key: cord-0162866-zyx070tj
authors: Bandyopadhyay, Shalmali; Chhetri, Maya; Delgado, Briceyda B.; Mavinga, Nsoki; Pardo, Rosa
title: Bifurcation and Multiplicity Results for Elliptic Problems with Subcritical Nonlinearity on the Boundary
date: 2021-05-25
journal: nan
DOI: nan
sha: 874c719fd32dcb00cf0676234cea9db87a99af03
doc_id: 162866
cord_uid: zyx070tj

We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degree theory.

We consider the following elliptic equation with nonlinear boundary condition −∆u + u = 0

in Ω ;

Reaction-diffusion equations involving nonlinear boundary conditions, appear naturally in applications. For example, limb development which incorporates both outgrowth due to cell growth as well as cell division and interactions between morphogens produced in several very specific zones of the limb bud, see [31] . Another known application is when a highly exothermic reaction takes place in a thin layer around a boundary [22] , this information is then used in cryosurgery (surgery using local application of intense cold to destroy damaged tissue) [17] .

Elliptic equations with nonlinear boundary conditions have been investigated extensively in recent years. Results on existence of positive solutions of problems with nonlinear boundary conditions can be found (without being exhaustive), using techniques such as, monotone methods and functional analysis in [1, 19] , concentration compactness method of Lions (see [24, 25] ) in [15] , bifurcation theory in [5, 6, 7, 26, 28, 30] , variational methods in [20, 29, 34] , and topological degree in [10, 13] .

Regarding the nonlinear eigenproblem (1.1), we refer to [14, 33, 29, 34, 26] where there are existence results with a parameter λ on the boundary for a pure power sublinear nonlinearity. When f is also a pure power and superlinear, we mention for instance [14] with a combination of interior and boundary reaction terms, and also [18] where the authors describe the profile near blowup time for solution of the associated parabolic problem.

To the best of our knowledge, there are so far no existence results with respect to the parameter λ on the boundary of problem (1.1) when the boundary nonlinearity f is superlinear and subcritical, but not necessarily a pure power. In this paper, we fill this gap by showing that there exists a positive weak solution for λ small (see Theorem 1.1), depending only on the behavior of f at infinity. Further, by imposing additional conditions on f to guarantee bifurcation from the trivial solution, nonexistence for large λ, and some necessary technical assumptions, we obtain global bifurcation and multiplicity results (see Theorem 1.2) .

Main focus of this paper is to study (1.1) when f is superlinear and subcritical, that is, there exists a constant b > 0 such that

By a weak solution of problem (1.1), we mean a pair (λ, u) ∈ (0, ∞) × H 1 (Ω) such that

for all ψ ∈ H 1 (Ω).

(1.2)

Moreover, one gets that the weak solution u is actually in C 2,α (Ω) ∩ C 1,α (Ω) if f satisfies the condition (H) ∞ (see Corollary 2.3) . Therefore, we use C(Ω) as our underlying space, and define the solution set as Σ := (λ, u) ∈ [0, +∞) × C(Ω) : (λ, u) is a weak solution of (1.1) .

The closure of the set of nontrivial solutions will be denoted by S . Let λ 0 , λ ∞ ∈ [0, +∞). We say that (λ 0 , 0) respectively, (λ ∞ , ∞) is a bifurcation point from the trivial solution (respectively, from infinity) if there exists a sequence (λ n , u n ) ∈ Σ such that λ n → λ 0 and u n C(Ω) → 0 (respectively, λ n → λ ∞ and u n C(Ω) → +∞) as n → +∞. Likewise, we say that a connected component C bifurcates from the trivial solution at (λ 0 , 0) if C is a maximal closed connected subset of S ∪ (λ 0 , 0) containing (λ 0 , 0). A connected component bifurcating from infinity can be defined similarly.

We state our first result on local bifurcation from infinity.

Then, there existsλ > 0 such that for all λ ∈ (0,λ], (1.1) has a positive weak solution u such that u C(Ω) → ∞ as λ → 0 + . Moreover, there exists a connected component C + ⊂ Σ, of positive weak solutions of (1.1), bifurcating from infinity at λ = 0, such that λ takes all values in (0,λ] along C + (see Fig. 1 ).

