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


We establish the existence of at least two solutions of the 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 cutoff function. Our main tools are fundamental elliptic regularity theory and the mountain pass theorem.

1 Introduction

We consider the following class of sublinear elliptic free boundary problems:

$$\begin{aligned} \textstyle\begin{cases} -\Delta u=\alpha \chi _{\{u>1\}}(x)f(x,(u-1)_{+})+\beta u^{-\gamma} & \text{in } \Omega \setminus G(u), \\ \vert \nabla u^{+} \vert ^{2}- \vert \nabla u^{-} \vert ^{2}=2& \text{on } G(u), \\ u>0 & \text{in } \Omega, \\ u=0 & \text{on } \partial \Omega . \end{cases}\displaystyle \end{aligned}$$

Here, \(\Omega \subset \mathbb{R}^{N}\) is a bounded domain, \(N\geq 2\), \(0<\gamma <1\), the boundary Ω has \(C^{2,a}\) regularity, \(G(u)=\partial \{u:u>1\}\), \(\alpha , \beta >0\) are parameters, and χ is an indicator function. Furthermore, \(\nabla u^{\pm}\) are the limits of u from the sets \(\{u:u>1\}\) and \(\{u:u\leq 1\}^{\circ}\) respectively, and \((u-1)_{+}=\max \{u-1,0\}\). The nonlinear term f is a locally Hölder continuous function \(f:\Omega \times \mathbb{R}\rightarrow [0,\infty )\) that satisfies the following conditions for all \(x\in \Omega \), \(t>0\):

$$\begin{aligned}& (f_{1}) \quad \text{For some } c_{0}, c_{1}>0, \bigl\vert f(x,t) \bigr\vert \leq c_{0}+c_{1} t^{p-1},\quad \text{where } 1< p< 2. \\& (f_{2})\quad f(x,t)>0. \end{aligned}$$

We shall prove the existence of two distinct nontrivial solutions of (1.1) for a sufficiently large α.

The case when \(f(x,t)=1\), \(\beta =0\) 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 \(\beta =0\) 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 \(\beta =0\) for \(2< p<\infty \) if \(N=2\), and \(2< p\leq 2^{*}=\frac {2N}{N-2}\) if \(N\ge 3\). 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 modeled by the free boundary (cf. Caffarelli and Friedman [6], Friedman and Liu [11], and Temam [19]).

Elliptic problems driven by a singular term have, of late, been of great interest. However, we shall discuss only the seminal work of Lazer and McKenna [16] from 1991 that opened a new door for the researchers in elliptic and parabolic PDEs. The problem considered in [16] was as follows:

$$\begin{aligned} \textstyle\begin{cases} -\Delta u=p(x)u^{-\gamma}& \text{in } \Omega, \\ u=0 &\text{on } \partial \Omega, \end{cases}\displaystyle \end{aligned}$$

where \(p>0\) is a \(C^{a}(\bar{\Omega})\) function, \(\gamma >0\), Ω is a bounded domain with a smooth boundary Ω of \(C^{2+a}\) regularity (\(0< a<1\)), and \(N\geq 1\). The authors in [16] proved that problem (1.3) has a unique solution \(u\in C^{2,a}(\Omega )\cap C(\bar{\Omega})\) 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 \(H_{0}^{1}(\Omega )\cap C^{2}(\bar{\Omega}\setminus G(u))\) and is a classical solution on \(\Omega \setminus G(u)\). This solution vanishes on Ω continuously and satisfies the free boundary condition in the following sense:

$$\begin{aligned} \lim_{\epsilon ^{+}\rightarrow 0} \int _{\{u=1+\epsilon ^{+} \}}\bigl(2- \vert \nabla u \vert ^{2} \bigr)\psi \cdot \hat{n}\,dS- \lim_{\epsilon ^{+}\rightarrow 0} \int _{\{u=1-\epsilon ^{+} \}} \vert \nabla u \vert ^{2}\psi \cdot \hat{n}\,dS&=0 \end{aligned}$$

\(\text{for all} \psi \in C_{0}^{1}(\Omega ,\mathbb{R}^{N})\) that are supported a.e. on \(\{u:u\neq 1\}\). Here is the outward drawn normal to \(\{u:1-\epsilon ^{-}< u<1+\epsilon ^{+}\}\) 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 \(\int _{\Omega}|\nabla u|^{2}\,dx\) will be often denoted by \(\|u\|^{2}\), where \(\|\cdot \|\) is the norm of an element in the Sobolev space \(H_{0}^{1}(\Omega )\).

We begin by defining a weak solution of problem (1.1).

Definition 1.1

A function \(u\in H_{0}^{1}(\Omega )\), \(u>0\) a.e. in Ω is said to be a weak solution of problem (1.1) if it satisfies the following:

$$\begin{aligned} 0={}& \int _{\Omega}\nabla u\cdot \nabla \varphi \,dx-\alpha \int _{\Omega} g\bigl(x,(u-1)_{+}\bigr))\varphi \,dx \\ &{}-\beta \int _{\Omega}u^{-\gamma}\varphi \,dx\quad \text{for all } \varphi \in H_{0}^{1}(\Omega ). \end{aligned}$$

We define the associated energy functional to problem (1.1) as follows:

$$\begin{aligned} E(u)={}&\frac{1}{2} \Vert u \Vert ^{2}+ \int _{\Omega}\bigl(\chi _{\{u>1\}}(x)-\alpha G \bigl(x,(u-1)_{+}\bigr)\bigr)\,dx \\ &{}- \frac{\beta}{1-\gamma} \int _{\Omega}\bigl(u^{+}\bigr)^{1-\gamma}\,dx\quad \text{for all } u\in H_{0}^{1}(\Omega ), \end{aligned}$$

where \(F(x,t)=\int _{0}^{t}f(x,t)\,dt\), \(t\geq 0\).

The functional E fails to be of \(C^{1}\) class due to the term \(\int _{\Omega}(u^{+})^{1-\gamma}\,dx\). Moreover, it is nondifferentiable due to the term \(\int _{\Omega}\chi _{\{u>1\}}(x)\,dx\). We shall first tackle the singular term by defining a cut-off function \(\phi _{\beta}\) as follows:

$$ \phi _{\beta}(u)= \textstyle\begin{cases} u^{-\gamma} & \text{if } u>u_{\beta}, \\ u_{\beta}^{-\gamma} & \text{if } u\leq u_{\beta}. \end{cases} $$

Here \(u_{\beta}\) is a solution of the following problem:

$$\begin{aligned} \begin{aligned}& -\Delta u=\beta u^{-\gamma} \quad \text{in } \Omega, \\ &u>0\quad \text{in } \Omega, \\ &u=0\quad \text{on } \partial \Omega . \end{aligned} \end{aligned}$$

The existence of \(u_{\beta}\) 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 Sect. 6). Note that we call (1.7) a singular problem. We denote \(\Phi _{\beta}(u)=\int _{0}^{u}\phi _{\beta}(t)\,dt\).

