ON THE HOLLMAN MCKENNA CONJECTURE: INTERIOR CONCENTRATION NEAR CURVES

. Consider the problem where ε > 0 is a parameter, Ω is a smooth bounded domain in R 2 and p > 2. Let Γ be a stationary non-degenerate closed curve relative to the weighted arc-length (cid:82) Γ Φ p +32 p 1 . We prove that for ε > 0 suﬃciently small, there exists a solution u ε of the problem, which concentrates near the curve Γ whenever d (Γ ,∂ Ω) > c 0 > 0 . As a result, we prove the higher dimensional concentration for a Ambrosetti-Prodi problem, thereby proving an aﬃrmative result to the conjecture by Hollman-McKenna [9] in two dimensions.

(Communicated by Manuel del Pino) Abstract. Consider the problem where ε > 0 is a parameter, Ω is a smooth bounded domain in R 2 and p > 2. Let Γ be a stationary non-degenerate closed curve relative to the weighted arc-length Γ Φ p+3 2p

1
. We prove that for ε > 0 sufficiently small, there exists a solution uε of the problem, which concentrates near the curve Γ whenever d(Γ, ∂Ω) > c 0 > 0. As a result, we prove the higher dimensional concentration for a Ambrosetti-Prodi problem, thereby proving an affirmative result to the conjecture by Hollman-McKenna [9] in two dimensions.
1. Introduction. In this paper, we are interested on the higher dimensional concentration of solutions for the elliptic problem of Ambrosetti-Prodi type. There has been a considerable interest in understanding the number of solutions of the elliptic problem −∆u = ζ(u) − tΦ 1 in Ω where t is a positive parameter, Ω is a smooth bounded domain in R N , Φ 1 is an eigenfunction of −∆ with Dirichlet boundary condition corresponding to the first eigenvalue λ 1 , and lim t→+∞ = ν, where (ν, µ) contains some eigenvalues of −∆ subject to Dirichlet boundary condition. Here, µ = +∞ and ν = −∞ are allowed.
A problem of this kind was first studied by Ambrosetti and Prodi [1] and by many authors, especially in the 1980's. We refer the readers to [2] and the references therein for a detailed bibliography on the topic. The main result is that if ζ(t) grows sub-critically at ±∞, then (1.1) has at least two solutions: one is a local minimum of the corresponding functional, the other being a solution of the mountain-pass type. If ζ(ξ) = ξ 2 and Ω is a unit square in R 2 , Bruer-McKenna-Plum [2] showed that (1.1) admits at least four solutions using a computer assisted proof. Dancer-Yan [3], considered ζ(ξ) = |ξ| p where 1 < p < N +2 N −2 and constructed solutions with sharp peaks near the local maximum points of Φ 1 or near the boundary. Using this idea, one can produce solutions with arbitrarily many peaks, as t → ∞, thereby proving Lazer-Mckenna conjecture [10]. Moreover, in [3], the asymptotic behavior, as t → +∞, of the mountain pass solution is studied, proving in particular that the single peak of this solution approaches the boundary of the domain. This type of concentration was already obtained by de Figueiredo-Santra-Srikanth [8] in the ball, proving that the mountain pass solution is non-radial. The paper in [8] was also motivated by [2] because the numerical approximation of the solutions helped them to guess the geometrical properties of the solutions and their Morse indices. Lazer-McKenna conjecture was later extended to different kind of nonlinearities by Dancer-Yan [4], Dancer-Santra [5] ; Li-Yan-Yang [11], [12] , Wei-Yan [18] for the critical case and del Pino-Munoz [6] for the exponential case and asymptotically linear case is due to Molle-Passaseo [17]. Moreover, there exists results due to del Pino-Kowalczyk-Wei [7] and Mahmoudi-Malchiodi-Montenegro [16] for the higher dimensional concentration for the nonlinear Schrödinger equation. For the Neumann case, see Malchiodi-Montenegro [14] [15] and [19].
