An elliptic problem of the Prandtl-Batchelor type with a singularity

We establish the existence of at least two solutions of the {\it Prandtl-Batchelor} like elliptic problem driven by a power nonlinearity and a singular term. The associated energy functional is nondifferentiable and hence the usual variational techniques do not work. We shall use a novel approach in tackling the associated energy functional by a sequence of $C^1$ functionals and a {\it cutoff function}. Our main tools are fundamental elliptic regularity theory and the mountain pass theorem.

The nonlinear term f is a locally Hölder continuous function f : Ω × R → [0, ∞) that satisfies the following conditions for all x ∈ Ω, t > 0.
(f 2 ) f (x, t) > 0. (1.2) We shall prove the existence of two distinct nontrivial solutions of (1.1) for a sufficiently large α.
The case when f (x, t) = 1, β = 0 is the well-known Prandtl-Batchelor problem, where the region {u : u > 1} represents the vortex patch bounded by the vortex line {u : u = 1} in a steady state fluid flow for N = 2 (cf. Batchelor [4,5]). This case has been studied by several authors, e.g. Caflisch [8], Elcrat and Miller [10], Acker [1], and Jerison and Perera [14]. We drew our motivation for studying the present problem in this paper from perera [18]. The problem studied by Perera [18] is the case when β = 0 in problem (1.1). The nonlinearity f includes the sublinear case of f (x, t) = t p−1 . Jerison and Perera [14] considered problem (1.1) with β = 0 for 2 < p < ∞ if N = 2, and 2 < p ≤ 2 * = 2N N − 2 if N ≥ 3. This problem has its application in the study of plasma that is confined in a magnetic field. The region there {u : u > 1} represents the plasma and the boundary of the plasma is modelled by the free boundary (cf. Caffarelli and Friedman [6], Friedman and Liu [11], and Temam [19]). Elliptic problems driven by a singular term have, off late, been of great interest. However, we shall discuss only the seminal work of Lazer and McKenna [16] from 1991 that has opened a new door for the researchers in elliptic and parabolic PDEs. The problem considered in [16] was as follows: where p > 0 is a C a (Ω) function, γ > 0, Ω is a bounded domain with a smooth boundary ∂Ω of C 2+a regularity (0 < a < 1), and N ≥ 1. The authors in [16] proved that problem (1.3) has a unique solution u ∈ C 2,a (Ω) ∩ C(Ω) such that u > 0 in Ω. Another noteworthy work, addressing the singularity driven elliptic problem is due to Giacomoni et al. [12]. Jerison and Perera [14] obtained a mountain pass solution of this problem for the superlinear subcritical case. Yang and Perera [20] addressed the problem for the critical case. Recently, Choudhuri and Repovš [9] established the existence of a solution for a semilinear elliptic PDE with a free boundary condition on a stratified Lie group. Furthermore, those readers looking to expand their knowledge on the techniques and trends of the topics in analysis of elliptic PDEs may refer to Papageorgiou et al. [17]. We shall prove that a solution of problem (1.1) is Lipschitz continuous of class H 1 0 (Ω) ∩ C 2 (Ω \ G(u)) and is a classical solution on Ω \ G(u). This solution vanishes on ∂Ω continuously and satisfies the free boundary condition in the following sense: for all ψ ∈ C 1 0 (Ω, R N ) that are supported a.e. on {u : u = 1}. Heren is the outward drawn normal to {u : 1 − ǫ − < u < 1 + ǫ + } and dS is the surface element. The novelty of this work, which separates it from the work of Perera [18], lies in the efficient handling of the singular term that disallows the associated energy functional to be C 1 at u = 0. This difficulty is the reason why one cannot directly apply the results from the variational set up. To handle this situation, we shall define a cut-off function.
Remark 1.1. Note that Ω |∇u| 2 dx will be often denoted by u 2 , where · is the norm of an element in the Sobolev space H 1 0 (Ω).
We begin by defining a weak solution of problem (1.1).
in Ω is said to be a weak solution of problem (1.1) if it satisfies the following: We define the associated energy functional to problem (1.1) as follows: The functional E fails to be of C 1 class due to the term Ω (u + ) 1−γ dx. Moreover, it is nondifferentiable due to the term Ω χ {u>1} (x)dx. We shall first tackle the singular term by defining a cut-off function φ β as follows: The existence of u β can be guaranteed by Lazer and McKenna [16]. Moreover, a solution of problem (1.7) is a subsolution of (1.1) (refer to Lemma 6.1 in Section 6). Note that, we call (1.7) a singular problem. We denote Φ β (u) = u 0 φ β (t)dt. Furthermore, the functional E is nondifferentiable and hence we approximate it by C 1 functionals. This technique is adopted from the work of Jerison and Pererra [14]. Working along similar lines, we now define a smooth function h : R → [0, 2] as follows: Clearly, H is a smooth and nondecreasing function such that We further define for δ > 0 Define, The functional E δ is of C 1 class. The main result of this paper is the following theorem.
The paper is organized as follows. In Section 2 we introduce the key preliminary facts. In Section 3 we prove a convergence lemma. In Section 4 we prove a free boundary condition. In Section 5 we prove two auxiliary lemmas. In Section 6 we prove a result on positive Radon measure. Finally, in Section 7 we prove the main theorem.