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 \(h:\mathbb{R}\rightarrow [0,2]\) as follows:

$$ h(t)= \textstyle\begin{cases} 0 & \text{if } t\leq 0, \\ \text{a positive function} & \text{if } 0< t< 1, \\ 0 & \text{if } t\geq 1, \end{cases} $$

and \(\int _{0}^{1}h(t)\,dt=1\). We let \(H(t)=\int _{0}^{t}h(t)\,dt\). Clearly, H is a smooth and nondecreasing function such that

$$ H(t)= \textstyle\begin{cases} 0 & \text{if } t\leq 0, \\ \text{a positive function}< 1 & \text{if } 0< t< 1, \\ 1 & \text{if } t\geq 1. \end{cases} $$

We further define for \(\delta >0\)

$$\begin{aligned} f_{\delta}(x,t)=H \biggl(\frac{t}{\delta} \biggr)f(x,t),\qquad F_{\delta}(x,t)= \int _{0}^{t}f_{\delta}(x,t)\,dt \quad \text{for all } t\geq 0. \end{aligned}$$


$$\begin{aligned} E_{\delta}(u)={}&\frac{1}{2} \Vert u \Vert ^{2} \\ &{}+ \int _{\Omega} \biggl[H \biggl( \frac{u-1}{\delta} \biggr)-\alpha F_{\delta}\bigl(x,(u-1)_{+}\bigr)-\beta \Phi _{ \beta}(u) \biggr]\,dx \quad \text{for all } u\in H_{0}^{1}(\Omega ). \end{aligned}$$

The functional \(E_{\delta}\) is of \(C^{1}\) class. The main result of this paper is the following theorem.

Theorem 1.1

Let conditions \((f_{1})-(f_{2})\) hold. Then there exist \(\Lambda , \beta _{*}>0\) such that for all \(\alpha >\Lambda \), \(0<\beta <\beta _{*}\) problem (1.1) has two Lipschitz continuous solutions, say \(u_{1},u_{2}\in H_{0}^{1}(\Omega )\cap C^{2}(\bar{\Omega}\setminus G(u))\), satisfying (1.1) classically in \(\bar{\Omega}\setminus G(u)\). These solutions also satisfy the free boundary condition in the generalized sense and vanish continuously on Ω. Furthermore,

  1. 1.

    \(E(u_{1})<-|\Omega |\leq -|\{u:u=1\}|<E(u_{2})\), where \(|\cdot |\) denotes the Lebesgue measure in \(\mathbb{R}^{N}\), hence \(u_{1}\), \(u_{2}\) are nontrivial solutions.

  2. 2.

    \(0< u_{2}\leq u_{1}\) and the regions \(\{u_{1}:u_{1}<1\}\subset \{u_{2}:u_{2}<1\}\) are connected where Ω is connected. The sets \(\{u_{2}>1\}\subset \{u_{1}>1\}\) are nonempty.

  3. 3.

    \(u_{1}\) is a minimizer of E (but \(u_{2}\) is not).

The paper is organized as follows. In Sect. 2 we introduce the key preliminary facts. In Sect. 3 we prove a convergence lemma. In Sect. 4 we prove a free boundary condition. In Sect. 5 we prove two auxiliary lemmas. In Sect. 6 we prove a result on positive Radon measure. Finally, in Sect. 7 we prove the main theorem.

2 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. [7, Theorem 5.1].

Lemma 2.1

Let u be a Lipschitz continuous function on the unit ball \(B_{1}(0)\subset \mathbb{R}^{N}\) satisfying the distributional inequalities

$$ \pm \Delta u\leq A \biggl(\frac {1}{\delta}\chi _{\{ \vert u-1 \vert < \delta \}}(x)H\bigl( \vert \nabla u \vert \bigr)+1 \biggr) $$

for constants \(A>0\), \(0<\delta \leq 1\), H is a continuous function obeying \(H(t)=o(t^{2})\) as \(t\to \infty \). Then there exists a constant \(C>0\) depending on N, A and \(\int _{{B_{1}}(0)}u^{2}\,dx\), but not on δ, such that

$$ \sup_{x\in B_{\frac{1}{2}}(0)} \bigl\vert \nabla u(x) \bigr\vert \leq C. $$

The following are the Palais–Smale condition and the mountain pass theorem.

Definition 2.1

(cf. Kesavan [15, Definition \(5.5.1\)])

Let V be a Banach space and \(J:V\rightarrow \mathbb{R}\) be a \(C^{1}\)-functional. Then J is said to satisfy the Palais–Smale \((PS)\) condition if the following holds: Whenever \((u_{n})\) is a sequence in V such that \((J(u_{n}))\) is bounded and \((J'(u_{n}))\rightarrow 0\) strongly in \(V^{*}\) (the dual space), then \((u_{n})\) has a strongly convergent subsequence in V.

Lemma 2.2

(cf. Alt and Caffarelli [3, Theorem 2.1])

Let J be a \(C^{1}\)-functional defined on a Banach space V. Assume that J satisfies the \((PS)\)-condition and that there exists an open set \(U\subset V\), \(v_{0}\in U\), and \(v_{1}\in X\setminus \bar{U}\) such that

$$ \inf_{v\in \partial U}J(v)>\max \bigl\{ J(v_{0}),J(v_{1}) \bigr\} . $$

Then J has a critical point at the level

$$ c=\inf_{\psi \in \Gamma} \max_{u\in \psi ([0,1])}{J(v)} \geq \inf _{u\in \partial U}J(u), $$

where \(\Gamma =\{\psi \in C([0,1]):\psi (0)=v_{0},\psi (1)=v_{1}\}\) is the class of paths in V joining \(v_{0}\) and \(v_{1}\).

Before we prove Lemma 3.1, we would like to give an a priori estimate of the parameter β.

3 Convergence lemma

We denote the first eigenvalue of \((-\Delta )\) by \(\alpha _{1}\) and the first eigenvector by \(\varphi _{1} \) (for an existence of \(\alpha _{1}\), \(\varphi _{1}\), refer to Kesavan [15]). Fix α to, say, \(\alpha _{0}\) and let β be any positive real number. On testing problem (1.1) with \(\varphi _{1}\), the following weak formulation has to hold if u is a weak solution of problem (1.1). Thus

$$\begin{aligned} \begin{aligned} \alpha _{1} \int _{\Omega}u\varphi _{1}\,dx&= \int _{\Omega}\nabla u \cdot \nabla \varphi \,dx=\alpha \int _{\Omega}f\bigl(x,(u-1)_{+}\bigr)\varphi _{1}\,dx+ \beta \int _{\Omega}\bigl(u^{+}\bigr)^{-\gamma}\varphi \,dx. \end{aligned} \end{aligned}$$

So there exists \(\beta _{*}>0\), which depends on the chosen fixed α, such that \(\beta _{*}t^{-\gamma}+\alpha f(x,(t-1)_{+})>\alpha _{1} t\) for all \(t>0\). This is a contradiction to (3.1). Therefore, \(0<\beta <\beta _{*}\).