Based on the numerical evidence, Hollman-McKenna [9] asked the following question whether there exist other types of concentrations for the superlinear Ambrosetti -Prodi problem. For example, is it possible to construct solutions which concentrates near an interior line or interior geodesic? Note all the results available for this problem are point concentrating solutions only. Numerical results performed in [9] strongly suggests that not only there are solutions which concentrate interior geodesic but also existence of layered solutions close to the boundary. The idea of our result relies on the fact that locally superlinear Ambrosetti-Prodi problem looks like a nonlinear Schrödinger equation.
In this paper, we discuss the conjecture related to the existence of interior concentrating solution of a superlinear Ambrosetti Prodi problem. Let Ω ⊂ R 2 be a smooth bounded domain and consider the problem where p > 2. By [3], there exists ε 0 > 0 such that for all ε ∈ (0, ε 0 ] there exists a negative solution u ε of (1.2) such that u ε > −Φ 1 p 1 and the expansion for every compact subsets of Ω as ε → 0. Furthermore, note that by the Hopf's lemma, the function Φ . Then given c > 0, there exists ε 0 > 0 such that for all ε ∈ (0, ε 0 ) satisfying the gap condition the problem (1.2) has a solution u ε which concentrates near Γ. Here λ > 0 is a number defined by (9.14).
The method is very constructive. We will extensively use the idea by del Pino-Kowalczyk-Wei [7] to prove the result. The advantage of the method is that we can not only prove the existence of solution for (1.2) but also obtain the profile of the solutions. Our method can be used in a symmetric domain to prove existence of solution which concentrate on interior spheres without any gap condition. The disadvantage of this method is that the estimates depend heavily on the exponential decay of the limiting problem and as a matter of fact the method cannot be extended to the exponential or jumping nonlinearities (see [5] and [6]). Remark 1. The idea of the proof is the following. First we shall study the weighted We find a sufficient condition for a curve to be a non-degenerate stationary curve of this functional. In the next section, we shall find an approximate solution (locally on a strip) near a curve and the corresponding error term. In section 4, we introduce the gluing procedure to get the global approximation. This will lead to a nonlinear coupled system. Solving one equation of the system we reduce the problem on the strip. To solve the reduced problem first we check the invertibility of the corresponding linear operator involved in the reduced equation in a proper space in section 5. In section 6, using the invertibility of the linear operator we solve the linear projection problem. Then we adjust the parameters in such a way that the coefficients of the approximate solutions in the strip become identically zero which will actually imply the existence of a true solution of the problem. This has been done in last few sections.

Study of the functional
In this section, we analyze the properties Let l be the total length of the curve Γ. Let γ(θ) be a natural parametrization of Γ with a positive orientation, where θ denotes the arc length parameter measured from a fixed point of Γ. Let ν(θ) denote the outer unit normal to Γ. Points y which are δ-close to Γ can be represented in the form whenever |t| < δ and 0 ≤ θ ≤ l. Any curve sufficiently close to Γ can be parametrized as where g is a smooth, l-periodic function which is uniformly small. Let Γ g be the curve defined in this way. Let us denote Φ 1 (t, θ) by Φ 1 (y) where y is given in (2.1).

BHAKTI BHUSAN MANNA AND SANJIBAN SANTRA
Here y → (t, θ) is a local diffeomorphism. Then the weighted length of this curve is given by the functional W(g) of g as (γ g (θ)) |γ + gν + g ν| dθ.
Since |γ | = 1 and ν = k(θ)γ , where k(θ) denotes the curvature of Γ, we have the expression for W(g) as and we have Hence Γ is a stationary if and only if Now let us calculate the second derivative W (0) h, h .