λ u C(Ω)λ We use uniform a-priori bounds results for asymptotically superlinear, subcritical nonlinearities, and re-scaling argument together with degree theory and bifurcation theory to prove Theorem 1.1. Such method was first used in [4] for a result in the Dirichlet boundary condition case. We remark that Theorem 1.1 is independent of the behavior of the nonlinearity f away from infinity.

Next, in order to discuss global bifurcation and multiplicity results, we impose additional conditions on the nonlinearity f . First, we assume conditions on f that guarantees bifurcation from the trivial solution, that is, f ∈ C 1 ([0, ∞)) satisfies the following:

Second, to discuss the bifurcation direction of weak solutions near the bifurcation point, following quantities play a crucial role. For ν > 1 as defined in (H) 0 , set

Finally, let µ 1 > 0 be the first Steklov eigenvalue and ϕ 1 ∈ H 1 (Ω) the corresponding nonnegative eigenfunction associated with the Steklov eigenvalue problem

See Remark 2.8 for the regularity and positivity of the nonnegative eigenfunction ϕ 1 . Now, we state the following theorem concerning global bifurcation and multiplicity result.

(1.5)

Then, there exists a connected component C + of positive weak solutions of (1.1) emanating from the trivial solution at the bifurcation point µ 1 f ′ (0) , 0 ∈ Σ possessing a unique bifurcation point from infinity at λ = 0. More precisely, if (λ, u λ ) ∈ C + , then the following holds: Furthermore, if R 0 < 0, then the bifurcation from the trivial solution at

, 0 is supercritical. In addition, there existsλ > µ 1 f ′ (0) such that problem (1.1) has at least two positive weak solutions for any λ ∈ µ 1 f ′ (0) ,λ , and at least one positive weak solution for λ = µ 1 f ′ (0) , and for λ =λ, as depicted in Fig. 2 We use novel approach of combining re-scaling argument used in the proof of local result, Theorem 1.1, and uniform a-priori bound together with the abstract local and global bifurcation theory [11, 32] , and degree theory to prove Theorem 1.2.

Analogous existence results, such as Theorems 1.1 and 1.2, when the nonlinear reaction term appears as a source in the equation complemented with homogeneous Dirichlet boundary condition, can be found, among others, in [3, 4] and the survey paper [23] .

Sections 2 deals with some preliminaries such as regularity of weak solutions, positivity, and uniform a-priori bounds. In Section 3, we prove Theorem 1.1 using re-scaling argument and degree theory. In Section 4, we collect results concerning bifurcation from the trivial solution using Rabinowitz's global bifurcation Theorem [32] . We also characterize the subcritical or supercritical nature of weak solutions near the bifurcation point. Finally, we prove Theorem 1.2 by combining bifurcation theory and degree theory.

Unless otherwise specified, solutions in this paper are understood as weak solutions, as defined in (1.2).

In this section, we discuss regularity and positivity of weak solutions of (1.1), and uniform a-priori bound result.

Here, we state and prove regularity results for some linear and nonlinear problems, which are relevant for our purposes. In particular, we prove that any weak solution of (1.1) is in fact Hölder continuous, see Proposition 2.2 and Corollary 2.3.

To analyze the existence and regularity of weak solutions of (1.1), we must set up the appropriate functional framework. To this end, we consider the following linear problem

where h ∈ L q (∂Ω) for q ≥ 1. It is known that for each q ≥ 1, (2.7) has a unique solution in W 1,m (Ω) and

see, for instance [28] for more details. We denote the solution operator corresponding to (2.7) by

It is known that the trace operator Lemma 2.1. Let N ≥ 2 and h ∈ L q (∂Ω) with q ≥ 1. Then, the unique solution v = Th of the linear problem (2.7) satisfies the following:

(ii) If q = N −1, then Γv ∈ L r (∂Ω) for all r ≥ 1 and the map S : L q (∂Ω) → L r (∂Ω) is continuous and compact for 1 ≤ r < ∞.

is continuous and compact.

In what follows, we will show that any weak solution u of our nonlinear problem (1.1) lies in fact in C α (Ω) for some α ∈ (0, 1). To accomplish this, we will establish regularity results for problems with nonlinearities satisfying (H) ∞ . Hereafter, we will use the same symbol to denote both the function and the associated Nemytskii operator. 