Lemma 3.1

Let conditions \((f_{1})-(f_{2})\) hold, \(\delta _{j}\rightarrow 0\) (\(\delta _{j}>0\)) as \(j\rightarrow \infty \), and let \(u_{j}\) be a critical point of \(E_{\delta _{j}}\). If \((u_{j})\) is bounded in \(H_{0}^{1}(\Omega )\cap L^{\infty}(\Omega )\), then there exists a Lipschitz continuous function u on Ω̄ such that \(u\in H_{0}^{1}(\Omega )\cap C^{2}(\bar{\Omega}\setminus G(u))\) and a subsequence such that

  1. (i)

    \(u_{j}\rightarrow u\) uniformly over Ω̄,

  2. (ii)

    \(u_{j}\rightarrow u\) locally in \(C^{1}(\bar{\Omega}\setminus \{u=1\})\),

  3. (iii)

    \(u_{j}\rightarrow u\) strongly in \(H_{0}^{1}(\Omega )\),

  4. (iv)

    \(E(u)\leq \lim \inf E_{\delta _{j}}(u_{j})\leq \lim \sup E_{\delta _{j}}(u_{j}) \leq E(u)+|\{u:u=1\}|\), i.e., u is a nontrivial function if \(\lim \inf E_{\delta _{j}}(u_{j})<0\) or \(\lim \sup E_{\delta _{j}}(u_{j})>0\).

Furthermore, u satisfies

$$ -\Delta u=\alpha \chi _{\{u>1\}}(x)g\bigl(x,(u-1)_{+}\bigr)+\beta u^{-\gamma} $$

classically in \(\Omega \setminus G(u)\), 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 \(0<\delta _{j}<1\). Consider the following problem:

$$\begin{aligned} \textstyle\begin{cases} -\Delta u_{j}=-\frac{1}{\delta _{j}}h ( \frac{u_{j}-1}{\delta _{j}} )+\alpha f_{\delta _{j}}(x,(u_{j}-1)_{+})+ \beta \phi _{\beta}(u_{j}) &\text{in } \Omega, \\ u_{j}>0 &\text{in } \Omega, \\ u_{j}=0 &\text{on } \partial \Omega . \end{cases}\displaystyle \end{aligned}$$

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 \((u_{j})\) is bounded in \(L^{\infty}(\Omega )\). Therefore, there exists \(C_{0}\) such that \(0\leq f_{\delta _{j}}(x,(u_{j}-1)_{+})\leq C_{0}\). Let \(\varphi _{0}\) be a solution of the following problem:

$$\begin{aligned} \textstyle\begin{cases} -\Delta \varphi _{0}=\alpha C_{0}+\beta u_{\beta}^{-\gamma} &\text{in } \Omega , \\ \varphi _{0}=0 &\text{on } \partial \Omega . \end{cases}\displaystyle \end{aligned}$$

Now, since \(h\geq 0\), we have that \(-\Delta u_{j}\leq \alpha C_{0}+\beta u_{\beta}^{-\gamma}=-\Delta \varphi _{0}\) in Ω. Therefore by the maximum principle,

$$\begin{aligned} 0\leq u_{j}(x)\leq \varphi _{0}(x) \quad \text{for all } x\in \Omega . \end{aligned}$$

From the argument used in the proof of Lemma 6.1, together with \(\beta _{*}>0\) and large \(\Lambda >0\), we conclude that \(u_{j}>u_{\beta}\) in Ω for all \(\beta \in (0,\beta _{*})\). Since \(\{u_{j}:u_{j}\geq 1\}\subset \{\varphi _{0}:\varphi _{0}\geq 1\}\), hence \(\varphi _{0}\) gives a uniform lower bound, say \(d_{0}\), on the distance from the set \(\{u_{j}:u_{j}\geq 1\}\) 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 \(\frac {d_{0}}{2}\) neighborhood of the boundary Ω. Obviously, \(0\leq h\leq 2\chi _{(-1,1)}\) and hence

$$\begin{aligned} \begin{aligned} \pm \Delta u_{j}&=\pm \frac{1}{\delta _{j}}h \biggl( \frac{u_{j}-1}{\delta _{j}} \biggr)\mp \alpha f_{\delta _{j}} \bigl(x,(u_{j}-1)_{+}\bigr)+ \beta u_{j}^{-\gamma} \\ &\leq \frac{2}{\delta _{j}}\chi _{\{|u_{j}-1|< \delta _{j}\}}(x)+ \alpha C_{0}+\beta u_{j}^{-\gamma} \\ &\leq \frac{2}{\delta _{j}}\chi _{\{|u_{j}-1|< \delta _{j}\}}(x)+ \alpha C_{0}+\beta u_{\beta}^{-\gamma}. \end{aligned} \end{aligned}$$

By Lazer and McKenna [16], for any subset K of Ω that is relatively compact in it, i.e., \(K\Subset \Omega \), we have that \(u_{\beta}\geq C_{K}\) for some \(C_{K}>0\). Therefore

$$\begin{aligned} \begin{aligned} \pm \Delta u_{j}&\leq \frac{2}{\delta _{j}}\chi _{\{|u_{j}-1|< \delta _{j} \}}(x)+\alpha C_{0}+\beta C_{K}^{-\gamma}. \end{aligned} \end{aligned}$$

Since \((u_{j})\) is bounded in \(L^{2}(\Omega )\) and by Lemma 2.1, it follows that there exists \(A>0\) such that

$$\begin{aligned} \sup_{x\in B_{\frac{r}{2}}(x_{0})} \bigl\vert \nabla u_{j}(x) \bigr\vert &\leq \frac{A}{r} \end{aligned}$$

for suitable \(r>0\) such that \(B_{r}(0)\subset \Omega \). Therefore, \((u_{j})\) is uniformly Lipschitz continuous on the compact subsets of Ω such that its distance from the boundary Ω is at least \(\frac{d_{0}}{2}\) 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 \(\frac{d_{0}}{2}\)-neighborhood of Ω. By the Eberlein–Šmulian theorem, we can conclude that \(u_{j}\rightharpoonup u\) in \(H_{0}^{1}(\Omega )\).

