Note that this implies that u ≡ 0 on ∂Ω. Using that p and q verify (1.2), we observe that s p+1 + s q+1 ≥ max{s p+1 , s q+1 } ≥ s 2 for s ≥ 0 and hence we get Now we use the continuity of the Sobolev trace embedding H 1 (Ω) → L 2 (∂Ω) to obtain the existence of a positive constant Λ such that Therefore we conclude that λ ≥ Λ since u ≡ 0 on ∂Ω. Now we recall that, since p > 1, Theorem 1 in [15] provides with a unique solution to the problem ∆u = u p in Ω, ∂u ∂ν = λu on ∂Ω, (2.1) for every λ > 0, which will be denoted by U λ (it is also worthy of mention that U λ > 0 in Ω, and is increasing and continuous in λ On the other hand, the unique positive solution U λ to (2.1) is a supersolution to (1.3). Since U λ stays bounded away from zero while V λ → 0 uniformly in Ω as λ → ∞, we also have V λ ≤ U λ when λ is large enough, so that a nonnegative nontrivial solution to (1.3) exists when λ is large enough. Therefore, we may define Λ * as the infimum of those λ 0 for which (1.3) has a nonnegative nontrivial solution for every λ > λ 0 . Thanks to Lemma 2.1 we have Λ * > 0. Let us next show that solutions do exist for every λ > Λ * . To this aim we momentarily change notation to make explicit the dependence of (1.3) on λ and denote it as (1.3) λ .
Take λ > Λ * . Thanks to the definition of Λ * , there exists a value µ with Λ * < µ < λ such that (1.3) µ admits a nontrivial nonnegative solution, which will be denoted by u µ . Observe that u µ is a subsolution to (1.3) λ since µ < λ, while, according to Lemma 2.1, u µ ≤ U µ ≤ U λ . Observing that U λ is a supersolution to (1.3) λ , we obtain at least a nonnegative nontrivial solution to this problem, as we wanted to show.
We now observe that by Lemma 2.1, U λ is a supersolution to (1.3) λ which controls every possible nonnegative nontrivial solution. Thus it is standard to obtain that (1.3) λ admits a maximal solution for every λ > Λ * , and the maximal solution is increasing in λ. To conclude the proof, we only need to show that there exists a nontrivial nonnegative solution when λ = Λ * as well. To this end we just have to take an arbitrary sequence λ n ↓ Λ * and consider the function where u λ is the maximal solution to (1.3) λ constructed above. Observe that by the monotonicity of u λ , this limit exists pointwise. Moreover, a standard compactness argument shows that the limit also holds in H 1 (Ω). Thus u will be a nonnegative solution to (1.3) Λ * , and we only have to rule out the possibility u = 0. Arguing by contradiction, assume that u = 0, and define the functions v n = u λn / u λn L 2 (∂Ω) , which verify in Ω, ∂v n ∂ν = λ n v n on ∂Ω.
Then, we have This implies that v n is bounded in H 1 (Ω) and therefore we may assume that v n → v weakly in H 1 (Ω) and strongly in L 2 (Ω), L q+1 (Ω) and L 2 (∂Ω). In particular, v L 2 (∂Ω) = 1. Moreover, and we obtain that v = 0, which is a contradiction since v L 2 (∂Ω) = 1. This concludes the proof.
To complete the proof of Theorem 1.1 we have to prove part (c).
The proof of Lemma 2.4 is based in the mountain pass theorem (cf. [3], [22], [25]). But observe first that the natural functional associated to solutions of (1.3) is given by and in order to obtain a well-defined and differentiable functional in H 1 (Ω) we should assume that p is subcritical. To get rid of this supplementary undesirable hypothesis, we take advantage of the fact that every solution verifies u ≤ U λ and modify the functional accordingly. Introduce the truncated functions and, for technical reasons, also and consider the truncated functional Notice that the subquadratic term |u| q+1 has not been truncated, since it is not necessary. It is standard that J is a C 1 functional in H 1 (Ω), whose critical points are weak solutions to the problem As a first step, let us check that weak solutions to (2.3) verify 0 ≤ u ≤ U λ and therefore are also weak solutions to (1.3).
