Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 15 (2007), pp. 107–126. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

POSITIVE SOLUTIONS FOR ELLIPTIC PROBLEMS WITH CRITICAL INDEFINITE NONLINEARITY IN BOUNDED DOMAINS

JACQUES GIACOMONI, JYOTSHANA V. PRAJAPAT, MYTHILY RAMASWAMY

Abstract. In this paper, we study the semilinear elliptic problem with critical nonlinearity and an indefinite weight function, namely

−Δu = λu + h(x)u^{(n+2)/(n−2)} in a smooth open bounded domain Ω ⊆ ℝ^{n}, n > 4 with Dirichlet boundary conditions and for λ ≥ 0. Under suitable assumptions on the weight function, we obtain the positive solution branch, bifurcating from the first eigenvalue λ_{1}(Ω). For n = 2, we get similar results for −Δu = λu + h(x)ϕ(u)e^{u} where ϕ is bounded and superlinear near zero.

1. Introduction

In this paper, we study the following (critical exponent) semilinear elliptic problem in an open bounded domain Ω ⊆ ℝ^{n} with smooth boundary

−Δu = λu + h(x)u^{\frac{n+2}{n−2}} in Ω

u > 0 in Ω; u = 0 on ∂Ω (1.1)

for dimensions n > 4, λ a nonnegative parameter and h a C^{2} function which changes sign. If n = 2, we are interested in the following corresponding critical problem

−Δu = λu + h(x)ϕ(u)e^{u} in Ω

u > 0 in Ω; u = 0 on ∂Ω. (1.2)

Concerning h, we assume the following hypotheses:

(H1) h belongs to C^{2}(Ω̄),

(H2) h could change sign: Denoting by Ω^{+} := {x ∈ Ω : h(x) > 0} and by

Ω^{−} := {x ∈ Ω : h(x) < 0}, we have Ω^{+} ≠ ∅ and ∂Ω ⊂ ({h > 0} ∪ {h < 0}). (H3) Γ := \overline{Ω^{+}} ∩ \overline{Ω^{−}} ⊂ Ω with ∇h(x) ≠ 0 for all x ∈ Γ. (H4) Ω^{0} := Ω\\overline{Ω^{+} ∪ Ω^{−}}, possibly empty, satisfies:

(1) \overline{Ω^{0}} ⊂ Ω^{−},

(2) ∂Ω^{0} ∩ ∂Ω^{+} = ∅, and (3) λ_{1}(Ω^{0}) > λ_{1}(Ω^{+}), where λ_{1}(·) is the first eigenvalue of −Δ in · with

2000 Mathematics Subject Classification. 35J60, 35B45, 35B33, 35B32.

Key words and phrases. Critical indefinite nonlinearity; bifurcation; a priori estimates.

\begin{matrix}c \\ ◯ \end{matrix}2007 Texas State University - San Marcos.

Published February 28, 2007.

Dirichlet boundary conditions.

Notice that either ∂Ω ⊂ {h > 0} or ∂Ω ⊂ {h < 0} since Γ ∩ ∂Ω is empty. We assume near each point x̄ ∈ S_{1} = {x ∈ Ω^{+}|∇h(x) = 0}, either one of the following flatness conditions holds: (H5a) h(x) = h(x̄)+∑_{i=1}^{n} a_{i}|x_{i} −x̄_{i}|^{β−1}(x−x̄)_{i} +R(x) with ∇R(x) = o(|x−x̄|^{β}),

a_{i} ≠ 0 , for all i and n − 2 < β < n if n ≥ 5. (H5b) c dist(x, S_{1})^{β−1} ≤ |∇h(x)| ≤ C dist(x, S_{1})^{β−1} for all x ∈ Ω^{+} with c, C > 0

and, c_{1}|x_{i}|^{β} ≤ x · ∇h(x + x̄) for x ∈ B_{σ_{0}}(x̄) with i ∈ (1, …, n), c_{1}, σ_{0} > 0 and

n − 2 < β < n for n ≥ 6
n − 2 < β < n − 1 for n = 5.

As h is C^{2}, using Taylor’s expansion,

x · ∇h(x + x̄) = ⟨2D^{2}h(x̄), x⟩ + o(|x|^{2}).

As β > n − 2 ≥ 2, this condition is likely to hold at minimum points of h but not at maximum points.

Concerning ϕ, we assume, as in [1], that

(H6) ϕ is bounded, C^{1}(ℝ, ℝ^{+}) and C_{1}u^{p′} ≤ ϕ(u) ≤ Cu^{p} in a neighborhood of 0,

with p′ ≥ p > 1 and C_{1}, C > 0. Moreover, ϕ(u) > C_{0} > 0, for ∥u∥ large. (H7) ϕ′ is bounded and such that ϕ + ϕ′ ≥ 0.

Under the above assumptions, we have the following result.

Theorem 1.1. (1) Suppose that assumptions (H1)-(H4) and (H5a) or (H5b) are satisfied. Then, there exists a continuum of positive solutions, C, to (1.1) in ℝ^{+} × C_{0}^{1}(Ω) bifurcating from λ_{1}(Ω) and satisfying

(i) Π_{ℝ}C = [0, λ_{∗}] where λ_{1}(Ω) ≤ λ_{∗} < λ_{1}(Ω^{+}).

(ii) If ∫_{Ω} hϕ_{Ω}^{1}^{\frac{2n}{n−2}} < 0, then λ_{1}(Ω) < λ_{∗} and there exist at least two solutions to (1.1) for λ ∈ (λ_{1}(Ω), λ_{∗}). Here ϕ_{Ω}^{1} is the first eigenfunction of Laplacian in Ω.

(2) Suppose that assumptions (H1)-(H4) and (H6)-(H7) are satisfied. Then, there exists a continuum of solutions, C, to (1.2) in ℝ^{+} × C_{0}^{1}(Ω) bifurcating from λ_{1}(Ω) and satisfying

(i) Π_{ℝ}C = [0, λ_{∗}] where λ_{1}(Ω) ≤ λ_{∗} < λ_{1}(Ω^{+}).

(ii) If ϕ(u) ∼ C_{2}u^{q} when u → 0^{+} and if ∫_{Ω} hϕ_{Ω}^{1}^{q} < 0, then λ_{1}(Ω) < λ_{∗} and there exist at least two solutions to (1.2) for λ ∈ (λ_{1}(Ω), λ_{∗}).

Furthermore, in both cases, there exists (0, u_{0}) ∈ C such that u_{0} > 0.

We remark that for n = 2, we can even assert that the branch extends beyond 0, as done in [1, Theorem 1.3].

If we consider star shaped domains as in [9], then we can relax the flatness condition as follows: (H5c) c dist(x, S_{1})^{β−1} ≤ |∇h(x)| ≤ C dist(x, S_{1})^{β−1} for all x ∈ Ω^{+} with c, C > 0

and n − 2 < β < n for n > 4.

Theorem 1.2. Suppose that assumptions (H1)-(H5c) are satisfied and Ω is star shaped. Then the same conclusions as in Theorem 1.1, (1), hold for n > 4.

Notice that we get at least one positive solution for both problems for λ > 0, near zero and at least 2 solutions for some range of λ. We prove theorems 1.1 and 1.2 using the bifurcation theory of Rabinowitz (see [17]). The crucial step here is to prove the existence of uniform a priori bound in L^{∞}(Ω) of solutions to (1.1) and (1.2) independent of λ in a compact set. Note that such a priori estimates are not obvious since the nonlinearity is critical and indefinite in sign in both equations (1.1) and (1.2). This problem has been studied when Ω^{0} is empty and h < 0 near the boundary, in [1] for dimension 2 and in [14] for dimension 5 and above, using uniform bounds. Our aim here is to extend these results to the case of nonempty Ω^{0} and h positive near the boundary and also to explore other possible flatness conditions.

Ouyang [21] has studied the bifurcation for equation (1.1) without the uniform bound and flatness assumptions. Here we are able to conclude that (0, u_{0}) lies in the branch for some u_{0} > 0 because of the bound.

Several results regarding a priori estimates are available concerning the case, h positive, for the subcritical case (i.e. u^{p} with 1 < p < \frac{n+2}{n−2} instead of u^{\frac{n+2}{n−2}} ) using different methods: blow up methods in [6] (for 1 < p < \frac{n+1}{n−1}), [15], moving plane methods in [13], for example.