3. Set up for the approximation.
holds in C 2 loc (Ω). Proof. Since the negative solution is obtained by monotone iteration, using the maximum principle one can show that the negative solution is unique. But −∆u = (|u| p − Φ 1 )ε −2 and hence the right hand side of (1.2) is equal to f (x) + o(1) uniformly on compact subset (using (1.3)). By the L q estimate, u ε is bounded in W 2,q loc for any q > 1. This gives the C 1,θ loc convergence for some 0 ≤ θ < 1. Using the Schauder estimates, u ε is bounded in C 2,θ loc (Ω). By the Arzela-Ascoli theorem u ε → −Φ 1 − ε α ψ 1 . We want to determine α > 0 and ψ 1 a C 4 function independent of α. Hence we have BHAKTI BHUSAN MANNA AND SANJIBAN SANTRA which implies that α = 2 and ψ 1 = .
Hence we obtain (3.1). Moreover, it is easy to check that if p > 3 where and Let v ε (x) = u ε (εx). Then for u ε satisfying (1.2), v ε satisfies where Ω ε = Ω/ε. Now given u ε a solution of (1.2), we look for a solution of (1.2) of the typẽ Then we can easily check that u satisfies the equation Let us now calculate the Laplace operator in the new co-ordinate system : first note that Hence The expression of the metric and the Laplace operator is given by Also near the curve Γ the equation (3.6) takes the form Now consider the change of variables the natural stretched coordinates associated to the curve Γ in S, where Then in the new coordinates, the equation (3.6) reduces to For further reference, it is convenient to expand this operator in the form and a i are smooth functions in (s, z) given by We consider a further change of variable in (3.8) that replaces the main order of the potential q by 1. Let where q(t, θ) := Φ 1 p 1 (γ(θ) + tν(θ)).
Let us now find the expression ofα(θ),β(θ) and its derivatives in terms of α(θ) and β(θ) and its derivatives up to order ε 2 . From the expression (3.2) we get Similarly we can showα

BHAKTI BHUSAN MANNA AND SANJIBAN SANTRA
Now for the next term we shall again use the C 2 loc -convergence. First note that Now for the first derivative we havẽ Similarly we can show thatβ and we obtain Expanding q(εs, εz) = q(0, εz) + q t (0, εz)εs Then the left hand side of the equation (3.8) takes the form Note that s − f = x/β, u −α =α(v − 1) and using the above expansions we have the equation satisfied by v in the new coordinates and Let w(x) denotes the unique positive solution of the following equation Then w decays exponentially at infinity as e − √ p|x| . Taking w(x) as the first approximate solution the error becomes where B 2 (w) turn out to be of O(ε 3 ). Gathering terms of order ε and ε 2 , we get Moreover, using the expansion (1.3) we have β .
Note that S 1 , S 3 are even, and S 2 , S 4 are odd functions of x. Note that G is a function of order ε 3 . We want to construct a further approximate solution which eliminates the term of order ε in the error. We obtain (3.23) We choose ϕ 1 to be the solution of the following equation: using integration by parts (multiply by w and xw x ), we have we obtain and the assumption that Γ is stationary amounts to saying that (3.28) is zero. Hence (k + p+3 2p Φ −1 1 Φ 1 ) = 0. Let the solution of (3.24) has the form where (3.30) Moreover, w 1 is the even function in x: In fact Now the new error becomes Let Z(x) be the first eigenfunction of the problem Then we find that We now consider our basic approximation near the curve Γ ε as The new error is and we have Hence the problem can be reduced to S(w + ϕ) = 0 and near the curve can be expanded in the following way and (3.41) Note that the set up here is only local. We will be able however to reduce the problem to one qualitatively similar to that of the above form in the infinite strip.
Also note that the function involved are expressed in the (x, z) variable defined on the strip S S = (x, z) : −∞ < x < +∞; 0 < z < l ε .