Lemma 2.5. Let u ∈ H 1 (Ω) be a weak solution to (2.3). Then In particular, u is a weak solution to (1.3).
Proof. Take u − = min{u, 0} as a test function in the weak formulation of (2 and then we conclude that and it follows that u ≤ U λ . This concludes the proof. The next step is to ensure that J has the desirable compactness properties. We will use in H 1 (Ω) the norm which is equivalent to the usual one. Then we have: Lemma 2.6. The functional J verifies the Palais-Smale condition.
Proof. Let u n ∈ H 1 (Ω) be a sequence such that J(u n ) ≤ C and J (u n ) → 0. First, let us see that u n is bounded in H 1 (Ω). Assume, by contradiction, that so that u n L 2 (∂Ω) → ∞. Define v n = u n / u n L 2 (∂Ω) . Then v n H 1 (Ω) is bounded and we can extract a subsequence, denoted again by v n , such that v n v weakly in H 1 (Ω) and strongly in L 2 (∂Ω) and in L q+1 (Ω). Observe that This implies v = 0, which contradicts v L 2 (∂Ω) = 1. This contradiction proves that u n is bounded in H 1 (Ω), thus we can extract a subsequence, still denoted by u n , such that u n u weakly in H 1 (Ω), strongly in L 2 (∂Ω) and L q+1 (Ω) and a.e. in Ω. Since f and g are bounded, we obtain by dominated convergence Hence, we conclude that u n → u strongly in H 1 (Ω), so that J verifies the Palais-Smale condition.
Finally, we need to check that J verifies the geometric conditions of the mountain pass lemma. Let us prove first that u = 0 is a strict local minimum of J.
Lemma 2.7. The functional J has a strict local minimum at u = 0.
Proof. We have J(0) = 0. Let us assume that there exists a sequence u n → 0 with J(u n ) ≤ 0. Then, taking v n = u n / u n L 2 (∂Ω) (note that u n ≡ 0 on ∂Ω since u n ≡ 0 in Ω and J(u n ) ≤ 0), we have for some positive constant C. Hence we can extract a subsequence, still denoted by v n , such that v n v weakly in H 1 (Ω), strongly in L 2 (∂Ω) and L q+1 (Ω) and a.e. in Ω. In particular, we have that v n L 2 (∂Ω) = 1, and then v = 0. On the other hand, which implies that Ω |v n | q+1 → 0, a contradiction.
We can finally proceed with the proof of Lemma 2.4. This will also conclude the proof of Theorem 1.1.
Proof of Lemma 2.4. Since J verifies the Palais-Smale condition we can find a second solution using the mountain pass theorem. In fact, using Lemma 2.7, we only need to check the existence of u ∈ H 1 (Ω) such that J(u) < 0. This is easily achieved by simply taking an arbitrary fixed function u ∈ H 1 (Ω) which does not vanish identically on ∂Ω and considering a large enough λ. Thus the mountain pass theorem implies the existence of a critical point u of J such that J(u) > 0. It follows that u is a nontrivial solution to (2.3) and by Lemma 2.5, u is a nontrivial nonnegative solution to (1.3).
Finally, notice that Theorem 1.3 (a), whose forthcoming proof is independent of the present one, implies that the maximal solution has negative energy for large enough λ. Indeed, it will be proven that if for some sequence λ n → ∞ we had J(u λn ) ≥ 0 then u λn → 0 pointwise in Ω, which is impossible since the maximal solution is increasing in λ. In particular, we can guarantee that the nonnegative nontrivial solution just constructed does not coincide with the maximal solution.
To summarize, we have shown the existence of Λ * * > 0 such that problem (1.3) admits at least two nontrivial nonnegative solutions for λ ≥ Λ * * . The proof is finished.