In the indefinite case, a priori bounds are more delicate. In [3], assuming (H1), (H2), (H3) and Ω^{0} = ∅ and using a blow up method as in [15], the authors prove the existence of a priori bound for 1 < p < \frac{n+2}{n−1}, for more general elliptic operators. The restriction of p is due to a Liouville theorem they prove. In [2], assuming the behaviour of h near \overline{Ω^{+}}∩\overline{Ω^{−}} instead of the nondegenerate assumption (H3), they get a priori bound for a class of problems close to (1.1). The proof is carried out using a blow up method in Ω^{+} and the existence of a supersolution in Ω^{0}. Note that some restrictions on the exponent p remain in this work. In [18], restricting to variational and finite Morse index solutions, they prove that uniform a priori bounds exist for all p between 1 and \frac{n+2}{n−2}. In [11], assuming (H1)-(H3), 1 < p < \frac{n+2}{n−2}, Ω^{0} = ∅ and λ = 0, the authors prove that solutions are uniformly bounded. In this work, they divide Ω into three regions and establish a priori estimates in each region. For this they combine the blow up method, Harnack inequality and moving plane method (in a neighborhood of Γ as in [13]). Then, the question is: is it true for p = \frac{n+2}{n−2}?

In [9], assuming (H1)-(H3), n ≥ 3, λ = 0, Ω is starshaped and an assumption similar to (H5), they answer positively. They use the same approach as in [11]. The crucial step is the uniform estimates in Ω_{δ}^{+} := {x ∈ Ω^{+} | dist(x, Γ) ≥ δ} based upon on the blow up analysis in [19] for critical superlinear problems in S^{n}. This blow up analysis and some Pohozaev identities are also used to estimate solutions near the boundary in [9]. In [8], a similar result is established for the same equation with similar hypotheses but in S^{n}. But, these estimates are proved only for λ = 0. Here we show that the estimates are also true and independent of λ ≠ 0 and positive in a compact interval and get the existence of solutions and multiplicity results for (1.1) via bifurcation theory, extending the earlier results of [14].

Concerning a priori estimates for solutions to (1.2), we use a similar method as for (1.1). Mainly, the difference is that to get a priori bound in Ω_{δ}^{+}, we use the blow up analysis of [5] instead of the blow up analysis of [19]. Furthermore, here we use the specificity of two-dimensional case to estimate solutions at boundary via a Kelvin transform as in [13] (see also [11]). We would like to point out that these a priori estimates are independent of λ in a compact set as done earlier in [1].

Unlike [14] and [1], here Ω^{0} may be nonempty and h may be either positive or negative near the boundary of Ω. For both problems the uniform estimate in Ω^{0} is obtained by constructing a supersolution and is only valid for λ < λ_{1}(Ω^{0}). When h > 0 near the boundary, the assumptions on the weight functions are suitably used to avoid the boundary blow up. Now, let us state the main results concerning a priori estimates for solutions to (1.1) and (1.2):

Proposition 1.3. Let us assume n > 4, (H1)-(H5a) or (H5b) or for star shaped domains, (H5c). Let Λ such that 0 ≤ Λ < λ_{1}(Ω^{+}). Then for any solution (λ, u) to (1.1), such that 0 ≤ λ ≤ Λ, we have

∥u∥_{L^{∞}(Ω)} ≤ C(Λ, n, h, Ω).

Similarly for (1.2), we have the estimate:

Proposition 1.4. Let us assume n = 2, (H1)-(H4) and (H6)-(H7). Let Λ be such that 0 ≤ Λ < λ_{1}(Ω^{+}). Then for any solution (λ, u) to (1.2), such that 0 ≤ λ ≤ Λ, we have

∥u∥_{L^{∞}(Ω)} ≤ C(Λ, h, Ω).

Finally, let us mention that using the same ideas, we can also deal with (1.1) and (1.2) in the case of Ω = ℝ^{n}. It has been done in [1], [14] with Ω^{0} = ∅. In these works, h(x) → −∞ when |x| → +∞. Then the problem becomes sublinear at infinity. So contrary to what we do in the present paper, no careful analysis near the boundary has to be done. But uniform decay at infinity has to be worked out thanks to the behaviour of h at infinity. Now, to conclude this introduction, let us present the outline of this paper:

In the second section, we give first a priori bound in

Ω_{δ}^{−} := {x ∈ Ω^{−} : dist(x, Γ ∪ Ω^{0}) ≥ δ}.

The estimate in this region is performed by a L^{q}-estimate and hypoellipticity arguments. The third section, deals with a neighborhood of Γ and Ω^{0}. We use the moving plane method which allows us to give a priori bound in a neighborhood of Γ for both problems as in [14] and [1] (see also [4]). After that, by exhibiting a supersolution, we get an a priori estimate in a neighborhood of Ω^{0}. Here the fact that λ remains uniformly below to λ_{1}(Ω^{0}) is crucial for getting the existence of the supersolution. Section 4 is on a priori bound in

Ω_{δ,η}^{+} := {x ∈ Ω_{δ}^{+} : dist(x, ∂Ω) ≥ η}

for some (δ, η > 0). As in [14], we extend the blow up analysis of [19] for λ ≥ 0, indicating the proofs with either of the flatness assumptions (H5a), (H5b) or (H5c).

Section 5 is concerned with ∂Ω. Concerning (1.1), the most difficult case is when h is positive near the boundary. In this case, we use a blow up method as in [9] to estimate solutions close to the boundary. If h is non positive, we use the maximum principle. This finalizes the proof of Propositions 1.3 and 1.4.

In the final section, we prove the main results. We would like to point out that this result holds also in the pure superlinear case (ie. h > 0 in \overline{Ω}).

2. A priori bound in Ω_{δ}^{−}

Here we obtain a priori bounds for the solution u of (1.1) and (1.2) in the region Ω_{δ}^{−}. Note that Ω_{δ}^{−} contains a neighborhood of ∂Ω if h < 0 near ∂Ω. The following L^{p} estimate is crucial, which in fact is true in both sets Ω^{−} and Ω^{+}.

Proposition 2.1. Let x_{0} ∈ Ω^{±}, ϵ > 0, Λ ∈ ℝ^{+}, and B_{ϵ}(x_{0}) ⊂⊂ Ω^{±}. Assume that λ ≤ Λ.

(i) For n = 2, if u is a solution of (1.2) then

∫_{B_{\frac{ϵ}{2}} (x_{0})} e^{u} ≤ C(ϵ, h, Λ).  (2.1)

(ii) For n ≥ 3, if u is a solution of (1.1), then there exists C = C(ϵ, Λ) such that

∫_{B_{\frac{ϵ}{2}} (x_{0})} u^{\frac{n+2}{n−2}} dx ≤ (\frac{C}{inf_{B_{ϵ}} |h|})^{\frac{n+2}{4}} .  (2.2)

This estimate follows by multiplying by the first eigenfunction of Laplacian in a ball and using integration by parts. The details are in [1] and [14] respectively.

We present the proof for 1.1 here. The proof for the other equation is similar. For a bound for u in Ω_{δ}^{−}, we need to consider 2 cases:

(i) Either h < 0 near ∂Ω, or

(ii) h > 0 near ∂Ω. In case (i), let us define a δ neighborhood of ∂Ω

G := {x ∈ Ω_{δ}^{−} : dist(x, ∂Ω) ≤ δ}.

For δ small enough, by (H3) and (H4), G ⊂ Ω_{δ}^{−}. Let A := {x ∈ Ω_{δ}^{−} : −Δu(x) < 0}. We split now the domain into three sets

Ω_{δ}^{−} = (Ω_{δ}^{−} \ G) ∪ (G \ A) ∪ (G ∩ A).

We will get the apriori estimate in the first set using the earlier integral estimate and in the second one, a pointwise estimate and then in the third set via maximum principle and the previous estimates.

For x ∈ Ω_{δ}^{−} \ G, there exists a ball B_{δ/2}(x) ⊂ Ω−_{\frac{δ}{2}} and the integral estimate (2.2) hold for u in B_{δ/4}(x). Then we use the following Lemma [16, Lemma 9.20].

Lemma 2.2. Let u ∈ W^{2,n}(Ω) with Lu ≥ f where L is a strictly elliptic second order operator and f ∈ L^{n}(Ω). For all B = B_{ϵ}(y) ⊂ Ω and q > 0, we have

sup_{B_{\frac{ϵ}{2}} (y)} u ≤ C(n, q, ϵ)(( ∫_{B}(u^{+})^{q})^{\frac{1}{q}} + ∥f∥_{L^{n}(B)}).  (2.3)

We combine this estimate for f = 0, q = \frac{n+2}{n−2} and L = Δ + λ in the ball B_{\frac{δ}{2}} (x) ⊂ Ω_{δ/2}^{−} (since x ∈ Ω_{δ}^{−}), together with the estimate (2.2) to conclude that

sup_{B_{\frac{δ}{8}} (x)} u ≤ C(n, λ, δ)\{\frac{1}{inf_{B_{δ/2}(x)} |h|}\}^{(n−2)/4}.  (2.4)

Note that here we can take any λ < Λ. Thus we have if x ∈ Ω_{δ}^{−} \ G,

u(x) ≤ C(n, Λ, δ)\{\frac{1}{inf_{Ω_{δ/2}^{−}} |h|}\}^{(n−2)/4}.  (2.5)

In the case x ∈ G \ A,

0 ≤ −Δu(x) = λu(x) + h(x)u^{\frac{n+2}{n−2}} (x),

and hence

−h(x)u^{\frac{n+2}{n−2}} (x) ≤ λu(x).