Then,

Proof. We assume N > 2, since the proof is trivial when N = 2. By definition of a weak solution and the trace operator, (2.9), v ∈ H 1 (Ω) and its trace

and by the continuity of the Nemytskii operator

Now we proceed with the bootstrap argument. For h Γv ∈ L q 0 (∂Ω) and

Then, using (2.11) and the continuity of the Nemytskii operator

If q i = N − 1 for some i ∈ N, then by Lemma 2.1 (ii), Γv ∈ L r (∂Ω) for r ≥ 1. By (2.11), h(Γv) ∈ L m for m ≥ 1 . Using the L q -estimates for second-order linear elliptic equations, we get that u is actually in W 1,s (Ω) for any s > 1, in particular for s > N. By the continuity of the embedding W 1,s (Ω) ֒→ C α (Ω) for s > N, one has that v ∈ C α (Ω), see e.g [9, p. 285] . Now suppose q i < N − 1. Then,

If r i > r i−1 , then

Hence, by induction {r i } is strictly increasing. Then, clearly {s i } and {q i } are strictly increasing as well.

Taking the limit as i goes to infinity and noting that r ∞ > 0, we have

This contradicts the boundedness of {q i }. Therefore, there exists i 0 ∈ N such that q i 0 ≥ N − 1 and hence v ∈ C α (Ω) for some α ∈ (0, 1), as desired. Furthermore, the estimate in Lemma 2.1 and (2.11) give 

for some α ∈ (0, 1) and some

Proof. Proposition 2.2 yields the proof for the first part.

Since u ∈ C α (Ω), f is locally Lipschtiz continuous, f (u) ∈ C α (∂Ω). The conclusion follows from Lemma 2.1 (iv).

Under additional assumption on the nonlinearity f , Corollary 2.3 can be rewritten in the following way.

Proof. Note that under conditions (H) 0 and (H) ∞ , for any ε > 0, there exists a constant C ε > 0 such that

In particular, there exists a constant C > 0 such that f (s) ≤ C(|s| + |s| p ). Hence, the conclusion follows from (2.12).

Next lemma shows that any nonnegative nontrivial solution of (2.7) is positive on Ω.

Proof. Clearly v > 0 in Ω by the strong Maximum Principle, see [1, p. 127 ]. Assume to the contrary that there exists an x 0 ∈ ∂Ω such that v(x 0 ) = 0. By the Hopf's Lemma ( [16, Lem. 3.4] More precisely, u is a weak solution of (1.1) for λ > 0 ⇐⇒ u = λS f (Γu) .

We end this subsection with a remark about the sign and regularity of the eigenfunction ϕ 1 corresponding to the first Steklov eigenvalue µ 1 of problem (1.4).

Remark 2.8. By the regularity of weak solutions, see [9, Thm. 9.26] , and repeating the arguments as in the proof of Corollary 2.3, the eigenfunction ϕ 1 corresponding to the first Steklov eigenvalue µ 1 of problem (1.4) is in C 2,α (Ω) ∩ C 1,α (Ω) with 0 < α < 1 (see also e.g. [27, Thm 8.12] ). Therefore, by Proposition 2.5 ϕ 1 > 0 on Ω.

Our main tool in the proof of Theorem 1.1 is degree theory, for which the following uniform a-priori bound is crucial. To state the result, consider

where p is as in (H) ∞ , and for a.e. x ∈ Ω, all σ ∈ R, and

While we are not aware of any paper that establishes uniform a-priori estimate for (2.14), the result below follows by adapting the proof for systems case in [13, Thm. 3.7] . Their proof is written for |ζ(x, σ)| ≤ c(1 + |σ| r ) for some 0 < r < p, but the same arguments can be used to prove the existence of a priori bound under condition (2.15). Proposition 2.9. There exists a constant M > 0 such that every positive solution u ∈ C(Ω) of (2.14) satisfies u C(Ω) ≤ M .

Our proof is motivated by [4] . In particular, we re-scale (1.1) in such a way that the transformed problem approaches a limiting problem of "pure power type" as λ → 0 + . Then, using λ ≥ 0 as the homotopy parameter, we obtain a positive weak solution of the re-scaled problem, hence of (1.1) for λ > 0 small.