3. Behavior as λ → ∞. In this section we analyze the behavior for large λ of nonnegative solutions to (1.3).
Proof of Theorem 1.2. (a) First, notice that by comparison we have u λ ≤ U , where U is the unique nonnegative solution to (1.5). The existence of such solution is implied by the results in [24], while the uniqueness follows by Theorem 1 in [14] (see also Remark 2 below). Then it is standard to obtain that for every sequence λ n → ∞, there exists a subsequence such that Once we show that u λ → ∞ on ∂Ω as λ → ∞, it will follow that V = U , as we want to show. Choose m > 0 and let V m be the unique solution to the Dirichlet If we denote by λ m = sup ∂Ω |∇v m |/v m , it clearly follows that v m is a subsolution to (1.3) for λ ≥ λ m . Since there exist arbitrarily large supersolutions, we achieve Assume that for a sequence λ n → ∞ we have λ where Ω n = {y ∈ R N : x n + λ −1 n y ∈ Ω}. We may assume with no loss of generality that x n → x 0 ∈ ∂Ω. Then, a usual straightening of ∂Ω near x 0 together with the fact that w n ∞ = 1, allow us to obtain bounds to pass to the limit and obtain that, for a subsequence, According to the discussion in page 15 of [16], this is impossible. Hence the upper inequality in (1.6) is proved.
To show the lower inequality, we assume that there exists a sequence λ n → ∞ such that λ 2 1−q n M n → 0, and consider and verifies z n ∞ = 1, we can, as before, pass to the limit to obtain that Now we observe that this is impossible, since by the strong maximum principle z < z(0) = 1, and Hopf's principle would imply − ∂z ∂y1 (0) > 0. Again we have a contradiction, and therefore we conclude the existence of a constant c such that M λ ≥ cλ −2/(1−q) when λ is large enough. This shows the lower inequality in (1.6).

Remark 2.
It is worth mentioning that the proof of Theorem 1 of [14] should be clarified in a specific technical step. Such proof deals with uniqueness of solutions to under the assumption that f is a continuous function which is increasing and such that f (t)/t p is increasing for large t and some p > 1. At some point, the strong comparison principle is invoked and this requires f to be locally Lipschitz (this of course does not hold in our present situation). This difficulty can be overcomed as follows: assume that u is the minimal solution to (3.1) and let v be another solution.
Then v λ → ∞ uniformly on ∂Ω when λ → ∞. It follows that v λ → U uniformly on compact subsets of Ω where U is the unique nonnegative solution to (1.5).
Taking v λ as a test function in the weak formulation of (1.3) we have with θ = (q + 1)/(p + 1) ∈ (0, 1) and a positive constant C which is independent of λ. Hence, v λ Now, we just observe that from (1.3) we also have Next notice that by comparison v λ ≤ z λ , where z λ is the harmonic function in Ω which coincides with v λ on ∂Ω. Thanks to Green's representation formula: where G(x, y) is the Green function of the domain Ω.
On the other hand and large λ. This implies that v λ → 0 uniformly in compacts of Ω as λ → ∞.
Consider now the auxiliary boundary value problem Since µ is an upper bound for v λ in {d( for every ξ such that d(ξ) = d 0 . In particular, Before completing the proof of Theorem 1.3 it is convenient to state some basic features on certain radial initial value problems which will be instrumental for both the present and the next Section. Consider first the Cauchy problem where c ≥ 0 is a parameter.  Another initial value problem of a different nature must be analyzed to properly understand the dead core formation in problem (1.3). Namely, where d ≥ 0 has the status of a parameter. It should be remarked that such a problem exhibits infinitely many nontrivial nonnegative solutions. However, it only admits a positive solution in r > d. This fact and related features concerning (4.6) are stated next.  b) It should be noticed that, following the notation given in the lemmas, u(r, c) |c=0 = u(r, d) d=0 and so ω(0) = ω 0 (0). For immediate use we fix the notation R 0 = ω(0) and u 0 (r) = u(r, d) |d=0 .
Further auxiliary problems playing an important rôle in next section are the radial Dirichlet problem for which the existence a unique nonnegative radial solution u =ũ(r, µ) for all µ ≥ 0 is well-known, and the associated singular version  In addition, i) For µ ≥ µ 0 := u 0 (R) there exists a unique 0 ≤ c(µ) < ω −1 (R) such that u(r, µ) = u(r, c(µ)) 0 ≤ r ≤ R.