Since u(x) > 0, we have the pointwise estimate

u(x) ≤ (\frac{λ}{inf_{Ω_{δ}^{−}} |h|})^{(n−2)/4} for all x ∈ G \ A.  (2.6)

Using the above estimates and recalling that u = 0 on ∂Ω, we have for points on ∂(G ∩ A),

u(x) ≤ M, M = max \{C(n, λ, δ)\{\frac{1}{inf_{Ω_{δ/2}^{−}} |h|}\}^{(n−2)/4}, (\frac{λ}{inf_{Ω_{δ}^{−}} |h|})^{(n−2)/4}\}. (2.7)

Define c(x) := λ + h(x)u(x)^{4/n−2} and consider the equation

Δv + c(x)v ≥ 0 in (G ∩ A).  (2.8)

Note that for x ∈ A, c(x) < 0 and that u − M is a solution of (2.8). Hence, by the weak maximum principle ([16, Theorem 9.1] with f ≡ 0), we have

u(x) − M ≤ 0 in (G ∩ A).

Combining all the cases, we have the local estimate in this case.

Now let us consider the case (ii), h > 0 near ∂Ω. Then, by (H3) since Γ ⊂ Ω, for δ small, dist(Ω_{δ/2}^{−}, ∂Ω) > δ. Therefore a straightforward application of Lemma 2.2 yields (2.4).

Thus we have the following local estimate in Ω_{δ}^{−}.

Proposition 2.3. Let 0 ≤ λ ≤ λ_{1}(Ω^{+}). Assume (H1)–(H3).

(i) For u, a solution of (1.2) and for 1 < p′ < ∞,

sup_{Ω_{δ}^{−}} u(x) ≤ C(λ_{1}(Ω^{+}), δ)(\frac{1}{inf_{Ω_{δ/2}^{−}} |h|})^{1/(p′−1)}.  (2.9)

(ii) For u, a solution of (1.1)

sup_{Ω_{δ}^{−}} u(x) ≤ C(n, λ_{1}(Ω^{+}), δ)(\frac{1}{inf_{Ω_{δ/2}^{−}} |h|})^{(n−2)/4}.  (2.10)

Note that since inf_{Ω_{δ/2}^{−}} |h| > 0, from Proposition 2.3, we get uniform a priori

bounds in Ω_{δ}^{−}.

3. A priori bound in a neighborhood of Γ and Ω^{0}

Let us start by giving a priori bound in a neighborhood of Ω^{0}. Let Ω_{δ}^{0} be a smooth δ-neighborhood of Ω^{0}. By (H4)-1 and (H4)-2 and the results in the second section (bounds in Ω_{δ}^{−}), solutions to (1.1) and (1.2) are uniformly bounded in ∂Ω_{δ}^{0} by a constant M. Now, Let ψ the solution to

−Δψ = Λψ in Ω_{δ}^{0}

ψ > 0 in Ω_{δ}^{0}; ψ = M on ∂Ω_{δ}^{0} (3.1)

which exists if Λ < λ^{1}(Ω^{0}) and δ small. By maximum principle and for λ ≤ Λ, any solution (λ, u) to (1.1) and (1.2) satisfies u ≤ ψ in Ω_{δ}^{0}.

Now, the estimates in a δ-neighborhood of Γ, Γ_{δ}, are similar to step 2 in [4] (for solutions to (1.1) just replace p by \frac{n+2}{n−2}). Note that for solutions to (1.2), we use the assumption (H7) to carry out the moving plane method. Precisely, to show that (taking the same notations as in step 2 in [4]) \frac{∂f}{∂x_{1}} ≤ 0 we use that ϕ(u) + ϕ′(u) ≥ 0 and bounded.

4. A Priori Bound in Ω_{δ,η}^{+}

We start with (1.1). This part is done in [14] and used the blow up analysis in [19]. Precisely, let a sequence (λ_{i}, u_{i}) be solutions of (1.1), such that 0 ≤ λ_{i} ≤ λ_{1}(Ω^{+}) and a sequence of local maxima x_{i} ∈ Ω_{δ}^{+} of u_{i}, such that u_{i}(x_{i}) → ∞ as i → ∞. By the a priori bounds on Ω_{δ}^{−}, Γ and Ω^{0}) obtained in the earlier 2 sections, and also in view of the bound to be obtained in the next section in a neighbourhood of ∂Ω, {u_{i}} remains uniformly bounded on the boundary of Ω_{δ,η}^{+}. Thus x_{i} has to converge to some point in the interior of Ω_{δ,η}^{+}. Note that the a priori bound near the boundary is independent of the results of this section. We postpone that because we need to use some results from this section, in particular, Proposition 4.1.

In the first subsection we recall the standard blow up argument to analyse u_{i}, in a small neighbourhood of x_{i} and various local estimates required later on. In the second subsection, we use these estimates to prove that a blow up point of u_{i} is necessarily a critical point of h. This motivates the flatness assumptions at the critical points of h. Using this assumption, we analyse the nature of the blow up points and show that in fact u_{i} does not blow up, i.e the sequence {u_{i}} is uniformly bounded.

4.1. Blow up points of {u_{i}}. We start by recalling several definitions and propositions which are extensions of some results in [19] to the case of λ > 0. As the details are in [14], we give here only an outline, giving details only when new ideas are involved. The following is a standard result in the blow up analysis (see for example [22]).

Proposition 4.1. Suppose that h ∈ C^{1}(Ω_{δ}^{+}) and there exist A_{1} and A_{2} such that in Ω_{δ}^{+},

h(x) ≥ \frac{1}{A_{1}} , ∥∇h(x)∥ ≤ A_{2}.

Then for every 0 < ε < 1, R > 1, there exist positive constants C_{0} and C_{1} depending on A_{1}, A_{2}, ε, R, λ and n such that if v is a positive solution of

−Δv(x) = λv(x) + h(x)v^{\frac{n+2}{n−2}} , v > 0  (4.1)

with max_{B} v > C_{0}, then there exists a finite number k = k(v) and a set S(v, C_{0}) = {x_{1}, …, x_{k}} ⊂ Ω_{δ}^{+} such that

(i) x_{j} are the local maxima of v and for μ_{j} = v(x_{j})^{− \frac{2}{n−2}} , \{B_{Rμ_{j}}(x_{j})\}_{1≤j≤k} are disjoint balls and

∥v(x_{j})^{−1}v(x_{j} + μ_{j}x) − δ_{j}(x)∥_{C^{2}(B_{2R}(0))} < ε,

where

δ_{j}(x) = (1 + h_{j}|x|^{2})^{\frac{2−n}{2}} with h_{j} = (n(n − 2))^{−1}h(x_{j})

is the unique solution of

Δδ_{j} + h_{j}δj^{\frac{n+2}{n−2}} = 0 in ℝ^{n},

δ_{j} > 0 in ℝ^{n}, δ_{j}(0) = 1, (ii) v(x) ≤ C_{1}(dist(x, S))^{−( \frac{n−2}{2} )}, x ∈ Ω_{δ}^{+}.

The above Proposition, in particular (ii), motivates the definition of an isolated blow up point.

Definition 4.2. A point x_{0} ∈ Ω′ is called an isolated blow up point of {u_{i}}, solutions of (1.1), if there exists 0 < r̄ < dist(x_{0}, ∂Ω′) and C > 0 and a sequence {x_{i}} tending to x_{0}, such that x_{i} is a local maximum of {u_{i}}, u_{i}(x_{i}) → ∞ and

u_{i}(x) ≤ C|x − x_{i}|^{−( \frac{n−2}{2} )} ∀x ∈ Br̄(x_{0})\\{x_{i}\}.

Since we will be interested in the blow up points staying away from each other, we also need to introduce the definition of a simple isolated blow up point.