We now prove that u satisfies the following equation:

$$ -\Delta u=\alpha \chi _{\{u>1\}}(x)f\bigl(x,(u-1)_{+}\bigr)+\beta u^{-\gamma} $$

in the set \(\{u\neq 1\}\). This will include the cases (i) \(0< u_{\beta}<1<u\), (ii) \(1< u_{\beta}< u\), (iii) \(0< u_{\beta}< u<1\). The cases (i)–(iii) do not pose any real mathematical obstacle. Let \(\varphi \in C_{0}^{\infty}(\{u>1\})\). Then \(u\geq 1+2\delta \) on the support of φ for some \(\delta >0\). Using the convergence of \(u_{j}\) to u uniformly on Ω, we have \(|u_{j}-u|<\delta \) for any sufficiently large \(j,\delta _{j}<\delta \). So \(u_{j}\geq 1+\delta _{j}\) on the support of φ. Testing (3.1) with φ yields

$$\begin{aligned} \int _{\Omega}\nabla u_{j}\cdot \nabla \varphi \,dx&= \alpha \int _{ \Omega}f(x,u_{j}-1)\varphi \,dx+\beta \int _{\Omega} u_{j}^{-\gamma} \varphi \,dx. \end{aligned}$$

On passing to the limit \(j\rightarrow \infty \), we get

$$\begin{aligned} \int _{\Omega}\nabla u\cdot \nabla \varphi \,dx&=\alpha \int _{\Omega}f(x,u-1) \varphi \,dx+\beta \int _{\Omega} u^{-\gamma}\varphi \,dx. \end{aligned}$$

To arrive at (3.9), we have used the weak convergence of \(u_{j}\) to u in \(H_{0}^{1}(\Omega )\) and the uniform convergence of the same in Ω. Hence u is a weak solution of \(-\Delta u=\alpha f(x,u-1)+\beta u^{-\gamma}\) 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 \(-\Delta u=\alpha f(x,u-1)+\beta u^{-\gamma}\) in \(\{u:u>1\}\). Similarly, on choosing \(\varphi \in C_{0}^{\infty}(\{u:u<1\})\), one can find a \(\delta >0\) such that \(u\leq 1-2\delta \). Therefore, \(u_{j}<1-\delta \). Using the arguments as in (3.8) and (3.9), we find that u satisfies \(-\Delta u=\beta u^{-\gamma}\) in the set \(\{u:u<1\}\).

Let us now see what is the nature of u in the set \(\{u:u\leq 1\}^{\circ}\). On testing (3.1) with any nonnegative function, passing to the the limit \(j\rightarrow \infty \), and using the fact that \(h\geq 0\), \(H\leq 1\), we can show that u satisfies

$$\begin{aligned} -\Delta u&\leq \alpha f\bigl(x,(u-1)_{+}\bigr)+\beta u^{-\gamma}\quad \text{in } \Omega \end{aligned}$$

in the distributional sense. Also, we see that u satisfies \(-\Delta u=\beta u^{-\gamma}\) in the set \(\{u:u<1\}\). Furthermore, \(\mu =\Delta u+\beta u^{-\gamma}\) is a positive Radon measure supported on \(\Omega \cap \partial \{u:u<1\}\) (refer to Lemma 6.2 in Sect. 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 \(u\in W_{\mathrm{loc}}^{2,p}(\{u:u\leq 1\}^{\circ})\), \(1< p<\infty \). Thus μ is supported on \(\Omega \cap \partial \{u:u<1\}\cap \partial \{u:u>1\}\) and u satisfies \(-\Delta u=\beta u^{-\gamma}\) in the set \(\{u:u\leq 1\}^{\circ}\).

To prove (ii), we show that \(u_{j}\rightarrow u\) locally in \(C^{1}(\Omega \setminus \{u:u=1\})\). Note that we have already proved that \(u_{j}\rightarrow u\) in the \(C^{2}\) norm in a neighborhood of Ω of Ω̄. Suppose that \(M\subset \subset \{u:u>1\}\). In this set M we have \(u\geq 1+2\delta \) for some \(\delta >0\). Thus, for sufficiently large j with \(\delta _{j}<\delta \), we have \(|u_{j}-u|<\delta \) in Ω, and hence \(u_{j}\geq 1+\delta _{j}\) in M. From (3.2) we derive that

$$ -\Delta u=\alpha f(x,u-1)+\beta u^{-\gamma} \quad \text{in } M. $$

Clearly, \(f(x,u_{j}-1)\rightarrow f(x,u-1)\) in \(L^{p}(\Omega )\) for \(1< p<\infty \) because f is a locally Hölder continuous function and \(u_{j}\rightarrow u\) uniformly in Ω. Our analysis says something stronger. Since \(-\Delta u=\alpha f(x,u-1)\) in M, we have that \(u_{j}\rightarrow u\) in \(W^{2,p}(M)\). By the embedding \(W^{2,p}(M)\hookrightarrow C^{1}(M)\) for \(p>2\), we have that \(u_{j}\rightarrow u\) in \(C^{1}(M)\). This shows that \(u_{j}\rightarrow u\) in \(C^{1}(\{u>1\})\). Working along similar lines we can also show that \(u_{j}\rightarrow u\) in \(C^{1}(\{u:u<1\})\).

We shall now prove (iii). Since \(u_{j}\rightharpoonup u\) in \(H_{0}^{1}(\Omega )\), we know that by the weak lower semicontinuity of the norm \(\|\cdot \|\),

$$ \Vert u \Vert \leq \lim \inf \Vert u_{j} \Vert . $$

It suffices to prove that \(\lim \sup \|u_{j}\|\leq \|u\|\). To achieve this, we multiply (3.2) with \(u_{j}-1\) and then integrate by parts. We shall also use the fact that \(th (\frac{t}{\delta _{j}} )\geq 0\) for any t. This gives

$$\begin{aligned} \begin{aligned} \int _{\Omega} \vert \nabla u_{j} \vert ^{2}\,dx\leq{}& \alpha \int _{\Omega}f\bigl(x,(u_{j}-1)_{+}\bigr) (u_{j}-1)_{+}\,dx \\ &{}- \int _{\partial \Omega}\frac{\partial u_{j}}{\partial \hat{n}}\,dS+ \beta \int _{\Omega}u_{j}^{-\gamma}(u_{j}-1)_{+}\,dx \\ \rightarrow{}& \alpha \int _{\Omega}f\bigl(x,(u-1)_{+}\bigr) (u-1)_{+}\,dx \\ &{}- \int _{ \partial \Omega}\frac{\partial u}{\partial \hat{n}}\,dS+\beta \int _{ \Omega}u^{-\gamma}(u-1)_{+}\,dx \end{aligned} \end{aligned}$$

as \(j\rightarrow \infty \). Here, is the outward drawn normal to Ω. We saw earlier that u is a weak solution to \(-\Delta u=\alpha f(x,u-1)+\beta u^{-\gamma}\) in \(\{u:u>1\}\). Let \(0<\delta <1\). We test this equation with the function \(\varphi =(u-1-\delta )_{+}\) and get

$$\begin{aligned} \int _{\{u>1+\delta \}} \vert \nabla u \vert ^{2}\,dx&=\alpha \int _{\Omega}f\bigl(x,(u-1)_{+}\bigr) (u-1- \delta )\,dx+ \beta \int _{\Omega}u^{-\gamma}(u-1-\delta )_{+}\,dx. \end{aligned}$$