The function d(µ) is decreasing, differentiable with d(0) = R, d(µ 0 ) = 0. In particular,ũ(·, µ) has B d as dead core. B) If on the contrary R ≥ R 0 then the large solution U possesses as a dead core. Moreover, each µ ≥ 0 has associated a unique ω −1 and soũ(·, µ) has B d as dead core. Furthermore, d = d(µ) is differentiable, decreasing, d(0) = R while d(µ) → ω −1 0 (R) as µ → ∞. C) The distance R − d of the dead core ofũ(r, µ) to the boundary ∂B R satisfies the asymptotic estimate, Proof of Lemma 4.1. We only deal now with c > 0, the more subtle case c = 0 being studied in Lemma 4.2. By observing that any solution u initially satisfies then u must be increasing wherever defined. Standard theory, see [23], then implies that a unique solution u(r, c) to (4.4) exists which is defined in a maximal interval [0, ω(c)), u(r, c) → ∞ as r → ω(c), while it is smooth with respect to c (a further direct argument then says that u(r, c) increases with c). Let us show that ω is finite (reference to c is now omitted). By observing that u(s) < u(r) in (4.11) we find that (cf. [24]) which, together with the equation in (4.4) implies that Multiplying by u and integrating yields The finiteness of ω then follows by letting r → ω in the previous expression. Continuous dependence of ω on c is more delicate. First, the uniqueness of nonnegative solution to (4.9) implies that ω = ω(c) is decreasing. Since standard theory states that ω is lower semicontinuous in c ( [23]) then ω(c) = lim c →c+ ω(c ). On the other hand, ω(c ) → ω(c) as c → c−, otherwise for a certain increasing c n → c. However, u(r, c n ) < U (r) for r < ω 0 , U being in this case the solution to (4.9) in B R with R = ω 0 . This is incompatible with the fact that u(r, c n ) diverges to ∞ at r = ω(c). Thus, the continuity of ω = ω(c) is shown.
Proof of Lemma 4.2. As mentioned above, problem (4.6) admits infinitely nonnegative solutions defined in r ≥ d (see [20]). Existence, uniqueness and estimates near r = d of a positive local increasing solution u to (4.6) is provided by Theorem 2.3 in [20] (see also [26] for existence). Ode's standard theory then allows us to obtain a global increasing continuation u(r, d) up to a maximal interval [d, ω 0 (d)). Now it follows from the equation that for r ∈ [d, ω 0 (d)) which together with (4.12) gives (4.14) This implies both (4.7) and that ω 0 (d) < ∞. On the other hand that ω 0 is continuous and increasing in d is shown as in Lemma 4.1, while (b) is somehow standard. The more subtle issue of the differentiability of u(r, d) with respect to d is solved by Theorem 2.6 in [20]. As a consequence of it, w(t, d) is regarded as a mapping with values in C 2 [0, η] and d ≥ 0 (η can be taken not depending on d thanks to (4.7)). Since r → (u(r, d), u (r, d)) takes values in R + × R + for r > d then standard results on smoothness on initial data hold from r = d + η ahead. Thus, a continuation argument shows that u(r, d) is globally smooth with respect to d. It also follows from [20] that z(r) = (∂u/∂d)(r, d) solves the initial value problem The proof is concluded.
Proof of Lemma 4.3. Parts A) and B) are essentially a direct consequence of Lemmas 4.1 and 4.2. As for C) observe that setting r = R in (4.14) gives The existence of a second radial nonnegative solution follows by means of the mountain pass theorem, as in Section 2, applied to the functional J defined there but in the space of radial functions in H 1 (B R ), which will be denoted by H 1 r (B R ). We notice that if the maximal solution had nonpositive energy, we could directly apply the mountain pass theorem as in the proof of Theorem 1.1. However, this is not the case in general, so that we still need to prove a further geometric property of J.