Definition 4.3. A point x_{0} is an isolated simple blow up point of {u_{i}}, solutions of (1.1), if it is an isolated blow up point such that for some ρ > 0 (independent of i), ṽ_{i} has precisely one critical point in (0, ρ) ∀ large i, where

ṽ_{i}(r) = r^{\frac{n−2}{2}} \overline{v}_{i}(r), \overline{v}_{i}(r) = \frac{1}{|∂B_{r}|} ∫_{∂B_{r}(x_{i})} u_{i}, r > 0.

The following is a corollary of Proposition 4.1.

Corollary 4.4. Let x_{0} be an isolated blow up point of {u_{i}}. Then one can choose R_{i} → ∞ first and then ε_{i} → 0^{+} depending on R_{i} and a subsequence {u_{i}} so that

(i) r_{i} = _{\frac{R_{i}}{(u_{i}(x_{i}))^{\frac{2}{n−2}}}} → 0 and x_{i} is the only critical point of u_{i}(x) in |x−x_{i}| < r_{i}.

(ii) ṽ_{i}(r) has a unique critical point in 0 < r < r_{i}.

Another important result, we will use in the following, is the Harnack inequality

Lemma 4.5 (A Harnack inequality). Let h satisfy

\frac{1}{A_{1}} ≤ h(x) ≤ A_{1} ∀x ∈ Ω_{δ}^{+}  (4.2)

and {u_{i}} satisfy (1.1) having 0 as an isolated blow up point. Then for any 0 < r < \frac{r̄}{3}, with r̄ as in Definition 4.2, we have the Harnack inequality

max_{B_{2r}\B_{r/2}} u_{i}(y) ≤ C min_{B_{2r}\B_{r/2}} u_{i}(y)  (4.3)

with a uniform C = C(n, λ, ∥h∥_{L^{∞}(Ω_{δ}^{+})}).

The proof of this lemma follows on the same lines as in [19], [7].

Now, we look for lower and upper bounds for u_{i}, in a fixed neighbourhood of the blow up point. The arguments for the lower bound are as in [19] (section 2 there). For the upper bound, we need to exploit specifically the extra linear term in our case, as in [14].

Proposition 4.6. Assume B_{2}(0) ⊂ Ω_{δ}^{+} and

A_{1} ≥ h(x) ≥ \frac{1}{A_{1}} , ∥∇h(x)∥ ≤ A_{2} ∀x ∈ B_{2}  (4.4)

for some positive constants A_{1}, A_{2}. Let u_{i} be solutions of (1.1) and x_{i} → 0 be an isolated blow up point with

u_{i}(x) ≤ \frac{A_{3}}{|x − x_{i}|^{\frac{n−2}{2}}} for all x ∈ B_{2}\\{x_{i}\}.  (4.5)

Then there exists a positive constant C = C(n, λ_{0}, A_{1}, A_{2}, A_{3}), such that up to a subsequence,

u_{i}(x) ≥ Cu_{i}(x_{i})(1 + h_{i}u_{i}(x_{i})^{\frac{4}{n−2}} |x − x_{i}|^{2})^{(2−n)/2} for all |x − x_{i}| ≤ 1, (4.6)

where h_{i} is as defined in Proposition 4.1. In particular, for any e ∈ ℝ^{n}, |e| = 1, we have

u_{i}(x_{i} + e) ≥ C^{−1}u_{i}(x_{i})^{−1}.  (4.7)

Proposition 4.7 (Upper bound). Let h and {u_{i}} satisfy the conditions in Proposition 4.6. Also, assume that x_{i} → 0 is an isolated simple blow up point as defined in Definition 4.3. Then there exists a positive constant C = C(n, λ_{0}, A_{1}, A_{2}, A_{3}, ρ) such that

u_{i}(x) ≤ Cu_{i}(x_{i})^{−1}|x − x_{i}|^{2−n} for all 0 < |x − x_{i}| ≤ 1.  (4.8)

In particular, u_{i}(x_{i} + e)u_{i}(x_{i}) = O(1), where e is a unit vector in ℝ^{n}.

The following lemma is crucial in our analysis.

Lemma 4.8. Under the assumptions of Proposition 4.7, let {(λ_{i}, u_{i})} be a sequence of solutions of (1.1) and x_{i} → 0 be an isolated simple blow up point. Then upto a subsequence, we have

(i) if λ_{i} goes to zero, then

u_{i}(x)u_{i}(x_{i}) → w(x) = \frac{a}{|x|^{n−2}} + g(x)

where a = h_{i}^{−(n−2)/2} and g is some harmonic function.

(ii) if λ_{i} goes to λ̃ > 0, then

u_{i}(x)u_{i}(x_{i}) → w(x) = \frac{αC_{n}}{|x|^{n−2}} + E(x) + φ(x)

where G(x) = αC_{n}|x|^{2−n} + E(x) is the unique solution in the sense of distribution for the equation

−ΔG = λG + αδ_{0} in B_{σ_{1}}(0)
G = 0 on ∂B_{σ_{1}}(0)

and φ is the unique C^{2} solution of the boundary value problem

−Δφ = λφ in B_{σ_{1}}(0)
φ = w on ∂B_{σ_{1}}(0).

Here σ_{1} is sufficiently small such that λ < λ_{1}(B_{σ_{1}}(0)).

The proofs are in [14]. There the following lemma is used to handle the linear term. This imposes a restriction n > 4. Now, we fix e ∈ ℝ^{n} such that |e| = 1. As in [7] (see Proposition 3.5), we have the following result.

Lemma 4.9. Let us suppose {(λ_{i}, u_{i})} is a sequence of solutions of (1.1) and x̄ = 0 is an isolated and simple blow up point. Suppose also that n > 4. Then, there exists a positive constant C = C(n, h, ρ) such that

λ_{i}u_{i}(x_{i})^{\frac{2(n−4)}{n−2}} ≤ Cu_{i}(x_{i})^{2}u_{i}(x_{i} + e)^{2} + o(1) .  (4.9)

4.1.1. Nature of Blow up points for solutions to (1.1). We need the following Identity.

Lemma 4.10 (Pohozaev Identity). Let v be a C^{2} solution of (1.1). Then for any B_{σ} ⊂ Ω_{δ}^{+},

∫_{∂B_{σ}} B(σ, x, v, ∇v) = \{λ ∫_{B_{σ}} v^{2} − \frac{λ}{2} ∫_{∂B_{σ}} σv^{2}\} + \frac{n − 2}{2n} ∫_{B_{σ}}(x · ∇h)v^{\frac{2n}{n−2}}

− \frac{σ(n − 2)}{2n} ∫_{∂B_{σ}} h(x)v^{\frac{2n}{n−2}} ,

where

B(σ, x, v, ∇v) := −\frac{(n − 2)}{2}v \frac{∂v}{∂ν} − \frac{σ}{2} |∇v|^{2} + σ|\frac{∂v}{∂ν} |^{2}

and ν denotes the unit outer normal vector field on ∂B_{σ}.

For a proof see for example [19], [14]. The earlier estimates and Pohozaev identity can be used to derive various conclusions about the possible blow up points of {u_{i}}. In particular, the following result is often used.

Lemma 4.11. For u(x) = \frac{a}{|x|^{n−2}} + b(x) where a > 0 and b(x) is a nonnegative differentiable function, with b(0) > 0, we have

B(σ, x, u, ∇u) < 0,

on ∂B_{σ} for all σ small.

The proof follows from direct computations. After suitable rescaling of the solutions, using the above identities, one can prove that in fact that the blow up points are simple, isolated and they have to be the critical points of h, as in [19, 14].

Proposition 4.12. Suppose that (H1)-(H5) are satisfied. Then any isolated blow up point is simple and is a critical point of h.

Proposition 4.13. Suppose that (H1)-(H5) are satisfied.The blow up points of {u_{i}}_{i} are isolated: More precisely, for ε > 0 and R > 1, there exists some positive constant r^{∗} = r^{∗}(n, ε, R, A_{1}, c_{1}, c_{2}, d, modulus of continuity of ∇h) such that for any solution u_{i} with max_{Ω′} u_{i} > C^{∗}, we have

|q_{l} − q_{j}| ≥ r^{∗} ∀1 ≤ l ≠ j ≤ k,

where q_{l} = q_{l}(u_{i}), k = k(u_{i}) are as in Proposition 4.1.

4.1.2. A priori estimates for solutions of (1.1).

Proposition 4.14. Assume (H1)-(H4). Further either (H5a) or (H5b) holds or (H5c) holds for Ω star shaped. Let {(λ_{i}, u_{i})}_{i} be a sequence of solutions of (1.1) with 0 ≤ λ_{i} → λ̃. Then, {u_{i}}_{i} is uniformly bounded in L^{∞}(Ω_{δ,η}^{+}).

Proof. Let us first consider the case when (H5a) holds.