Integrating \((u-1-\delta )_{-}\Delta u=\beta u^{-\gamma}(u-1-\delta )_{-}\) over Ω yields

$$\begin{aligned} \int _{u< 1-\delta} \vert \nabla u \vert ^{2}\,dx&=-(1- \delta ) \int _{\partial \Omega}\frac{\partial u}{\partial \hat{n}}\,dS+\beta \int _{\Omega}u^{- \gamma}(u-1-\delta )_{-}\,dx. \end{aligned}$$

On adding (3.12) and (3.13) and passing to the limit \(\delta \rightarrow 0\), we get

$$\begin{aligned} \int _{\Omega} \vert \nabla u \vert ^{2}\,dx=&\alpha \int _{\Omega}f\bigl(x,(u-1)_{+}\bigr) (u-1)_{+}\,dx \\ &{}- \int _{\partial \Omega}\frac{\partial u}{\partial \hat{n}}\,dS+\beta \int _{\Omega}u^{-\gamma}(u-1)_{+}\,dx. \end{aligned}$$

Note that we have used \(\int _{\{u:u=1\}}|\nabla u|^{2}\,dx=0\). Invoking (3.14) and (3.11), we get

$$\begin{aligned} \lim \sup \int _{\Omega} \vert \nabla u_{j} \vert ^{2}\,dx&\leq \int _{\Omega} \vert \nabla u \vert ^{2}\,dx. \end{aligned}$$

This proves (iii).

We shall now prove (iv). Consider

$$\begin{aligned} E_{\delta _{j}}(u_{j})={}& \int _{\Omega} \biggl(\frac{1}{2} \vert \nabla u_{j} \vert ^{2}+H \biggl(\frac{u_{j}-1}{\delta _{j}} \biggr)\chi _{\{u\neq 1}\}-\alpha F_{ \delta _{j}}\bigl(x,(u_{j}-1)_{+} \bigr)-\beta u_{j}^{-\gamma}(u_{j}-1)_{+} \biggr)\,dx \\ &{}+ \int _{\{u=1\}}H \biggl(\frac{u_{j}-1}{\delta _{j}} \biggr)\,dx. \end{aligned}$$

Since \(u_{j}\rightarrow u\) in \(H_{0}^{1}(\Omega )\) and \(H (\frac{u_{j}-1}{\delta _{j}} )\chi _{\{u\neq 1\}}\), \(F_{\delta _{j}}(x,(u_{j}-1)_{+})\) are bounded and converge pointwise to \(\chi _{\{u:u>1\}}\) and \(F(x,(u-1)_{+})\), respectively, it follows that the first integral in (3.16) converges to \(E(u)\). Moreover,

$$ 0\leq \int _{\{u:u=1\}}H \biggl(\frac{u_{j}-1}{\delta _{j}} \biggr)\,dx \leq \bigl\vert \{u:u=1\} \bigr\vert . $$

This proves (iv). □

4 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 \(\vec{\varphi}\in C_{0}^{1}(\Omega ,\mathbb{R}^{N})\) such that \(u\neq 1\) a.e. on the support of φ⃗. Multiplying \(\nabla u_{j}\cdot \vec{\varphi}\) to (3.2) and integrating over the set \(\{u:1-\epsilon ^{-}< u<1+\epsilon ^{+}\}\) gives

$$\begin{aligned} \begin{aligned} &\int _{\{u:1-\epsilon ^{-}< u< 1+\epsilon ^{+}\}} \biggl[-\Delta u_{j}+ \frac{1}{\delta _{j}}h \biggl(\frac{u_{j}-1}{\delta _{j}} \biggr) \biggr]\nabla u_{j}\cdot \vec{ \varphi}\,dx \\ &\quad = \int _{\{u:1-\epsilon ^{-}< u< 1+\epsilon ^{+}\}}\bigl(\alpha f_{\delta _{j}}\bigl(x,(u_{j}-1)_{+} \bigr)+ \beta u_{j}^{-\gamma}\bigr)\nabla u_{j}\cdot \vec{\varphi}\,dx. \end{aligned} \end{aligned}$$

The term on the left-hand side of (4.1) can be expressed as follows:

$$\begin{aligned} &\nabla \cdot \biggl(\frac{1}{2} \vert \nabla u_{j} \vert ^{2}\vec{\varphi}-( \nabla u_{j} \cdot \vec{\varphi})\nabla u_{j} \biggr)+\nabla u_{j} \cdot (\nabla \vec{\varphi}\cdot \nabla u_{j}) \\ &\quad {}-\frac{1}{2} \vert \nabla u_{j} \vert ^{2} \nabla \cdot \vec{\varphi}+\nabla H \biggl(\frac{u_{j}-1}{\delta _{j}} \biggr)\cdot \vec{\varphi}. \end{aligned}$$

Using this, we integrate by parts to obtain

$$\begin{aligned} \begin{aligned} &\int _{\{u:u=1+\epsilon ^{+}\}\cup \{u=1-\epsilon ^{-}\}} \biggl[ \frac{1}{2} \vert \nabla u_{j} \vert ^{2}\vec{\varphi}-(\nabla u_{j} \cdot \vec{\varphi})\nabla u_{j}+H \biggl(\frac{u_{j}-1}{\delta _{j}} \vec{ \varphi} \biggr) \biggr]\cdot \hat{n}\,dx \\ &\quad = \int _{\{u:1-\epsilon ^{-}< u< 1+\epsilon ^{+}\}} \biggl(\frac{1}{2} \vert \nabla u_{j} \vert ^{2}\vec{\varphi}-(\nabla u_{j} \cdot \vec{\varphi}) \nabla u_{j} \biggr)\,dx \\ &\qquad {}+ \int _{\{u:1-\epsilon ^{-}< u< 1+\epsilon ^{+}\}} \biggl[H \biggl( \frac{u_{j}-1}{\delta _{j}} \biggr)\nabla \cdot \vec{\varphi}+\alpha f_{ \delta _{j}}\bigl(x,(u_{j}-1)_{+} \bigr)\nabla u_{j}\cdot \vec{\varphi} \\ &\qquad {}+\beta u_{j}^{- \gamma} \nabla u_{j}\cdot \vec{\varphi} \biggr]\,dx. \end{aligned} \end{aligned}$$