Lemma 5.1. Let λ > Λ * . Then either problem (1.3) has two nonnegative nontrivial radial solutions or the maximal solution u λ is a local minimum of the functional J in H 1 r (B R ). Proof. We may assume that the maximal solution u λ is the only nonnegative radial solution to (1.3). Fix λ 1 such that Λ * ≤ λ 1 < λ. Let us check that the function is a subsolution to (2.3), where ε > 0 is chosen to have u λ1 > ε on ∂B R (this is possible thanks to (5.1)). To see this, observe that on ∂B R provided that ε ≤ (λ − λ 1 )u λ1 /λ on ∂B R , which is certainly possible if ε is small enough. Thus u λ1 − ε is a subsolution to (2.3). We recall that the unique positive solution to (2.1), denoted by U λ , is a radial supersolution, verifying u ≤ U λ . We truncate again the nonlinearities f (x, u) and g(x, u) as follows: whereF andG are primitives off andg respectively. Sincef andg are bounded, it follows thatJ is coercive. Also,J is weakly sequentially lower semicontinuous, so that there exists a global minimum u ∈ H 1 r (B R ). Arguing as in Lemma 2.5, we can show that u ≤ u ≤ U λ , thus u is a weak solution to Indeed, u is a nonnegative solution to (1.3). To see this fact, assume that inf B R u < 0, then there exists a point , which is a contradiction. Thus u is nonnegative, and by assumption, u = u λ , so that u λ is a global minimum ofJ in H 1 r (B R ). We claim that u λ is also a local minimum of J in the C( On the other hand, by setting H = {v ∈ H 1 (B R ) : u ≤ v ≤ u}, H + = H ∩ {v : v ≥ 0} and noticing that u λ ≥ 0 we find that inf HJ = inf H +J while, similarly, inf H J = inf H + J due to the fact that J(v) ≥ J(v + ) for all v ∈ H 1 (B R ). Therefore, u λ is a minimum of J in the ball of center u λ and radius δ in the C(B R ) topology. Thus, u λ is also a local minimum of J in H 1 r (B R ) (see for instance Lemma 6.4 in [13]). This finishes the proof.
Remark 5. Observe that the radial symmetry of the solutions is only used to ensure that u λ1 > 0 on ∂B R , since u λ1 assumes its maximum and is constant there. The above proof is indeed "nonradial".
We can now conclude the proof of existence of a second nonnegative radial solution as in the proof of Lemma 2.4, with the use of the mountain pass lemma. Observe that, when J(u λ ) ≤ 0, we can argue exactly as in that proof to obtain the second solution. Thus only the case when J(u λ ) > 0 needs to be considered. Assume that u λ is the only radial nonnegative solution for some λ > Λ * . Then, according to Lemma 5.1, u λ is a local minimum of J. Since J(0) = 0 < J(u λ ), we can apply again the mountain pass theorem to obtain a second nonnegative solution, which is a contradiction. This concludes the proof of existence.
Before performing the proof of parts (b) and (c) in Theorem 1.4, we need some further auxiliary facts.
To show the preceding Lemmas we requiere an additional result that we state next. Its proof is a minor modification of the one of Theorem 1.1 in [19], and will be omitted.
Proof of Lemma 5.2. Given positive and small η, ε, there exist positive u 0 , M , δ, µ 0 such that for all R − r ≤ δ, µ ≥ µ 0 . Now observe that by conveniently diminishing δ and enlarging µ 0 we achieve provided R − r ≤ δ and µ ≥ µ 0 . Thus On the other hand, for µ ≥ µ 0 . Since for fixed δ > we have that Taking ε → 0+, η → 0+ we then obtain The complementary estimate is accomplished in an entirely similar way by employing instead the lower inequality in (5.8).  [21] (see also the proof of Theorem 1.2) that as µ → ∞, where ν stands for the outward unit normal at ∂Ω. Moreover, the same approach permits showing in addition that as µ → 0+.
Proof of Lemma 5.3. Let us proceed to prove estimate (5.4) (the notation h(u) = u p + u q will be kept in what follows). According to Lemma 4.1 the solution u = u(r, c) to (4.4) can be differentiated with respect to c and v(r) = ∂u ∂c (r, c) solves the initial value problem where, in view of part A) in Lemma 4.3 u(r, c) =ũ(r, µ) for µ = µ(c) → ∞ as c → ω −1 (R). Therefore we are having in mind that v(r) = v(r, µ) (very often, explicit reference either to c or µ will be avoided below whenever possible). In addition, it should be remarked that and U is the positive solution to (4.9).