(i) If λ̃ > 0, the analysis follows from Proposition 5.5 (iii) of [14] . Using Pohozaev identity in a ball around a blow up point, a contradiction is derived there. So λ̃ cannot be positive. (ii) If λ̃ = 0 and n ≥ 5, we get a contradiction by using Pohozaev identity as in the Appendix in [14]. Combining both, we conclude that the solutions are uniformly bounded, if (H5a) holds.

Below we recall the proof for λ̃ = 0 and n > 4 when (H5a) holds because some estimates will be needed for the other cases. We will follow the arguments in [19, Theorem 4.4], adapted here for the bounded domain Ω_{δ,η}^{+}. In the proof, we need

the following Pohozaev identity: If u is a C^{2}-solution to −Δu = λu + h(x)u^{\frac{n+2}{n−2}} in B_{σ} for some σ > 0, then

∫_{B_{σ}}(∇h)u^{\frac{2n}{n−2}} = \frac{2n}{n − 2} ∫_{∂B_{σ}}(\frac{∂u}{∂ν}∇u− \frac{1}{2}|∇u|^{2}ν)+∫_{∂B_{σ}} \frac{λ}{2}u^{2}ν+∫_{∂B_{σ}} hu^{\frac{2n}{n−2}} ν (4.10)

where ν is the unit outward normal. Let {y_{i}} ∈ Ω_{δ,η}^{+}. Without loss of generality, we can assume that y_{i} → 0. Then, for a small ball B_{σ} around y_{i}, by the above Pohozaev identity, we have

∫_{B_{σ}} ∇h(x)(u_{i}(x))^{\frac{2n}{n−2}} = I_{1} + I_{2} + I_{3}

where

I_{1} = \frac{2n}{n − 2} ∫_{∂B_{σ}}(\frac{∂u_{i}}{∂ν} ∇u_{i} − \frac{1}{2}|∇u_{i}|^{2}ν),

I_{2} = \frac{λ_{i}}{2} ∫_{∂B_{σ}} u_{i}^{2}ν, I_{3} = ∫_{∂B_{σ}} h(x)(u_{i}(x))^{\frac{2n}{n−2}} ν.

We will estimate I_{k}’s as follows: First, we observe that

|I_{1}| ≤ C ∫_{∂B_{σ}} |∇u_{i}|^{2}.

Then, we need to evaluate ∫_{∂B_{σ}} |∇u_{i}|^{2} for a suitable value of σ. For this, let A := B_{\frac{1}{2}} \B_{σ_{1}} such that σ_{1} < \frac{1}{2}. Let η_{i} a cut off function such that

η_{i}(x) = \begin{cases}1 & if x ∈ A_{i} = \{x, σ_{1} + ϵ_{i} ≤ |x| ≤ \frac{1}{2} − ϵ_{i}\} \\ 1 − \frac{|x|}{ϵ_{i}} & if x ∈ A\A_{i} \end{cases}

where ϵ_{i} := u_{i}(y_{i})^{−1}. Now, multiplying the equation (1.1) by η_{i}u_{i}, we get

∫_{A}(∇u_{i} · ∇(η_{i}u_{i})) = λ_{i} ∫_{A} u_{i}^{2}η_{i} + ∫_{A} hui^{\frac{2n}{n−2}} η_{i}.

It follows that

\begin{aligned}∫_{A_{i}} |∇u_{i}|^{2} & ≤ ∫_{A} |∇u_{i}|^{2}η_{i} \\ & ≤ λ_{i} ∫_{A} u_{i}^{2} + ∫_{A} hui^{\frac{2n}{n−2}} + \frac{1}{2}| ∫_{A\A_{i}} ∇(u_{i}^{2}) · ∇η_{i}|, \end{aligned}  (4.11)

∫_{A} hui^{\frac{2n}{n−2}} ≤ \frac{C}{(u_{i}(y_{i}))^{\frac{2n}{n−2}}} ,  (4.12)

\begin{aligned}| ∫_{A\A_{i}} ∇(u_{i}^{2}) · ∇η_{i}| & ≤ C( ∫_{A\A_{i}} |∇(u_{i}^{2})|^{2})^{1/2} × ( ∫_{A\A_{i}} |∇η_{i}|^{2})^{1/2} \\ & ≤ \frac{C}{(u_{i}(y_{i}))^{2}} × \frac{1}{(u_{i}(y_{i}))^{n−1}} = \frac{C}{(u_{i}(y_{i}))^{n+1}} . \end{aligned}  (4.13)

Hence, from (4.11), (4.12) and (4.13), we get

∫_{A_{i}} |∇u_{i}|^{2} ≤ \begin{cases}\frac{C}{(u_{i}(y_{i}))^{\frac{2n}{n−2}}} & for n ≥ 6, \\ \frac{C}{(u_{i}(y_{i}))^{\frac{8}{3}}} & for n = 5. \end{cases}

Now, taking σ_{i} ∈ [σ_{1} + ϵ_{1}, \frac{1}{2} − ϵ_{1}] such that

∫_{∂B_{σ_{i}}} |∇u_{i}|^{2} = min_{σ∈[σ_{1}+ϵ_{1}, \frac{1}{2} −ϵ_{1}]} ∫_{∂B_{σ}} |∇u_{i}|^{2},

we have

∫_{∂B_{σ_{i}}} |∇u_{i}|^{2} ≤ \begin{cases}_{\frac{C}{(u_{i}(y_{i}))^{\frac{2n}{n−2}}}} × \frac{1}{( \frac{1}{2} −σ_{1}−2ϵ_{i})} & for n ≥ 6, \\ _{\frac{C}{(u_{i}(y_{i}))^{\frac{8}{3}}}} × \frac{1}{( \frac{1}{2} −σ_{1}−2ϵ_{i})} & for n = 5 \end{cases}

Therefore,

|I_{1}|, |I_{3}| ≤ \begin{cases}\frac{C}{(u_{i}(y_{i}))^{\frac{2n}{n−2}}} & for n ≥ 6, \\ \frac{C}{(u_{i}(y_{i}))^{\frac{8}{3}}} & for n = 5 \end{cases}  (4.14)

Also we have for all n > 4,

|I_{3}| ≤ \frac{C}{(u_{i}(y_{i}))^{2n/(n−2)}} .

Then

∫_{B_{σ}} ∇h(x)(u_{i}(x))^{\frac{2n}{n−2}} ≤ \begin{cases}\frac{C}{u_{i}(y_{i})^{\frac{2n}{n−2}}} & for n ≥ 6, \\ \frac{C}{(u_{i}(y_{i}))^{\frac{8}{3}}} & for n = 5 \end{cases}

Now, we follow the arguments in [19, Theorem 4.4 and Corollary 4.1] to prove Step 1: |y_{i}| = O(_{\frac{1}{u_{i}(y_{i})^{\frac{2}{n−2}}}} ) so that y_{i}u_{i}(y_{i})^{\frac{2}{n−2}} = ξ_{i} → ξ,

Step 2: Multiplying (4.10) by u_{i}(y_{i})^{\frac{2(β−1)}{n−2}} ), and using estimates on I_{i} and rescaling

arguments, we get ∫_{ℝ^{n}} ∇h(z + ξ)\frac{dz}{(1+k^{2}|z|^{2})^{n}} = 0 which contradicts (H5a). Let us consider the case when (H5b) holds. In this case, we use Lemma 4.10,

∫_{∂B_{σ}} B(σ, x, v, ∇u_{i}) = \{λ_{i} ∫_{B_{σ}} u_{i}^{2} − \frac{λ_{i}}{2} ∫_{∂B_{σ}} σu_{i}^{2}\} + \frac{n − 2}{2n} ∫_{B_{σ}}(x · ∇h)ui^{\frac{2n}{n−2}}

− \frac{σ(n − 2)}{2n} ∫_{∂B_{σ}} h(x)ui^{\frac{2n}{n−2}} ,

from which together with (H5b), σ < σ_{0} and λ_{i} ≥ 0 it follows that

\begin{aligned}∫_{B_{σ}}(x · ∇h)ui^{\frac{2n}{n−2}} & ≤ −\frac{2n}{n − 2} ∫_{∂B_{σ}} B(σ, x, v, ∇u_{i}) + σ ∫_{∂B_{σ}} h(x)ui^{\frac{2n}{n−2}} \\ & + \frac{λ_{i}}{2} ∫_{∂B_{σ}} σu_{i}^{2}. \end{aligned}  (4.15)

Then, from (4.12) and (4.15) we get

\begin{aligned}∫_{B_{σ}} |x|^{β}ui^{\frac{2n}{n−2}} & ≤ ∫_{B_{σ}}(x · ∇h)ui^{\frac{2n}{n−2}} \\ & ≤ K_{1} ∫_{∂B_{σ}} u_{i}|∇u_{i}| + K_{2} ∫_{∂B_{σ}} |∇u_{i}|^{2} + O(u_{i}(y_{i})^{− \frac{2n}{n−2}} ). \end{aligned} (4.16)

The two boundary integrals can be estimated like I_{1} in (4.14). Hence, we get for n > 5,

∫_{B_{r_{i}}} |x|^{β}ui^{\frac{2n}{n−2}} ≤ O(u_{i}(y_{i})^{− \frac{2n−2}{n−2}} ).  (4.17)

Now using a change of variable and using Proposition (4.1), (i), we see that the left hand side integral is positive but the right hand side goes to 0, since β < n, a contradiction. Similarly the case n = 5 can be handled.