First, let us extend f to R by setting f (s) = f (|s|) for s ∈ R. Now consider the problem

Note that for λ > 0, u is a solution of (3.16) if and only if w = λ

We observe that lim

due to superlinear condition at infinity (H) ∞ for s 0 = 0, and by the continuity of f at s 0 = 0. Therefore, we can definef at λ = 0 by settingf (0, s) := b|s| p . Therefore, since f is Lipschitz continuous, so isf : [0, +∞) × R → [0, +∞) defined above.

Then the goal is to study the following re-scaled problem for λ ≥ 0 Our strategy to proceed with the proof of Theorem 1.1 is as follows: 1) we show that the limiting problem (3.19) , corresponding to λ = 0, has a positive solution using the Leray-Schauder degree, 2) show that the re-scaled problem (3.18) has a positive solution using 1) and λ ≥ 0 as the homotopy parameter, then 3) return to the original problem via the re-scaling.

To set up for the Leray-Schauder degree, we formulate the problem (3.18) in an abstract setting in terms of the compact and Nemytskii operators. For this, we define the compact mapF : [0, +∞) × C(Ω) → C(Ω) given bỹ

wheref (λ, ·) denotes the Nemytskii operator corresponding tof (λ, ·), and S is as defined in Remark 2.7. It follows from Remark 2.7 that (λ, w) is a weak solution of (3.18) ⇐⇒F(λ, w) = w .

First we establish the following result regarding the limiting problem (3.19) . Proof. Suppose to the contrary that for each r > 0, there exists θ ∈ [0, 1] such that the operator equation

has a solution w ∈ C(Ω) with w C(Ω) = r, that is, w is a solution of −∆w + w = 0

in Ω ; ∂w ∂η = θ b|w| p on ∂Ω .

(3.20)

Clearly w = 0 since w C(Ω) = r > 0. Hence w > 0 in Ω by Proposition 2.5. Now, let 0 < ε < µ 1 be fixed. Since p > 1, there exists r * > 0 such that bs p < εs for 0 < s ≤ r * . Then there exists θ r * ∈ [0, 1] and a solution w r * > 0 of (3.20) such that w r * C(Ω) = r * , and w r * satisfies bw p r * < εw r * whenever w r * C(Ω) = r * . Using ϕ 1 ≥ 0 as the test function and the fact that θ r * ∈ [0, 1], we have

a contradiction since ε < µ 1 . Thus there exists r > 0 such that for all θ ∈ [0, 1] and all w ∈ C(Ω) with w C(Ω) = r, w = θF(0, w). Therefore, using θ ∈ [0, 1] as a homotopy parameter, we get Then, we observe that the operator equation

Step 1: We show that there exists t 0 > 0 such that (3.21) does not have a solution for t ≥ t 0 . For this, let µ > µ 1 be fixed. Then there exists t 0 > 0 such that bs p + t > µs + t − t 0 for t ≥ 0. Suppose by contradiction that there exists t 1 ≥ t 0 such that w ≥ 0 is a solution of (3.21). Using ϕ 1 ≥ 0 as the test function, we get

which is a contradiction since µ > µ 1 . This establishes Step 1, which implies that for all a > 0, w =F(0, w) + t 0 z for all w ∈ C(Ω) with w C(Ω) = a for any a > 0. Hence, for any a > 0, we have deg(I −F (0, w) + t 0 z, B a , 0) = 0 . Therefore, there exists a solution of w =F(0, w), or equivalently a weak solution of (3.19), say w 0 ∈ B R \ B r . Using the fact that w 0 C(Ω) > r > 0, it follows from Proposition 2.5 that w 0 > 0 in Ω.

Now we use λ ≥ 0 as homotopy parameter to establish the following existence result for the re-scaled problem (3.18) . Proof. (a) Suppose not. Then there exist sequences λ n ≥ 0 with λ n → 0 and w n ∈ C(Ω) such thatF(λ n , w n ) = w n and w n C(Ω) = r (or w n C(Ω) = R). Since w n is bounded andF is compact, (λ n , w n ) → (0, w) for some w ∈ C(Ω) with w C(Ω) = r or w C(Ω) = R, a contradiction to Lemma 3.1 or Lemma 3.2, respectively. Hence there existsλ > 0 satisfying (a).