Preliminaries
An important result that will be used to pass to the limit in the proof of Lemma 3.1 is the following theorem due to Caffarelli et al. |∇u(x)| ≤ C.
The following are the Palais-Smale condition and the mountain pass Theorem.
Then J is said to satisfy the Palais-Smale (P S) condition if the following holds: Whenever (u n ) is a sequence in V such that (J(u n )) is bounded and (J ′ (u n )) → 0 strongly in V * (the dual space), then (u n ) has a strongly convergent subsequence in V .
Then J has a critical point at the level Before we prove Lemma 3.1, we would like to give an a priori estimate of the parameter β.

Convergence lemma
We shall denote the first eigenvalue of (−∆) by α 1 and the first eigenvector by ϕ 1 (for an existence of α 1 , ϕ 1 refer to Kesavan [15]). Fix α to, say, α 0 and let β be any positive real number. On testing problem (1.1) with ϕ 1 the following weak formulation has to hold if u is a weak solution of problem (1.1). Thus So there exists β * > 0 which depends on the chosen fixed α, such that This is a contradiction to (3.1). Therefore, 0 < β < β * .
) and a subsequence such that (i) u j → u uniformly overΩ, , the free boundary condition is satisfied in the generalized sense and u vanishes continuously on ∂Ω. If u is nontrivial, then u > 0 in Ω, the region {u : u < 1} is connected, and the region {u : u > 1} is nonempty.
Proof of Lemma 3.1. Let 0 < δ j < 1. Consider the following problem: The nature of the problem being a sublinear one and driven by a singularity allows us to conclude by an iterative technique that the sequence ( Therefore by the maximum principle, From the argument used in the proof of Lemma 6.1, together with β * > 0 and large Λ > 0, we conclude that u j > u β in Ω for all β ∈ (0, β * ). Since {u j : u j ≥ 1} ⊂ {ϕ 0 : ϕ 0 ≥ 1}, hence ϕ 0 gives a uniform lower bound, say d 0 , on the distance from the set {u j : u j ≥ 1} to ∂Ω. Furthermore, u j is a positive function satisfying the singular problem in a d 0 -neighborhood of ∂Ω. Thus (u j ) is bounded with respect to the C 2,a norm. Therefore, it has a convergent subsequence in the C 2 -norm in a d 0 2 neighborhood of the boundary ∂Ω. Obviously, 0 ≤ h ≤ 2χ (−1,1) and hence (3.5) By Lazer and McKenna [16], for any subset K of Ω that is relatively compact in it, i.e. K ⋐ Ω, we have that u β ≥ C K for some C K > 0. Therefore Since, (u j ) is bounded in L 2 (Ω) and by Lemma 2.1 it follows that there exists A > 0 such that for suitable r > 0 such that B r (0) ⊂ Ω. Therefore, (u j ) is uniformly Lipschitz continuous on the compact subsets of Ω such that its distance from the boundary ∂Ω is at least d 0 2 units. Thus by the Ascoli-Arzela theorem applied to (u j ), we have a subsequence, still denoted the same, such that it converges uniformly to a Lipschitz continuous function u in Ω with zero boundary values and with strong convergence in C 2 on a d 0 2 -neighborhood of ∂Ω. By the Eberlein-Šmulian theorem we can conclude that u j ⇀ u in H 1 0 (Ω). We now prove that u satisfies the following equation The cases (i)−(iii) do not pose any real mathematical obstacle. Let ϕ ∈ C ∞ 0 ({u > 1}). Then u ≥ 1+2δ on the support of ϕ for some δ > 0. Using the convergence of u j to u uniformly on Ω, we have |u j −u| < δ for any sufficiently large j, δ j < δ. So u j ≥ 1 + δ j on the support of ϕ. Testing (3.1) with ϕ yields On passing to the limit j → ∞ we get To arrive at (3.9), we have used the weak convergence of u j to u in H 1 0 (Ω) and the uniform convergence of the same in Ω. Hence u is a weak solution of −∆u = αf (x, u − 1) + βu −γ in {u > 1}. Since f, u are continuous and Lipschitz continuous respectively, we conclude by the Schauder estimates that it is also a classical solution of −∆u = αf (x, u − 1) + βu −γ in {u : u > 1}. Similarly, on choosing ϕ ∈ C ∞ 0 ({u : u < 1}), one can find a δ > 0 such that u ≤ 1 − 2δ. Therefore, u j < 1 − δ. Using the arguments as in (3.8) and (3.9), we find that u satisfies −∆u = βu −γ in the set {u : u < 1}. Let us now see what is the nature of u in the set {u : u ≤ 1} • . On testing (3.1) with any nonnegative function, passing to the limit j → ∞, and using the fact that h ≥ 0, H ≤ 1, we can show that u satisfies −∆u ≤ αf (x, (u − 1) + ) + βu −γ in Ω (3.10) in the distributional sense. Also, we see that u satisfies −∆u = βu −γ in the set {u : u < 1} . Furthermore, µ = ∆u + βu −γ is a positive Radon measure supported on Ω ∩ ∂{u : u < 1} (refer to Lemma 6.2 in Section 6). From (3.10), the positivity of the Radon measure µ and the usage of Section 9.4 in Gilbarg and Trudinger [13], we conclude that u ∈ W 2,p loc ({u : u ≤ 1} • ), 1 < p < ∞. Thus µ is supported on Ω ∩ ∂{u : u < 1} ∩ ∂{u : u > 1} and u satisfies −∆u = βu −γ in the set {u : u ≤ 1} • . To prove (ii), we show that u j → u locally in C 1 (Ω \ {u : u = 1}). Note that we have already proved that u j → u in the C 2 norm in a neighborhood of ∂Ω ofΩ. Suppose that M ⊂⊂ {u : u > 1}. In this set M we have u ≥ 1 + 2δ for some δ > 0. Thus, for sufficiently large j with δ j < δ, we have |u j − u| < δ in Ω and hence u j ≥ 1 + δ j in M . From (3.2) we derive that is a locally Hölder continuous function and u j → u uniformly in Ω. Our analysis says something stronger. Since −∆u = αf (x, u − 1) in M , we have that u j → u in W 2,p (M ). By the embedding W 2,p (M ) ֒→ C 1 (M ) for p > 2, we have that u j → u in C 1 (M ). This shows that u j → u in C 1 ({u > 1}). Working along similar lines we can also show that u j → u in C 1 ({u : u < 1}). We shall now prove (iii). Since u j ⇀ u in H 1 0 (Ω), we know that by the weak lower semicontinuity of the norm · , u ≤ lim inf u j .
It suffices to prove that lim sup u j ≤ u . To achieve this, we multiply (3.2) with u j − 1 and then integrate by parts. We shall also use the fact that th t δ j ≥ 0 for any t. This gives as j → ∞. Here,n is the outward drawn normal to ∂Ω. We saw earlier that u is a weak solution to −∆u = αf (x, u − 1) + βu −γ in {u : u > 1}. Let 0 < δ < 1. We test this equation with the function ϕ = (u − 1 − δ) + and get On adding (3.12) and (3.13) and passing to the limit δ → 0, we get (3.14) Note that we have used {u:u=1} |∇u| 2 dx = 0. Invoking (3.14) and (3.11), we get lim sup This proves (iii).