Moreover, we will assume the following condition on f and e as 4. The gluing procedure. Let w(y) denote the first approximation constructed near the curve in the coordinate y in R 2 . Let δ < δ0 100 be a fixed number. We consider a smooth cut-off function η δ (t) where t ∈ R such that η δ (t) = 1 if t < δ and = 0 if t > 2δ. Denote as well η ε δ (s) = η δ (ε|s|), where s is the normal coordinate to Γ ε . We define our first global approximation to be simplỹ w(y) = η ε 3δ (t)w extended globally as 0 beyond the 6δ ε -neighborhood of Γ ε . Denote S(u) = ∆u + |u + u ε (εy)| p − |u ε (εy)| p (4.1) for u =w +φ, whereφ is globally defined in R 2 . Then S(w +φ) = 0 if and only if We plan to decomposeφ in the following waỹ Hence we obtain a pair (ψ, φ) which satisfies the nonlinear coupled system: Note that the operatorL in the strip S may be taken as any compatible extension outside 6δ ε neighborhood of the curve. Now we want to reduce the problem to a problem on the strip S. To do this we solve, given a small φ we solve for ψ. Noww is exponentially decaying whenever |s| > δ ε , where s is a normal coordinate of Γ ε , then the problem has a unique bounded solution ψ whenever h ∞ < +∞. Moreover, by the Schauder estimates Now we assume that φ satisfies the estimate for |s| > δ ε , for some constant γ > 0. Since the remainderÑ has terms involving p > 1, the direct application of fixed point theorem yields a unique solution and the nonlinear operator satisfies the Lipschitz condition of the form . Hence we are reduced to the problem on the strip S for a φ ∈ H 2 (S) satisfying (4.9). By L 2 we mean the operator which coincides with L in the region |s| < 10δ ε . We shall define this operator next. The operatorL for |s| < 10δ ε is given in coordinates (x, z) by formula (3.40). We extend it for functions φ defined in the entire strip S, in terms of (x, z), as follows: where χ(r) is a smooth function function with χ(r) = 1 if r < 10δ and zero outside r > 20δ. Here L 1 is the operator defined by (3.40). Now we consider the projected problem in H 2 (S) : given f and e satisfying φ ∈ H 2 (S) c, d ∈ L 2 (S) such that φ(x, z)Z(x)dx = 0 (4.14) for 0 < z < l ε and We will prove that this problem has a unique solution whose norm is controlled by the L 2 -norm, not of the whole E 1 but rather that of E 12 . After this has been done, our task is to adjust the parameters f and e in such a way that c and d are identically zero. As we will see, this turns out to be equivalent to solving a nonlocal, nonlinear coupled second-order system of differential equations for the pair (c, d) under periodic boundary conditions. As we will see, this system is solvable in a region where the bounds (3.42) and (3.43) hold. In order to solve (4.12)-(4.14), we need to investigate the invertibility of L 2 in an L 2 − H 2 setting under periodic boundary and orthogonality conditions. 5. Invertibility of the operator L 2 . We study the following problem First we consider a simpler problem Lemma 5.1. Then there exists a constant C > 0, independent of ε, such that the problem (5.5)-(5.7) admits a solution φ satisfying the estimate Proof. Let us consider the Fourier decomposition of h and φ of the form Then for i = 1, 2 we obtain with the orthogonality condition as Now we consider the bilinear form Moreover, as (5.13) holds we must have, for some constant independent of i and k. Furthermore, from (5.11) In particular, we have from (5.11), φ ik satisfies Hence we obtain, Now we consider the following problem of φ ∈ H 2 (S) where h ∈ L 2 (S) and c, d ∈ L 2 (0, l).
We consider the problem with (5.13). Then using the integration by parts, we obtain the problem is solvable if while differentiation in x does not change.
Then problem (4.12)-(4.14) reduces to Hereĥ(x, z) = h(x, a −1 (z)) and the operatorB 3 is defined by using the above formulas to replace the z-derivatives by z -derivatives and the variable z by a −1 (z ) in the operator B 3 . The key point is the following: the operator is small in the sense that B 4 (ϕ) L 2 (Ŝ) ≤ Cδ ϕ L 2 (Ŝ) .