By using (ii), the integral on the left of equation (4.3) converges to

$$\begin{aligned} \int _{\{u:u=1+\epsilon ^{+}\}\cup \{u=1-\epsilon ^{-}\}} \biggl( \frac{1}{2} \vert \nabla u \vert ^{2}\varphi -(\nabla u\cdot \vec{\varphi}) \nabla u \biggr)\cdot \hat{n}\,dS+ \int _{\{u:u=1+\epsilon ^{+}\}} \vec{\varphi}\cdot \hat{n}\,dS. \end{aligned}$$

Equation (4.4) is further equal to

$$\begin{aligned} \int _{\{u:u=1+\epsilon ^{+}\}} \biggl(1-\frac{1}{2} \vert \nabla u \vert ^{2} \biggr)\vec{\varphi}\cdot \hat{n}\,dS- \int _{\{u:u=1-\epsilon ^{-}\}} \frac{1}{2} \vert \nabla u \vert ^{2}\vec{\varphi}\cdot \hat{n}\,dS. \end{aligned}$$

This is because \(\hat{n}=\pm \frac {\nabla u}{|\nabla u|}\) on the set \(\{u:u=1+\epsilon ^{\pm}\}\cup \{u:u=1-\epsilon ^{\pm}\}\). By using (iii), the first integral on the right-hand side of (4.3) converges to

$$\begin{aligned} \int _{\{u:1-\epsilon ^{-}< u< 1+\epsilon ^{+}\}} \biggl(\frac{1}{2} \vert \nabla u \vert ^{2}\nabla \cdot \vec{\varphi}-\nabla u D\vec{\varphi}\cdot \nabla u \biggr)\,dx, \end{aligned}$$

whereas the second integral of (4.3) is bounded by

$$\begin{aligned} \int _{\{u:1-\epsilon ^{-}< u< 1+\epsilon ^{+}\}}\bigl( \vert \nabla \cdot \vec{\varphi} \vert +C \vert \vec{\varphi} \vert \bigr)\,dx \end{aligned}$$

for some constant \(C>0\). The last two integrals (4.6)–(4.7) vanish as \(\epsilon ^{\pm}\rightarrow 0\) since \(|\text{supp}(\vec{\varphi})\cap \{u:u=1\}|=0\). Therefore we first let \(j\rightarrow \infty \) and then we let \(\epsilon ^{\pm}\rightarrow 0\) in (4.3) to prove that u satisfies the free boundary condition.

Using \((f_{1})\),

$$\begin{aligned} E_{\delta}(u)&\geq \int _{\Omega} \biggl\{ \frac{1}{2} \vert \nabla u \vert ^{2}- \alpha \biggl(c_{0}(u-1)_{+}+ \frac{c_{1}}{p}(u-1)_{+}^{p} \biggr)- \frac{\beta}{1-\gamma} u^{1-\gamma} \biggr\} \,dx. \end{aligned}$$

Clearly, since \(1< p<2\), we have that \(E_{\delta}\) is bounded from below and coercive. Thus \(E_{\delta}\) satisfies the \((PS)\) condition (see Definition 2.1). It is easy to see that every \((PS)\) 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 \(E_{\delta}\) has a minimizer, say, \(u_{1}^{\delta}\). By \((f_{2})\), we have \(F(x,t)>0\) for all \(x\in \Omega \) and \(t>0\). Thus, for any \(u\in H_{0}^{1}(\Omega )\) with \(u>1\) on a set of positive measure, we have

$$\begin{aligned} \int _{\Omega}F\bigl(x,(u-1)_{+}\bigr)\,dx&>0. \end{aligned}$$

Therefore, \(E(u)\rightarrow -\infty \) as \(\alpha \rightarrow \infty \). Thus, there exists \(\Lambda >0\) such that for all \(\alpha >\Lambda \) we have

$$\begin{aligned} m_{1}(\alpha )&=\inf_{u\in H_{0}^{1}(\Omega )}\bigl\{ E(u)\bigr\} < - \vert \Omega \vert . \end{aligned}$$


$$ \delta _{0}(\alpha )=\min \biggl\{ \frac{ \vert m_{1}(\alpha ) \vert }{2\alpha c_{0} \vert \Omega \vert }, \biggl( \frac{pc_{0}}{c_{1}} \biggr)^{\frac{1}{p-1}} \biggr\} . $$

5 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.

Lemma 5.1

For all \(\alpha >\Lambda \), \(0<\beta <\beta _{*}\), \(\delta <\delta _{0}(\alpha )\), the functional \(E_{\delta}\) has a minimizer \(u_{1}^{\delta}>0\) that satisfies

$$\begin{aligned} E_{\delta}\bigl(u_{1}^{\delta}\bigr)&\leq m_{1}(\alpha )+2\alpha \delta c_{0} \vert \Omega \vert < 0. \end{aligned}$$


Since \(E_{\delta}\) is bounded below and satisfies the \((PS)\) condition, it possesses a minimizer \(u_{1}^{\delta}\). Also, since \(H (\frac{t-1}{\delta} )\leq \chi _{(1,\infty )}(t)\) for all t, we have

$$\begin{aligned} \begin{aligned} E_{\delta}(u)-E(u)&\leq \alpha \int _{\Omega}\bigl[F\bigl(x,(u-1)_{+} \bigr)-F_{ \delta}\bigl(x,(u-1)_{+}\bigr)\bigr]\,dx \\ &=\alpha \int _{\Omega} \int _{0}^{(u-1)_{+}} \biggl[1-H \biggl( \frac{t}{\delta}f(x,t) \biggr) \biggr]\,dt\,dx \\ &\leq \alpha \int _{\Omega} \int _{0}^{\delta}f(x,t)\,dt\,dx \\ &\leq \alpha \biggl(c_{0}\delta +\frac{c_{1}}{p}\delta ^{p} \biggr) \vert \Omega \vert \quad \text{by } (f_{1}). \end{aligned} \end{aligned}$$

Further, for \(\delta <\delta _{0}(\alpha )\) we obtain (5.1). Since \(E_{\delta}(u_{1}^{\delta})<0=E_{\delta}(0)\), this implies that \(u_{1}^{\delta}\) is a nontrivial solution of problem (3.2). This solution is positive since it is a minimizer. □

We shall now prove that the functional \(E_{\delta}\) has a second nontrivial critical point, say \(u_{2}^{\delta}\).

Lemma 5.2

For any \(\alpha >\Lambda \) and \(0<\beta <\beta _{*}\), there exists a constant \(c_{3}(\alpha )\) such that for all \(\delta <\delta _{0}(\alpha )\) the functional \(E_{\delta}\) has a second critical point \(0< u_{2}^{\delta}\leq u_{1}^{\delta}\) that obeys

$$ c_{3}(\alpha )\leq E_{\delta}\bigl(u_{2}^{\delta} \bigr)\leq \frac{1}{2} \bigl\Vert u_{1}^{ \delta} \bigr\Vert ^{2}+ \vert \Omega \vert . $$

Furthermore, \(\emptyset \neq \{u_{2}^{\delta}:u_{2}^{\delta}>1\}\subset \{u_{1}^{ \delta}:u_{1}^{\delta}>1\}\).