Now let (H5c) hold for a star shaped domain Ω. The results of the previous sections imply that the blow up points in Ω^{+} are finite in number, say {p_{1}, p_{2}, …, p_{m}}. Using again the analysis from [14, Proposition 5.5 (iii)], λ̃ cannot be positive. Let λ̃ = 0. Now assuming (H5c) and Ω is star shaped, we show that there cannot be any blow up points, using the ideas in [9] (see section 3 there).

We apply Pohozaev identity to

Ω_{ε} = Ω \ ∪_{i=1}^{m}B_{ε}(p_{i})

for a fixed ε, to obtain

\{λ_{i} ∫_{Ω_{ε}} u_{i}^{2} − \frac{λ_{i}}{2} ∫_{∂Ω_{ε}}(x · ν)u_{i}^{2}\}

+ \frac{n − 2}{2n} ∫_{Ω_{ε}}(x · ∇h)ui^{\frac{2n}{n−2}} − \frac{(n − 2)}{2n} ∫_{∂Ω_{ε}} h(x)ui^{\frac{2n}{n−2}} (x · ν)

= ∫_{∂Ω_{ε}}(\frac{n − 2}{2} \frac{∂u_{i}}{∂ν} u_{i} − \frac{1}{2}|∇u|^{2}(x · ν) + \frac{∂u_{i}}{∂ν} (x · ∇u_{i}))

Let y_{i} tend to p_{1}. Now multiply both sides by u_{i}(y_{i})^{2} and take the limit as i → ∞. From Lemma 4.8, since λ_{i} → 0, we have

u_{i}(x)u_{i}(y_{i}) → w(x) = \frac{a_{1}}{|x − p_{1}|^{n−2}} + ⋯ + \frac{a_{m}}{|x − p_{m}|^{n−2}} + g(x)

with some harmonic g(x). Since w is smooth outside each B_{ε}(p_{i}), we have

u_{i}^{2}(y_{i}) ∫_{Ω_{ε}}(x · ∇h)ui^{\frac{2n}{n−2}} = \frac{1}{u_{i}^{\frac{2n}{n−2} −2}} ∫_{Ω_{ε}}(x · ∇h)(u_{i}(y_{i})u_{i}(x))^{\frac{2n}{n−2}} → 0.

Also using λ_{i} → 0,

λ_{i} ∫_{Ω_{ε}}(u_{i}(y_{i})u_{i}(x))^{2} → 0.

Thus in the limit

∫_{∂Ω_{ε}} \frac{1}{2}|∇w|^{2}(x · ν) = ∑_{i=}^{m} ∫_{∂B_{σ}(p_{i})} (\frac{1}{2}|∇w|^{2}(x · ν) − \frac{n − 2}{2} \frac{∂w}{∂ν} w − \frac{∂w}{∂ν} (x · ∇w))

Since Ω is star shaped, (x · ν) > 0 on ∂Ω. Furthermore |∇w|^{2} > 0 on ∂Ω . Hence at least for some i,

I = ∫_{∂B_{σ}(p_{i})} (\frac{1}{2}|∇w|^{2}(x · ν) − \frac{n − 2}{2} \frac{∂w}{∂ν} w − \frac{∂w}{∂ν} (x · ∇w)) > 0.

Now we apply Pohozaev identity on B_{σ}(p_{i}) and multiply by u_{i}(y_{i})^{2} and pass to the limit to get

I = lim(u_{i}^{2}(y_{i}) ∫_{B_{σ}}(x · ∇h)ui^{\frac{2n}{n−2}} )

The same calculations as in the proof of [14, Proposition 5.5 (iii)] show that

u_{i}^{2} ∫_{B_{σ}}(x · ∇h)ui^{\frac{2n}{n−2}} ≤ u_{i}^{2} ∫_{B_{σ}} |x|^{β}ui^{\frac{2n}{n−2}}

= \frac{1}{u_{i}^{\frac{2β}{n−2} −2}} ( ∫_{0}^{R_{i}} \frac{r^{β+n−1}}{(1 + (h_{i})^{2}r^{2})^{2n}} dr + o(1)) → 0

if n − 2 < β. Thus for λ_{i} approaching to 0, the solutions cannot blow up and hence are uniformly bounded. These prove the proposition 1.3.

4.2. A priori bound for solutions to (1.2) in Ω_{δ,η}^{+}. In the following, we assume (H1)-(H4), (H6), n = 2, 0 ≤ λ ≤ Λ. We suppose also that the solutions to (1.2) are uniformly bounded on ∂Ω_{δ,η}^{+}. These bounds are to be proved in the next section independent of the bounds obtained in this section. We recall the following result from [5].

Theorem 4.15 (Blow-up analysis (Brezis-Merle)). Assume u_{n} a sequence of solutions to

−Δu_{n} = V_{n}(x)e^{u_{n}} in Ω  (4.18)

where Ω is a bounded domain and V_{n}, u_{n} satisfy

(i) V_{n} ≥ 0,

(ii) ∥V_{n}∥_{L^{p}(Ω)} ≤ C_{1}, ∥e^{u_{n}}∥_{L^{p′} (Ω)} ≤ C_{2} with p (1 < p ≤ ∞) and p′ conjugate

and C_{1}, C_{2} positive.

Then, there exists a subsequence (u_{n_{k}}) satisfying the following alternative:

(i) Either u_{n_{k}} is bounded in L_{loc}^{∞}(Ω), (ii) Or u_{n_{k}} → −∞ uniformly in any compact subset of Ω. (iii) Or the blow up set S (relative to u_{n_{k}}) is finite non empty and u_{n_{k}} → −∞

in Ω/S. In addition, V_{n_{k}}e^{u_{n_{k}}} converges in the sense of measures in Ω to ∑ α_{i}δ(a_{i}) with α_{i} ≥ \frac{4π}{p′}, ∀ i and S = ∪_{i}\{a_{i}\}.

Indeed, arguing by contradiction: Let (λ_{n}, u_{n}) a sequence of solutions to (1.2) such that |λ_{n}| ≤ Λ and ∥u_{n}∥_{L^{∞}(Ω_{δ,η}^{+})} → +∞. let V_{n} := (λ_{n}u_{n}e^{−u_{n}} + ϕ(u_{n})) which

is clearly bounded in L^{∞}(Ω_{δ,η}^{+}) independently of n. Since λ_{n} ≥ 0 and using theorem 4.15 and Lemma 2.1, we get the contradiction.

5. A priori bound in a neighborhood of ∂Ω

By assumptions (H2), (H3) and (H4), either h < 0 or h > 0 near the boundary of Ω. We distinguish these two cases in the proof.

Consider first that h < 0 in a neighborhood of ∂Ω. Since a δ neighbourhood of the boundary, N_{δ}(∂Ω) ⊂ Ω_{δ}^{−} for δ > 0 small enough and since solutions to (1.1) and (1.2) are uniformly bounded in Ω_{δ}^{−} by the results of section 2, solutions to (1.1) and (1.2) are uniformly bounded in a neighborhood of ∂Ω.

Now, let us see the more delicate case: h > 0 near ∂Ω. We apply different arguments for (1.1) and (1.2). Concerning (1.1), we use a blow up analysis as in [9]. For (1.2), we use the moving plane method as in [13].

5.1. Blow up analysis near the boundary for (1.1). We will follow the main steps in [9] to analyse the behaviour of a possible blowing up sequence at boundary ∂Ω. Let {λ_{i}, u_{i}} a sequence of solutions to (1.1). If {u_{i}} is not bounded, then from Proposition 4.1, we can assert the existence of the sets

S_{i} := {x : x is a local maximum of u_{i}}

and for every C, define S̃_{i}(C), a subset of S_{i}, consisting of points satisfying: u_{i}(x) ≥ C and for any two points p, q ∈ S̃_{i}(C),

u_{i}(p)d(p, q)^{\frac{n−2}{2}} ≥ C.