Free boundary condition
We shall now show that u satisfies the free boundary condition in the generalized sense (refer to condition (1.4)). We choose ϕ ∈ C 1 0 (Ω, R N ) such that u = 1 a.e. on the support of ϕ. multiplying ∇u j · ϕ to (3.2) and integrating over the set {u : 1 − ǫ − < u < 1 + ǫ + } gives (4.1) The term on the left-hand side of (4.1) can be expressed as follows: Using this, we integrate by parts to obtain This is becausen = ± ∇u |∇u| on the set {u : u = 1 + ǫ ± } ∪ {u : u = 1 − ǫ ± }. By using (iii) the first integral on the right-hand side of (4.3) converges to whereas the second integral of (4.3) is bounded by for some constant C > 0. The last two integrals (4.6)-(4.7) vanish as ǫ ± → 0 since |supp( ϕ) ∩ {u : u = 1}| = 0. Therefore we first let j → ∞ and then we let ǫ ± → 0 in (4.3) to prove that u satisfies the free boundary condition.
Clearly, since 1 < p < 2, we have that E δ is bounded from below and coercive. Thus E δ satisfies the (P S) condition (see Definition 2.1). It is easy to see that every (P S) sequence is bounded by coercivity and hence contains a convergent subsequence by a standard argument-we extract weakly convergent subsequence and show that this weak limit is the strong limit of possibly, a different subsequence. Let us show that E δ has a minimizer, say, u δ 1 . By (f 2 ), we have F (x, t) > 0 for all x ∈ Ω and t > 0. Thus for any u ∈ H 1 0 (Ω) with u > 1 on a set of positive measure, we have Therefore, E(u) → −∞ as α → ∞. Thus, there exists Λ > 0 such that for all α > Λ we have

Auxiliary lemmas
We shall now establish the existence of the first solution of problem (1.1) which also is a minimizer for the functional E. Let us begin with the following lemma.
We shall now prove that the functional E δ has a second nontrivial critical point, say u δ 2 .
Let W = {x ∈ Ω : u(x) = u β (x)}. Since W is a measurable set, it follows that for any η > 0 there exists a closed subset V of W such that |W \ V | < η. Further assume that |W | > 0. We now define a test function ϕ ∈ C 1 c (R N ) such that Since u is a weak solution to (1.1), we have This is a contradiction. Therefore, |W | = 0 which implies that W = ∅. Hence, u > u β in Ω.

Proof of the main theorem
Finally we are in a position to prove Theorem 1.1.