Choose some \(\delta <\delta _{0}(\alpha )\). Consider

$$\begin{aligned}& h_{\delta}(x,t)=\frac{1}{\delta}h \biggl( \frac{\min \{t,u_{1}^{\delta}(x)\}-1}{\delta} \biggr),\qquad H_{\delta}(x,t)= \int _{0}^{t}h_{\delta}(x,t)\,dt,\\& \tilde{f}_{\delta}(x,t)=f_{\delta}\bigl(x,\bigl(\min \bigl\{ t,u_{1}^{\delta}(x)\bigr\} -1\bigr)_{+}\bigr),\qquad \tilde{F}_{\delta}(x,t)= \int _{0}^{t}\tilde{f}_{\delta}(x,t)\,dt. \end{aligned}$$

Further, we set

$$ \tilde{E}_{\delta}(u)= \int _{\Omega} \biggl[\frac{1}{2} \vert \nabla u \vert ^{2}+H_{ \delta}(x,u)-\alpha \tilde{F}_{\delta}(x,u)-\beta \phi _{\beta}(u) \biggr]\,dx,\quad u\in H_{0}^{1}(\Omega ). $$

The functional \(\tilde{E}_{\delta}\) is of \(C^{1}\) class and its critical points coincide with the weak solutions of the following problem:

$$\begin{aligned} \begin{aligned} \textstyle\begin{cases} -\Delta u=-h_{\delta}(x,u)+\alpha \tilde{f}_{\delta}(x,u)+\beta \phi _{\beta}(u)& \text{in } \Omega, \\ u=0 &\text{on } \partial \Omega . \end{cases}\displaystyle \end{aligned} \end{aligned}$$

By the elliptic (Schauder) regularity, a weak solution of (5.3) is also a classical solution. Also, by the maximum principle, we have that \(u\leq u_{1}^{\delta}\). Thus u is a weak solution of problem (3.3) and hence is a critical point of \(\tilde{E}_{\delta}\) with \(\tilde{E}_{\delta}(u)=E_{\delta}(u)\). We shall now show that \(\tilde{E}_{\delta}\) has a critical point, say \(u_{2}^{\delta}\), that satisfies

$$\begin{aligned} m_{2}(\alpha )\leq \tilde{E}_{\delta} \bigl(u_{2}^{\delta}\bigr)\leq \frac{1}{2} \bigl\Vert u_{1}^{\delta} \bigr\Vert ^{2}+ \vert \Omega \vert \quad \text{for some } m_{2}(\alpha )>0. \end{aligned}$$

This enables us to conclude that \(E_{\delta}(u_{2}^{\delta})=\tilde{E}_{\delta}(u_{2}^{\delta})>0>E_{ \delta}(u_{1}^{\delta})\), which in turn will imply that \(u_{2}^{\delta}>0\) and different from \(u_{1}^{\delta}\).

By the mountain pass theorem (see Lemma 2.2), the functional \(\tilde{E_{\delta}}\) that is coercive (owing to its sublinear nature) satisfies the \((PS)\) condition. Clearly, for any \(t\leq 1\), we have

$$ \tilde{f}_{\delta}(x,t)=f_{\delta}(x,0) $$


$$ \tilde{f}_{\delta}(x,t)\leq c_{0}+c_{1}\bigl(\min \bigl\{ t,u_{1}^{\delta}(x)\bigr\} -1\bigr)_{+}^{p-1} \leq c_{0}+c_{1}(t-1)^{p-1} \quad \text{for } t>1. $$

By \((f_{1})\), we get

$$ \tilde{F}_{\delta}(x,t)\leq c_{0}(t-1)_{+}+ \frac{c_{1}}{p}(t-1)_{+}^{p} \leq \biggl(c_{0}+ \frac{c_{1}}{p} \biggr) \vert t \vert ^{q} $$

for all t with \(q>2\) if \(N=2\) and \(2< q\leq \frac{2N}{N-2}\) if \(N\geq 3\). We observe that

$$\begin{aligned} \tilde{E}_{\delta}(u)&\geq \int _{\Omega} \biggl[\frac{1}{2} \vert \nabla u \vert ^{2}- \alpha \biggl(c_{0}+\frac{c_{1}}{p} \biggr) \vert u \vert ^{q}-\beta \vert u \vert ^{1- \gamma} \biggr]\,dx \end{aligned}$$
$$\begin{aligned} &\geq \frac{1}{2} \Vert u \Vert ^{2}-\lambda c_{4} \biggl(c_{0}+\frac{c_{1}}{p} \biggr) \Vert u \Vert ^{q}-\beta c_{5} \Vert u \Vert ^{1-\gamma}. \end{aligned}$$

By the embedding result \(H_{0}^{1}(\Omega )\hookrightarrow L^{q}(\Omega )\) for \(q>2\), the integral in (5.5) is positive if \(\|u\|=r\), i.e., when \(u\in \partial B_{r}(0)\) for sufficiently small \(r>0\), where \(B_{r}(0)=\{u\in H_{0}^{1}(\Omega ):\|u\|< r\}\). Furthermore, since \(\tilde{E}_{\delta}(u_{1}^{\delta})=E_{\delta}(u_{1}^{\delta})<0= \tilde{E}_{\delta}(0)\), we choose \(r<\|u_{1}^{\delta}\|\), and then applying the mountain pass theorem (Lemma 2.2), we get a critical point \(u_{2}^{\delta}\) of \(\tilde{E}_{\delta}\) with

$$ \tilde{E}_{\delta}\bigl(u_{2}^{\delta}\bigr)=\inf _{\psi \in \Gamma} \max_{u\in \psi ([0,1])}\tilde{E}_{\delta}(u) \geq m_{2}( \alpha ), $$

where \(\Gamma =\{\psi \in C([0,1],H_{0}^{1}(\Omega )):\psi (0)=0,\psi (1)=u_{1}^{ \delta}\}\) is the class of paths joining 0 and \(u_{1}^{\delta}\). For the path \(\psi _{0}(t)=tu_{1}^{\delta}\), \(t\in [0,1]\), we have

$$\begin{aligned} \tilde{E}_{\delta}\bigl(\psi _{0}(t)\bigr)&\leq \int _{\Omega} \biggl( \frac{1}{2} \bigl\vert \nabla u_{1}^{\delta} \bigr\vert ^{2}+H_{\delta} \bigl(x,u_{1}^{\delta}\bigr) \biggr)\,dx \end{aligned}$$