(b) Now using λ ∈ [0,λ] as the homotopy parameter, it follows from part (a) that

In particular, it follows from (3.23) that for all λ ∈ [0,λ]

This complete the proof of Lemma 3.3. Lemma 3.3 implies that the re-scaled problem (3.18) has a nontrivial solution w λ ∈ C(Ω) for all λ ∈ [0,λ] satisfying r < w λ C(Ω) < R. Moreover, since f is nonnegative and satisfies (H) ∞ , so doesf and hence w λ > 0 in Ω by Proposition 2.5. Now we return to the original problem (1.1). Using the re-scaling

we can conclude that (1.1) has a positive solution (λ, u) for λ ∈ (0,λ]. Also, since w λ C(Ω) > r > 0, it follows that u C(Ω) → +∞ as λ → 0 + . We use the following Leray-Schauder continuation theorem to establish the last part of Theorem 1.1. 

In this section, we will prove that there exists a connected set of positive weak solutions C + of (1.1) bifurcating from the trivial solution at λ = µ 1 f ′ (0) , and bifurcating from infinity at λ = 0. Furthermore, we discuss the direction of bifurcation of positive weak solutions at ( µ 1 f ′ (0) , 0). Finally, we prove Theorem 1.2.

We first show that the condition (H) 0 guarantees solutions bifurcating from the trivial solution. The proof is similar to the case of bifurcation from infinity, see for instance [5, Proposition 3.1] . We provide the proof below for completeness. , and {u n } satisfies, up to a subsequence,

for some β ∈ (0, 1).

Proof. Suppose that λ n → λ for some λ ∈ R and set v n := u n ||u n || C(Ω) .

Observe that v n is a weak solution of the problem

on ∂Ω . → 0 in C(Ω) as n → ∞. Therefore, the right-hand side of the second equation in (4.24) is bounded in C(Ω). Hence, by the elliptic regularity, v n ∈ W 1,s (Ω) for any s > 1, in particular for s > N. Then, the Sobolev embedding theorem implies that ||v n || C α (Ω) is bounded by a constant C that is independent of n. Then, the compact embedding of C α (Ω) into C β (Ω) for 0 < β < α yields, up to a subsequence, v n → Φ ≥ 0 in C β (Ω). Since ||v n || C(Ω) = 1, we have that ||Φ|| C(Ω) = 1. Hence, Φ ≡ 0.

Using the weak formulation of equation (4.24), passing to the limit, and taking into account that λ n → λ for some λ ∈ R and v n → Φ, we obtain that Φ is a weak solution of the equation

Then, it follows that λf ′ (0) = µ 1 , the first Steklov eigenvalue, and Φ = ϕ 1 > 0 is its corresponding eigenfunction, ending the proof.

Now, we will show that µ 1 f ′ (0) , 0 is a bifurcation point from the trivial solution of positive weak solutions of (1.1). That is, there exists a sequence (λ n , u n ) ∈ Σ such that λ n → µ 1 f ′ (0) , u n > 0 on Ω, and that ||u n || C(Ω) → 0. In particular, we have the following result.

Assume that the nonlinearity f ∈ C 1 ([0, ∞)) satisfies hypothesis (H) 0 . Then, there exists a connected component C + ⊂ Σ of positive weak solutions of (1.1) emanating from the trivial solution at µ 1 f ′ (0) , 0 ∈ R×C(Ω). Moreover, C + is unbounded in R × C(Ω).

Proof. The proof follows from the general results on bifurcation from the trivial solutions given in [32, Thm. 1.3] . More precisely, there exists a connected component C + ⊂ Σ of positive weak solutions of (1.1) emanating from the trivial solution at µ 1 f ′ (0) , 0 ∈ R × C(Ω) and, the branch C + either meets another bifurcation point from the trivial solution, or it is unbounded in R × C(Ω). Since f ≥ 0 satisfies (H) 0 , it follows from Lemma 2.1 (iv) and Proposition 2.5 that the branch contains only positive solutions. From the Crandall-Rabinowitz Theorem, see [11] , C + can neither meet another bifurcation point from zero (that is, another point µ ′ f ′ (0) , 0 for another Steklov eigenvalue µ ′ ), nor can meet µ 1 f ′ (0) , 0 again, so the branch is unbounded in R × C(Ω).