Fix now 0 < r 0 < R and we get dw dt = −r N −1 v, and so w = w(t) satisfies the initial value problem where the value of t 0 corresponding to r 0 will be suitably chosen in the course of the proof. Next, given a positive and small η, there exists u 0 > 0 such that for u ≥ u 0 . By proceeding in a similar way as in the proof of Lemma 5.2 and by employing Lemma 5.4 it is possible to ensure that for small ε > 0 there exist a small δ > 0 and a large µ 0 , such that for R − r ≤ δ and µ ≥ µ 0 .
On the other hand, it follows from (5.9) that t ∼ 1 R N −1 (R − r), as r → R. Thus, by reducing δ if necessary we have As a consequence of the preceding assertions we achieve that w(t) satisfies and D = D(ε, η) and b = b(µ) are given by We now introduce in problem (5.12) the change and so z(τ ) defines the solution to where, The solution to (5.13) is explicitly given by where Therefore, since τ * → −∞ as µ → ∞ we achieve On the other hand v(R, µ) = w(0) ≤ w + (0) = z(τ * ), together with Thus and so By letting ε → 0+ and η → 0+ in the precedent expression we finally arrive at where θ is given in (5.5).
The complementary asymptotic estimate is obtained by using in problem (5.10) the lower estimate for h (ũ) given in (5.11). We find in this way the lower estimate The analysis then proceeds in the same lines as in the lower estimate. This concludes the proof of (5.4).
To show the asymptotic estimate (5.6) first observe that Lemma 4.3 ensure us thatũ(r, µ) = u(r, d) with µ → ∞ as d → ω −1 0 (R). On the other hand, smoothness of u(r, d) with respect to d provided in Lemma 4.2 implies that v(r) = ∂u ∂d (r, d) satisfies the initial value problem Then, the argument in the preceding proof of (5.4) can be repeated to achieve (5.6).
We now come to the conclusion of the proof of Theorem 1.4.
Proof of Theorem 1.4 (b). According to Theorem 1.2, there are only two possibilities for a sequence of nonnegative radial solutions u n to (1.3) corresponding to λ n → ∞ (passing to a subsequence): either u n (R) → ∞ or u n (R) → 0.
Let us first prove that in the first case we necessarily have u n = u λn , that is, the maximal solution is the only family of nonnegative radial solutions which becomes unbounded as λ → ∞. To this aim, we consider the solution u =ũ(r, µ) to problem (4.8) and the function which is C 1 in µ ≥ 0 (Lemmas 4.1, 4.2 and 4.3). We claim that the following assertions hold: ii) dz dµ > 0 for µ ≥ µ 0 and large µ 0 .
Observe also for its use below that the constructed solution z λ verifies as λ → +∞ (5.25) for some positive constant C, as can be easily seen from the previous discussion.
We finally conclude the proof of Theorem 1.4.
Proof of Theorem 1.4 (c). Let us see that the solution obtained in part (a) is not radial. For this aim, choose a function ψ ∈ C 1 0 (B R ) with ψ > 0 in B R , and for some γ > 0, define φ(x) = λ −β ψ(λγ(x − x 0 )) where x 0 ∈ ∂B R is fixed. Let us check that J(tφ) < 0 for some positive t if λ is large enough. Notice that tφ ≤ U λ if λ is large, so that where A, B, C and D are positive constants. If γ < B/A, we can select a value of t, which we will fix and denote by t 0 , such that J(t 0 φ) < 0 if λ is large enough.
Observe also that sup This solution is different from the maximal solution u λ since J(u λ ) < 0 for large enough λ. Finally, for the radial solution z λ constructed in part (b) above, we have as λ → ∞, thanks to (5.25). According to (5.26), the solution v just obtained cannot be radial. This finishes the proof. 6. Some remarks on multiplicity of solutions. Theorem 1.1 provides two nonnegative solutions to (1.3), the maximal solution u = u λ , which exists for all λ ≥ Λ * , and an extra solution with nonnegative energy u = v λ whose existence is only ensured for large λ. As will be described next, additional solutions can be built from u λ and v λ when ∂Ω possesses more than a single connected component.