ON THE HOLLMAN MCKENNA CONJECTURE 5613
This last estimate is a rather straightforward consequence of the fact that ε|s| < 20δ ε wherever the operatorB 3 is supported, and the other terms are even smaller when ε is small. Thus by reducing δ if necessary, we apply the invertibility result of Lemma 1. The result thus follows by transforming the estimate for φ into a similar one for ϕ via a change of variables. This concludes the proof. 6. The nonlinear projected problem. In this section we will solve the problem (4.12)-(4.14) under periodic boundary and orthogonality conditions in S. Here whenever this operator is well-defined, namely, for φ satisfying (4.9). A first elementary but crucial observation is that the term in (3.37) Hence (4.12) reduces to as we can absorb the term χE 11 into d(εz)χZ. Note that Furthermore, the Lipschitz dependence of the term of error E 12 on the parameters f and e for the norms defined in (3.42)-(3.43). We have the validity of the estimate The operator T has the following useful property: Let h has support contained in |x| ≤ 20δ ε . Then φ = T (h) satisfies the estimate In fact, since B 3 is supported on |x| < 20δ ε and so do the terms involving c and d, then φ satisfies for |x| ≥ 20δ ε an equation of the form where o(1) → 0 as ε → 0. For |x| ≥ 20δ ε , we can use a barrier of the form φ( The lower bound follows in a similar fashion. The bound for ∇φ follows using the standard elliptic estimates. The operator ψ(φ) satisfies and by Lipschitz condition we have These facts will allow us to construct a region where the contraction mapping principle applies. As we have said, for a certain constant C > 0. Now we consider a closed bounded set B by Then we claim for C > 0 large but fixed, the map A maps B into B is a contraction.
There exists a constant C > 0 large but fixed such that for sufficiently small ε and all (f, e) satisfying (3.42)-(3.43), the problem (4.12)-(4.14) has a unique solution φ depending on (f, e) satisfying Moreover, φ is Lipschitz continuous with respect to the variables f and e.

Proof. Let us analyze the Lipschitz character of the nonlinear operator involved in
] By the Sobolev embedding theorem we obtain ] while using estimate φ ∈ B, (6.4), (6.7) and the fact that the area of S δ is of order O( δ ε ), and the Sobolev embedding, we get As a result we obtain N 2 (φ) L 2 (S) ≤ C(ε 3 2 + ε 3 ). Furthermore, by the Lipschitz property we have (6.10) Moreover, . Hence we conclude that Hence we obtain For the second estimate in (6.9) we use the fact that But ψ satisfies an equation of the form L 2 (ψ) = h with h compactly supported.
Hence ψ belongs to B. As a result, A is a contraction mapping thanks to (6.12). We conclude that map A has a unique fixed point in B.
The error E 2 and the operator T itself carry the functions f and e as parameters. A tedious but straightforward analysis of all terms involved in the differential operator and in the error yield that this dependence is indeed Lipschitz with respect to the H 2 -norm (for each fixed ε).
In the operator, consider, for instance, the following only term involving f : Then we have Then using the Young's and interpolation inequality and as a result we obtain For the other terms the analysis follows in a simpler way. Emphasizing the dependence on f , we obtain that the linear operator T satisfies We recall that we have the Lipschitz dependence (6.3). Moreover, the operator N also has Lipschitz dependence on (f, e). It can be easily checked that for φ ∈ B we have, Hence from the fixed point theory we obtain φ f1,e1 − φ f2,e2 ≤ Cε  ) against w x and Z, respectively. It is therefore of crucial importance to carry out computations of the terms R E 1 w x dx and R E 1 Zdx. Using the fact that Zw x dx = 0 we have (ϕ 1 solving (3.24)) Let us first consider S(w + ϕ 1 )w x dx: we shall use the expression (3.34) and the fact that S 3 is even and B 2 (w) = O(ε 3 ) to get Differentiating the equation (3.31) we get and multiplying the equation (3.31) by w 1x we get Integrating by parts we have and using equation (3.32), we obtain Note that w 1 = 1−p 2p xw x − 1 p w, to obtain Since and finally we have Using the Taylor's theorem and (3.30) Furthermore, and L 0 (eZ)w x dx = 0.
Summarizing, we obtain The next computation correspond to the projection onto Z of the error.