In this subsection, we discuss sufficient conditions for the bifurcation from the trivial solution to be either subcritical (to the left) or supercritical (to the right). Following lemma is key in determining the direction of bifurcation from the trivial solution at µ 1 f ′ (0) , 0 . Lemma 4.3. Assume that the nonlinearity f ∈ C 1 ([0, ∞)) satisfies the hypothesis (H) 0 . Consider a sequence of positive weak solutions u n of (1.1) corresponding to the parameters λ n such that λ n → µ 1 f ′ (0) and u n C(Ω) → 0. Then, we have

where R 0 and R 0 are defined in (1.3) , and ν > 1 as defined in (H) 0 .

Proof. Using the weak formulation of (1.1) with ϕ 1 as the test function, we get

Consequently, we get

The proof will be completed in several steps.

Step 1: By Theorem 4.2, there exists a connected component C + of positive weak solutions of (1.1) bifurcating from the trivial solution at the bifurcation point µ 1 f ′ (0) , 0 and that C + is unbounded in R × C(Ω).

Step 2: At this step, we show that (1.1) has no positive weak solution for λ > µ 1 K , where K > 0 is as given in the hypothesis (1.5). Indeed, let u be a positive weak solution of (1.1) for some λ > 0. Then, using ϕ 1 ≥ 0 as the test function, we get

This yields λ ≤ µ 1 K . Hence there exists no positive weak solution u of (1.1) for λ > µ 1 K , completing the proof of this step.

Step 3: Here, we show that C + from Step 1 contains weak positive solutions that bifurcate from infinity at λ = 0, and establish (1.6).

By

Step 1-Step 2, if (λ, u) ∈ C + then u C(Ω) → 0 as λ → µ 1 f ′ (0) , and C + is bounded in the λ-direction. Hence, there exists a sequence (λ n , u n ) ∈ C + such that λ n ∈ (0, K) and u n C(Ω) → ∞. By choosing a subsequence if necessary, there exists a sequence (λ n , u n ) ∈ C + with the property that λ n →λ and u n C(Ω) → ∞. It suffices to showλ = 0.

Assume to the contrary thatλ > 0. For a 0 > 0, let [a 0 , b 0 ] be any fixed compact interval withλ ∈ (a 0 , b 0 ). By Proposition 2.9, for any λ ∈ [a 0 , b 0 ], there exists a uniform constant M = M(a 0 , b 0 ) > 0 such that for every (λ, w) with λ ∈ [a 0 , b 0 ] and w a positive weak solution of the re-scaled problem (3.17) λ , we have w C(Ω) ≤ M .

Here we recall from Section 3 that for any λ > 0, u is a positive weak solution of (1.1) if and only if w = λ 1 p−1 u is a weak solution of (3.17). Hence,

which contradicts that u n C(Ω) → ∞ with λ n →λ > 0. Henceλ = 0. As a conclusion, necessarily, C + contains a unique bifurcation point from infinity at λ = 0 and (1.6) holds. Then, (1.1) has a positive weak solution for any λ ∈ 0, µ 1 f ′ (0) . This completes Step 3. Now, set λ := sup{λ > 0 : (λ, u) ∈ C + }.

Then,λ < ∞ by Step 2.

Step 4: Assuming R 0 < 0, we prove the existence of two positive weak solutions for each λ ∈ µ 1 f ′ (0) ,λ . It follows from Theorem 4.4 (ii), that the bifurcation is supercritical at the bifurcation point µ 1 f ′ (0) , 0 from the trivial solution. Note that since

, λ and u 0 be a positive weak solution corresponding to λ 0 . Now, let λ ∈ µ 1 f ′ (0) , λ 0 be fixed. We show that there exist two distinct positive weak solutions of (1.1) corresponding to λ using degree theory. For this, first we extend f to R by setting f (t) = 0 for t < 0. First solution corresponding to λ:

First we note that, since f is Lipschitz continuous, there exists c ∈ R such that λf (s) + cs is nondecreasing on [0, M ′ ], where M ′ > M, and M > 0 is given by Proposition 2.9. Now let θ ∈ [0, 1] and β > µ 1 . For a given u ∈ C(Ω), define the operator T θ :

where v is given by −∆v + v = 0 in Ω ; ∂v ∂η + θc v = θ(λf (u) + cu) + (1 − θ)(βu + + 1) on ∂Ω , and f θ (u) := θ(λf (u) + cu) + (1 − θ)(βu + + 1). We note that T θ is compact by Remark 2.7, and fixed point of the operator T 1 is a weak solution of (1.1).