Furthermore, we have

u_{i}(x)d(x, S̃_{i}(C))^{\frac{n−2}{2}} ≤ K(C).

for a constant K(C) depending on C. We will show that for C large enough,

N_{η}(∂Ω) ∩ S̃_{i}(C) = ∅,  (5.1)

where N_{η}(∂Ω) is a η-neighborhood of ∂Ω. This will complete the proof in case of (1.1). The proof is carried out as follows: Step 1 We show that S̃_{i}(C) is discrete. Step 2 N_{η}(∂Ω) ∩ S̃_{i}(C) = ∅ for η > 0 small enough and C large enough. Step 1: Let us define for a fixed constant C, p_{i}, q_{i} ∈ S̃_{i}(C) such that (thanks to Proposition 4.1)

σ_{i} := d(p_{i}, q_{i}) = inf_{p,q∈S̃_{i}(C)} d(p, q)

d_{i} := d(p_{i}, ∂Ω) ≤ d(q_{i}, ∂Ω)

m_{i} := u(p_{i})^{− \frac{2}{n−2}} .

Arguing by contradiction, we assume that σ_{i} ,m_{i} → 0 and p_{i} tend to p_{0}, a point on the boundary, when i → +∞. As in [9], we distinguish three main cases:

(1) m_{i} = o(d_{i}) and σ_{i} = O(d_{i}),

(2) m_{i} = o(d_{i}) and d_{i} = o(σ_{i}),
(3) d_{i} = O(m_{i}).

In case 1, we use the rescaling function

v_{i}(x) := σi^{\frac{n−2}{2}} u_{i}(σ_{i}x + p_{i}).

By definition of σ_{i}, v_{i} has only isolated blow-up points. Moreover, since σ_{i} = o(d_{i}), we can argue similarly as in subsection 3.1 (see in particular Proposition 4.12) to get that {v_{i}} is bounded. For this, note that if the sequence {v_{i}} is not bounded, then there exists at least two blow up points with finite distance between them.

In case 2, we use the rescaling function

v_{i}(x) := di^{\frac{n−2}{2}} u_{i}(d_{i}x + p_{i}).

Since d_{i} = o(σ_{i}) and m_{i} = o(d_{i}), we are in the situation of one isolated and simple blow up point in a half space (because D_{i} = \frac{Ω−p_{i}}{d_{i}} → H := \{x, x_{n} ≥ −1\} up to some standard geometric transformations). So we cannot use directly the blow up analysis from Section 3.1. Since the proof is similar to section 2.2 in [9], we just sketch the proof: From

−Δv_{i} = λ_{i}d_{i}^{2}v_{i} + h̃vi^{\frac{n+2}{n−2}}

where h̃(x) = h(d_{i}x + p_{i}), we prove that

v_{i}(x)v_{i}(0) → w(x) := a(\frac{1}{|x|^{n−2}} − \frac{1}{|x + e|^{n−2}} + \frac{x_{n} + 1}{2^{n−1}})  (5.2)

with a = lim [\frac{n(n−2)}{h(p_{i})}] and e = (0, …, 2).

The contradiction is based on Lemma 4.10 (Pohozaev identity) in D_{i} ∩B_{R} where R is large. Indeed, multiplying this pohozaev identity by v_{i}(0)^{2}, we have

v_{i}(0)^{2} ∫_{∂(B_{R}∩D_{i})} B(R, x, v_{i}, ∇v_{i})

= v_{i}(0)^{2}\{λ_{i}d_{i}^{2} ∫_{B_{R}∩D_{i}} |v_{i}|^{2} − \frac{λ_{i}}{2} d_{i}^{2} ∫_{∂(B_{R}∩D_{i})} x.νv_{i}^{2}\}

+ v_{i}(0)^{2}d_{i} \frac{n − 2}{2n} ∫_{B_{R}∩D_{i}}(x · ∇h)v^{\frac{2n}{n−2}}

− v_{i}(0)^{2} \frac{n − 2}{2n} ∫_{∂(B_{R}∩D_{i})} x.νh(d_{i}x + p_{i})vi^{\frac{2n}{n−2}} ,

where ν is the unit outward normal. Now, when i → +∞, we have

v_{i}(0)^{2} ∫_{∂(B_{R}∩D_{i})} B(R, x, v_{i}, ∇v_{i}) → I_{R} = ∫_{∂(B_{R}∩H)} B(R, x, w, ∇w),  (5.3)

Moreover, I_{R} tends to −∞ when R → +∞ (see [9] p.76). Furthermore,

v_{i}(0)^{2} \frac{λ_{i}}{2} d_{i}^{2} ∫_{∂(B_{R}∩D_{i})} v_{i}^{2}x.ν → 0  (5.4)

which follows from (5.2) for any R fixed. Now we claim that

v_{i}(0)^{2}d_{i} \frac{n − 2}{2n} ∫_{B_{R}∩D_{i}}(x · ∇h)v^{\frac{2n}{n−2}} → 0.  (5.5)

We need to consider 2 cases: ∇h(p_{0}) ≠ 0 or ∇h(p_{0}) = 0. Suppose that p_{i} → p_{0} satisfying ∇h(p_{0}) ≠ 0. First,

\begin{matrix}d_{i} ∫_{B_{\frac{1}{2}}} ∇h(d_{i}x + p_{i})vi^{\frac{2n}{n−2}} \\ = \frac{2n}{n − 2} ∫_{∂B_{\frac{1}{2}}} (\frac{∂v_{i}}{∂ν} ∇v_{i} − \frac{1}{2}|∇v_{i}|^{2})ν + \frac{2n}{n − 2} ∫_{∂B_{\frac{1}{2}}} \frac{λ_{i}}{2} v_{i}^{2}ν + ∫_{∂B_{\frac{1}{2}}} hvi^{\frac{2n}{n−2}} ν. \end{matrix} (5.6)

Multiplying both sides of (5.6) by v_{i}(0)^{2} and using (4.9) to estimate λ_{i} terms, we get

d_{i}v_{i}(0)^{2} ≤ C.

from which (5.5) follows. If ∇h(p_{0}) = 0, we argue as in [9] (see p. 75).(5.5) now follows. From (5.3), (5.4), (5.5) and λ_{i} ≥ 0 we get the contradiction in case 2.

Finally, in case 3, we define

v_{i}(x) := mi^{\frac{n−2}{2}} u_{i}(m_{i}x + p_{i}).

Using Proposition 4.1, we see that {v_{i}} is bounded in B_{r}(0) with r small enough. Then, using results from subsection 3.1 we can show that (i) either {v_{i}} is weakly convergent to v_{0} satisfying

\begin{matrix}−Δv_{0} = h(p_{0})v0^{\frac{n+2}{n−2}} \\ v_{0}(0) = 1 v_{0}|_{∂H} ≡ 0, \end{matrix}  (5.7)

where H := {x : x_{n} ≥ −b} for some b > 0 (since d_{i} = O(m_{i})), (ii) or there exists z̃_{i} ∈ D_{i} such that d(z̃_{i}, ∂D_{i}) → 0 and v_{i}(z̃_{i}) → +∞ when

i → +∞. In the first case, we get the contradiction since there is no solution to (5.7). In the second case, let z_{i} the preimage of z̃_{i} and r_{i} = d(z_{i}, ∂Ω). Using the definition of σ_{i}, w_{i}(x) := u_{i}(r_{i}x + p_{i}) has only one possible blow up point in a half space. In this case, we argue as in case 2 to get the contradiction. If {w_{i}} is bounded, we use again (5.7) to get the contradiction. Thus, we have proved that S̃_{i} is discrete in a neighborhood of ∂Ω for C large. Step 2: To show that S̃_{i} ∩ N_{η}(∂Ω) = ∅, we have just to repeat the arguments in case 2 and case 3. This completes the proof of a priori bound in a neighborhood of ∂Ω for (1.1).

5.2. A priori bound in a neighborhood of ∂Ω for (1.2). Here we deal with (1.2). We get the a priori bound using the moving plane method which implies that solutions to (1.2) are nonincreasing near the boundary. Precisely, we proceed as in [13]. Let (λ, u) be a solution to (1.2). We prove

(1) First an integral estimate:

∫_{Ω} e^{u}ϕ_{1} ≤ C  (5.8)

where ϕ_{1} is the first eigenfunction of −Δ with Dirichlet boundary conditions satisfying ∥ϕ_{1}∥_{∞} = 1.