We begin by establishing that u 0 > ǫϕ 1 for sufficiently small ǫ > 0. Clearly, u 0 − ǫϕ 1 satisfies −∆(u 0 − ǫϕ 1 ) + (u 0 − ǫϕ 1 ) = 0 in Ω . Now, using the hypothesis (1.5), and the facts that λ > µ 1 f ′ (0) , u 0 C(Ω) < M ′ and f is continuous, we get

for ǫ > 0 sufficiently small. Then

Therefore, by Proposition 2.5, u 0 > ǫϕ 1 for ǫ > 0 sufficiently small.

where ǫ > 0 to be chosen sufficiently small later such that in particular u 0 > ǫϕ 1 in Ω.

First, we justify that the degree deg(I − T θ , Y, 0) is well defined and independent of θ ∈ [0, 1]. That is, u = T θ u for any u on the boundary of Y , ∂Y . We note that if u ∈ ∂Y , then either u C(Ω) = M ′ or u = ǫϕ 1 . Now, if u C(Ω) = M ′ , then by Proposition 2.9, u = T θ u for any θ ∈ [0, 1].

On the other hand, if u = ǫϕ 1 is a solution of u = T θ u for θ = 0, then β > µ 1 yields the contradiction βǫϕ 1 + 1 = ∂ǫϕ 1 ∂η = µ 1 ǫϕ 1 < βǫϕ 1 .

Thus, u = T θ u when u = ǫϕ 1 . Now, by repeating arguments in Step 2 with λf (u) replaced by βu + + 1 and using β > µ 1 , we see that u = T 0 u for any u ∈ Y . Then, using θ ∈ [0, 1] as a homotopy parameter, we conclude that Second solution corresponding to λ:

We construct the second positive weak solution distinct from u 2 by the method of sub-and supersolutions. Using the facts that f (0) = 0 and f ′ (0) > 0, we verify that u = ǫϕ 1 is a subsolution of (1.1) for ǫ ≈ 0. Indeed, we observe that since λ > µ 1 f ′ (0) is fixed, ξ(s) := µ 1 s − λf (s) satisfies ξ(0) = 0 and ξ ′ (0) < 0, then ξ(s) < 0 for s ≈ 0. Therefore, for all 0 ≤ ψ ∈ H 1 (Ω), the following holds for ǫ ≈ 0

Note that u 0 ∈ Y since ǫϕ 1 < u 0 < M < M ′ for sufficiently small ǫ > 0. It follows from [8] that min(u 2 , u 0 ) is a strict supersolution of (1.1). Since u 0 , u 2 ∈ Y , u = ǫϕ 1 < min(u 2 , u 0 ) on Ω. Hence, there exists a positive weak solution u 1 of (1.1) corresponding to the fixed λ satisfying ǫϕ 1 ≤ u 1 < u 2 on Ω by Proposition 2.5. This completes Step 4.

Step 5: At this step, we prove the existence of a solution for λ = λ. For each λ ∈ ( µ 1 f ′ (0) ,λ), problem (1.1) admits a positive weak solution u λ . Using Proposition 2.9, (4.28) for λ ∈ [ µ 1 f ′ (0) ,λ], and Proposition 2.4, there exists a uniform constant C > 0 such that u λ C α (Ω) ≤ C for any λ ∈ ( µ 1 f ′ (0) ,λ). By compact embeddings, u λ has a subsequence that converges to (say), uλ in C β (Ω) as λ →λ, where β < α.

Moreover,

,λ .

By the reflexivity of H 1 (Ω), u λ has a subsequence that converges weakly to (say), uλ in H 1 (Ω) as λ →λ. On the other hand, since u λ → uλ ∈ C β (Ω) and f is locally Lipschitz, then f (u λ ) → f (uλ) in C β (Ω) as λ →λ.