(2) For any x_{0} in ∂Ω,
u ≤ C in B_{δ}(x_{0}) ∩ Ω  (5.9)

where C and δ do not depend on x_{0}. Proof of (5.8): Multiplying (1.2) by ϕ_{1} and integrating by parts, we get

λ_{1} ∫_{Ω} uϕ_{1} = λ ∫_{Ω} uϕ_{1} + ∫_{Ω} h(x)ϕ(u)e^{u}ϕ_{1}.  (5.10)

Using that {x : h(x) ≥ δ} = Ω_{δ}^{+} ⊃ N_{ϵ}(∂Ω), where N_{ϵ}(∂Ω) is a ϵ-neighborhood of ∂Ω, and that

u(x) ≤ C in Ω\(N_{ϵ}(∂Ω))  (5.11)

with C not depending on u, we have

λ ∫_{Ω} uϕ_{1} + ∫_{Ω} h(x)ϕ(u)e^{u}ϕ_{1} ≥ (λ_{1}(Ω) + a) ∫_{Ω} uϕ_{1} − C_{0}  (5.12)

where C_{0} and a > 0 do not depend on u. From (5.10) and (5.12), we get ∫_{Ω} uϕ_{1} ≤ C and using in addition (5.11), we get (5.8). Proof of (5.9): This is done by moving plane method as in section 3 for points in Γ_{δ}. We give the details here as some arguments are different. To apply moving plane method in a neighbourhood of x_{0}, we need that ∂Ω ∩ B_{δ}(x_{0}) is convex. If it is not convex, we make a kelvin transform. Precisely, without generality, we can assume x_{0} = 0 and the unit outward normal belongs to the x_{1}-axis. Then, taking y_{0} in the x_{1}-axis such that B_{ϵ_{0}}(y_{0}) ∩ Ω = ∅ and ∂B_{ϵ_{0}}(y_{0}) tangent in x_{0} to ∂Ω, we define the inversion I(y_{0}) defining in ℝ^{2}\{y_{0}} as

I(y_{0}) : x → y = y_{0} + \frac{|y_{0}|^{2}}{|x − y_{0}|^{2}} (x − y_{0})

and

v(x) = u(y_{0} + |y_{0}|^{2} \frac{(x − y_{0})}{|x − y_{0}|^{2}} ) for x in Ω^{∗} = I(y_{0})^{−1}(Ω) ⊂ B_{ϵ_{0}}(y_{0})

which satisfies

−Δv = \frac{|y_{0}|^{4}}{|x − y_{0}|^{4}} (λv + h(y_{0} + |y_{0}|^{2} \frac{(x − y_{0})}{|x − y_{0}|^{2}} )ϕ(v)e^{v}).

Now, we define

f(x, v) = \frac{|y_{0}|^{4}}{|x − y_{0}|^{4}} (λv + h(y_{0} + |y_{0}|^{2} \frac{(x − y_{0})}{|x − y_{0}|^{2}} )ϕ(v)e^{v}).

Therefore, since v = 0 on I(y_{0})^{−1}(∂Ω) which is strictly convex near x_{0}, the moving plane method can be carried out if

\frac{∂f(x, v)}{∂ν} ≤ 0  (5.13)

Let

Σ_{ϵ} = {x ∈ Ω^{∗} | dist(x, ∂Ω^{∗}) ≤ ϵ}.

Then, we prove (5.13) in Σ_{ϵ} ∩ B_{ϵ_{1}}(x_{0}) as follows:

\begin{aligned}\frac{∂f(x, v)}{∂ν} & = |y_{0}|^{4}(λv + hϕ(v)e^{v})∂_{ν} \frac{1}{|x − y_{0}|^{4}} \\ & + \frac{|y_{0}|^{4}}{|x − y_{0}|^{4}} ∂_{ν}h(y_{0} + |y_{0}|^{2} \frac{(x − y_{0})}{|x − y_{0}|^{2}} )ϕ(v)e^{v} \\ & ≤ −\frac{C_{1}}{|x − y_{0}|^{5}} |y_{0}|^{4}(hϕ(v)e^{v}) + \frac{C_{2}∥∇h∥_{∞}}{|x − y_{0}|^{6}} |y_{0}|^{6}ϕ(v)e^{v} \\ & = \frac{|y_{0}|^{4}}{|x − y_{0}|^{5}} (−C_{1}h + C_{2}∥∇h∥_{∞} \frac{|y_{0}|^{2}}{|x − y_{0}|})ϕ(v)e^{v} \\ & ≤ \frac{|y_{0}|^{4}}{|x − y_{0}|^{5}} (−C_{1}δ + C(|I(y_{0})x − y_{0}|)ϕ(v)e^{v} \\ & ≤ \frac{|y_{0}|^{4}}{|x − y_{0}|^{5}} (−C_{1}δ + C(|y_{0}| + ϵ + ϵ_{1}))ϕ(v)e^{v} ≤ 0 \end{aligned}  (5.14)

since h ≥ δ > 0, C_{1} > 0 and for |y_{0}|, ϵ_{1}, ϵ small enough. Now, we proceed as in [13] (see also [11]): the unit outward directions in x ∈ B_{ϵ_{1}}(x_{0}) ∩ ∂Ω^{∗} forms a cone centered at (1, 0) with a positive angle θ. Let

I = \{ν ∈ ℝ^{2} : ν.(1, 0) ≥ |ν| cos θ, |ν| ≤ \frac{1}{2}ϵ_{1}\}

be a piece of cone and I_{x} = {x − ν | ν ∈ I}. Then by above arguments, we have

v(y) ≥ v(x) for x ∈ B_{\frac{ϵ_{1}}{2}} (x_{0}) ∩ Ω^{∗}, y ∈ I_{x} ∩ Ω^{∗}.

Using (5.8), we get

∫_{Ω^{∗}} f(x, v) dist(x, ∂Ω^{∗})dx ≤ C.

Therefore, for x ∈ B_{\frac{ϵ_{1}}{2}} (x_{0}) ∩ Ω^{∗},

C ≥ ∫_{Ω^{∗}} f(y, v) dist(y, ∂Ω^{∗})dy

≥ k ∫_{I_{x}} ϕ(v(y))e^{v(y)} dist(y, ∂Ω^{∗})dy
≥ k_{1}ϕ(v(x))e^{v(x)}

since ϕ(·)e^{·} is increasing. Thus, v(x) ≤ C for x ∈ B_{\frac{ϵ_{1}}{2}} (x_{0}) ∩ Ω^{∗} which implies (5.9) for some δ depending on |y_{0}|. The proof is now completed for solutions to (1.2).

6. Existence of solutions to (1.1) and (1.2)

Proof of Theorem 1.1 and 1.2. First, note that from previous sections, we have uniform a priori bound of solutions to (1.1) in [0, Λ] × L^{∞}(Ω) with Λ < λ_{1}(Ω^{0}). From assumption (H4)-3 and bootstrap arguments, we have that any solution (λ, u) to (1.1) such that 0 ≤ λ ≤ λ_{1}(Ω^{+}) satisfies:

∥u∥_{C_{0}^{1}(Ω)} ≤ C.  (6.1)

Multiplying (1.1) by ϕ_{Ω^{+}}^{1} and integrating by parts, we get that any non trivial solution (λ, u) satisfies λ ≤ λ_{1}(Ω^{+}). Thus, (6.1) is also true for any nontrivial solution (λ, u) with λ ≥ 0.

Now, the existence of C follows from global bifurcation theory of Rabinowitz (see [17]). Furthermore, from (6.1), C reaches {λ = 0} and Π_{ℝ}C = [0, λ∗] with λ∗ < λ_{1}(Ω^{+}). This completes the proof of 1-(i) and 2-(i). To prove 1-(ii) and 2-(ii), we have just to prove that the bifurcation from λ_{1}(Ω) is supercritical (i.e. goes towards the right). It implies that λ∗ > λ_{1}(Ω). To get the supercritical branching, we use standard arguments: From ∫_{Ω} hϕ_{Ω}^{1}^{\frac{2n}{n−2}} < 0 and the main result from [12], we show that the unique curve of solutions to (1.1) emanating from λ_{1}(Ω) is defined only for λ > λ_{1}(Ω) (see [21] for more details). To prove that the solution (0, u_{0}) is such that u_{0} > 0, we can follow the arguments from the proof of Theorem 1.2, section 6 of [14]. This completes the proof of Theorem 1.1 and 1.2. □

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Jacques Giacomoni MIP-CEREMATH/bat C, Manufacture des Tabacs, Allée de Brienne 21, 31000 Toulouse, France

   E-mail address: giacomo4@yahoo.fr

Jyotshana V. Prajapat School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India

   E-mail address: jyotsna@math.tifr.res.in

Mythily Ramaswamy TIFR Center, IISc. Campus, Bangalore 560 012, India

   E-mail address: mythily@math.tifrbng.res.in