On the existence of positive solutions of nonlinear elliptic boundary value problems

Nonlinear elliptic equations with nonlinear boundary conditions

An introduction to nonlinear functional analysis and elliptic problems, volume 82 of Progress in Nonlinear Differential Equations and their Applications

Positive solutions for some semi-positone problems via bifurcation theory

Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity

Equilibria and global dynamics of a problem with bifurcation from infinity

Infinite resonant solutions and turning points in a problem with unbounded bifurcation

Maximal and minimal weak solutions of an elliptic problem with nonlinear boundary condition

Functional analysis, Sobolev spaces and partial differential equations

Infinitely many stability switches in a problem with sublinear oscillatory boundary conditions

Bifurcation from simple eigenvalues

A priori estimates and existence of positive solutions of semilinear elliptic equations

Existence for an elliptic system with nonlinear boundary conditions via fixed-point methods

A convex-concave problem with a nonlinear boundary condition

Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions

Elliptic partial differential equations of second order

Mathematical model of thermal effects of blinking in human eye

The profile near blowup time for solution of the heat equation with a nonlinear boundary condition

Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions

Two sequences of solutions for indefinite superlinear-sublinear elliptic equations with nonlinear boundary conditions

Function spaces

Multidimensional reaction diffusion equations with nonlinear boundary conditions

On the existence of positive solutions of semilinear elliptic equations

The concentration-compactness principle in the calculus of variations. The limit case

The concentration-compactness principle in the calculus of variations. The limit case

Bifurcation of positive solutions to scalar reactiondiffusion equations with nonlinear boundary condition

Generalized eigenproblem and nonlinear elliptic equations with nonlinear boundary conditions

Bifurcation from infinity for reaction-diffusion equations under nonlinear boundary conditions

Nonlinear elliptic problems with superlinear reaction and parametric concave boundary condition

Bifurcation for an elliptic problem with nonlinear boundary conditions

A mathematical model for outgrowth and spatial patterning of the vertebrate limb bud

Some global results for nonlinear eigenvalue problems

On a concave-convex elliptic problem with a nonlinear boundary condition

An elliptic equation with an indefinite sublinear boundary condition

Acknowledgements: This project was approved by MSRI Summer Research for Women in Mathematics (SWiM) Program for summer 2020. The visit was postponed due to Covid-19, but authors are grateful to MSRI for bringing together this group for collaboration. R. Pardo was partially supported by grant PID2019-103860GB-I00, MICINN, Spain, and by UCM-BSCH, Spain, GR58/08, Grupo 920894.

where we have used the definition of R 0 (see (1. 3)), that ϕ 1 > 0 on ∂Ω and the fact that un un C(Ω) → ϕ 1 uniformly on ∂Ω ( see Proposition 4.1).Passing to the limit in (4.26) and using (4.27), we obtain the first inequality of (4.25). The second inequality is trivial and the third is obtained likewise. Now, we can state the following result with regarding subcritical or supercritical bifurcations from the trivial solution. (i) (Subcritical bifurcations). If R 0 > 0, then the bifurcation of positive weak solutions from the trivial solution at λ = µ 1 f ′ (0) is subcritical, i.e. λ < µ 1 f ′ (0) for every positive solution (λ, u) of (1.1) with (λ, u C(Ω) ) in a neighborhood of ( µ 1 f ′ (0) , 0).(ii) (Supercritical bifurcations). If R 0 < 0, then the bifurcation of positive weak solutions from the trivial solution at λ = µ 1 f ′ (0) is supercritical, i.e. λ > µ 1 f ′ (0) for every positive solution (λ, u) of (1.1) with (λ, u C(Ω) ) in a neighborhood of ( µ 1 f ′ (0) , 0). Proof. Consider a sequence of positive weak solutions u n of (1.1) corresponding to the parameters λ n such that λ n → µ 1 f ′ (0) and u n C(Ω) → 0. Observe that, by (4.25), conditions R 0 > 0 and R 0 < 0 imply that µ 1 f ′ (0) > λ n and µ 1 f ′ (0) < λ n , respectively, for sufficiently large n. This completes the proof.Then, by taking limits in the weak formulation of u λ as λ →λ, we getHence uλ is a positive weak solution of (1.1)λ. Therefore, (1.1) has at least two positive weak solutions for λ ∈ µ 1 f ′ (0) ,λ , and at least one positive weak solution for λ =λ. Finally, since the connected set C + bifurcates to the right at µ 1 f ′ (0) , 0 and bifurcates from infinity at λ = 0, C + must cross the hyperplane λ = µ 1 f ′ (0) at a point distinct from u = 0. Hence, the problem (1.1) has a positive weak solution for λ = µ 1 f ′ (0) . This completes the proof of Theorem 1.2.