since \(H_{\delta}(x,t)\) is nondecreasing in t and \(\tilde{F}_{\delta}(x,t)\geq 0\) for all t by condition \((f_{2})\). Since

$$\begin{aligned} H_{\delta}\bigl(x,u_{1}^{\delta}(x)\bigr)= \int _{0}^{u_{1}^{\delta}} \frac{1}{\delta}h \biggl( \frac{t-1}{\delta} \biggr)\,dt=H \biggl( \frac{u_{1}^{\delta}(x)-1}{\delta} \biggr)\leq 1, \end{aligned}$$

it follows by (5.7) and (5.8) that

$$\begin{aligned} \begin{aligned} \tilde{E}_{\delta} \bigl(u_{2}^{\delta}\bigr)&\leq \max_{u\in \psi _{0}([0,1])} \tilde{E}_{\delta}(u)\leq \int _{ \Omega} \biggl(\frac{1}{2} \bigl\vert \nabla u_{1}^{\delta} \bigr\vert ^{2}+1 \biggr)\,dx \\ &=\frac{1}{2} \bigl\Vert u_{1}^{\delta} \bigr\Vert ^{2}+ \vert \Omega \vert . \end{aligned} \end{aligned}$$


6 Positive Radon measure

We shall now prove two more results that will be needed in the last section.

Lemma 6.1

Let \(0<\beta <\beta _{*}\). Then a solution of the problem

$$\begin{aligned} \textstyle\begin{cases} -\Delta v=\beta v^{-\gamma}& \textit{in } \Omega , \\ v>0& \textit{in } \Omega , \\ v=0& \textit{on } \partial \Omega , \end{cases}\displaystyle \end{aligned}$$

say \(u_{\beta}\), satisfies \({u}_{\beta}< u\) a.e. in Ω, where u is a solution of problem (1.1).


Let \(u\in H_{0}^{1}(\Omega )\) be a positive solution of problem (1.1) and \(u_{\beta}>0\) be a solution of problem (6.1). For any \(0<\beta <\beta _{*}\), define a weak solution \(u_{\beta}\) of problem (6.1) as follows:

$$\begin{aligned} \begin{aligned} 0&= \int _{\Omega}\nabla u_{\beta}\cdot \nabla \varphi \,dx-\beta \int _{ \Omega}u_{\beta}^{-\gamma}\varphi \,dx \quad \text{for all } \varphi \in H_{0}^{1}(\Omega ). \end{aligned} \end{aligned}$$

By the Schauder estimates, we have \(u\in C^{2,a}(\Omega )\), and by Lazer and McKenna [16] we have \(u_{\beta}\in C^{2,a}(\Omega )\cap C(\bar{\Omega})\). We shall show that \(u\geq {u}_{\beta}\) a.e. in Ω. We let \(\tilde{\Omega}=\{x\in \Omega :u(x)<{u}_{\beta}(x)\}\). Thus, from the weak formulations satisfied by u, \({u}_{\beta}\) and testing with the function \(\varphi =(u_{\beta}-u)_{+}\), we have

$$\begin{aligned} \begin{aligned} 0\leq{}& \int _{\Omega}\nabla (u_{\beta}-u)\cdot \nabla (u_{\beta}-u)_{+}\,dx \\ ={}&- \alpha \int _{\Omega}\chi _{\{u>1\}}g\bigl(x,(u-1)_{+} \bigr) (u_{\beta}-u)_{+}\,dx \\ &{}+\beta \int _{\Omega}\bigl(u_{\beta}^{-\gamma}-u^{-\gamma} \bigr) (u_{\beta}-u)_{+}\,dx \leq 0. \end{aligned} \end{aligned}$$

Thus, \(\|(u_{\beta}-u)_{+}\|=0\) and hence \(|\tilde{\Omega}|=0\). However, since the functions u, \(u_{\beta}\) are continuous, it follows that \(\tilde{\Omega}=\emptyset \). Hence, by (6.3), we obtain \(u\geq \underline{u}_{\beta}\) in Ω.

Let \(W=\{x\in \Omega :u(x)=u_{\beta}(x)\}\). Since W is a measurable set, it follows that for any \(\eta >0\) there exists a closed subset V of W such that \(|W\setminus V|<\eta \). Further assume that \(|W|>0\). We now define a test function \(\varphi \in C_{c}^{1}(\mathbb{R}^{N})\) such that

$$ \varphi (x)= \textstyle\begin{cases} 1& \text{if } x\in V, \\ 0< \varphi < 1& \text{if } x\in W\setminus V, \\ 0& \text{if } x\in \Omega \setminus W. \end{cases} $$

Since u is a weak solution to (1.1), we have

$$\begin{aligned} \begin{aligned} 0={}& \int _{\Omega}-\Delta u\varphi \,dx-\beta \int _{V}u^{-\gamma}\,dx- \beta \int _{W\setminus V}u^{-\gamma}\varphi \,dx \\ &{}- \int _{V}f\bigl(x,(u-1)_{+}\bigr)\,dx- \int _{W\setminus V}f\bigl(x,(u-1)_{+}\bigr)\varphi \,dx \\ ={}&- \int _{V}f\bigl(x,(u-1)_{+}\bigr)\,dx- \int _{W\setminus V}f\bigl(x,(u-1)_{+}\bigr) \varphi \,dx< 0. \end{aligned} \end{aligned}$$

This is a contradiction. Therefore, \(|W|=0\), which implies that \(W=\emptyset \). Hence, \(u>u_{\beta}\) in Ω. □

Lemma 6.2

Function u is in \(H_{\mathrm{loc}}^{1,2}(\Omega )\) and the Radon measure \(\mu =\Delta u+\beta u^{-\gamma}\) is nonnegative and supported on \(\Omega \cap \{u:u<1\}\) for \(\beta \in (0,\beta _{*})\).


We follow the proof due to Alt and Caffarelli [2]. Choose \(\delta >0\), \(\beta \in (0,\beta _{*})\), and a test function \(\varphi ^{2}\chi _{\{u:u<1-\delta \}}\), where \(\varphi \in C_{0}^{\infty}(\Omega )\). Therefore,

$$\begin{aligned} \begin{aligned} 0={}&- \int _{\Omega}\nabla u\cdot \nabla \bigl(\varphi ^{2}\min \{u-1+ \delta ,0\}\bigr)\,dx \\ &{}+\beta \int _{\Omega}u^{-\gamma}\varphi ^{2}\min \{u-1+ \delta ,0\}\,dx \\ ={}& \int _{\Omega \cap \{u:u< 1-\delta \}}\nabla u\cdot \nabla \bigl(\varphi ^{2}(u-1+ \delta )\bigr)\,dx \\ &{}+\beta \int _{\Omega \cap \{u:u< 1-\delta \}}u^{-\gamma}\bigl( \varphi ^{2}(u-1+ \delta )\bigr)\,dx \\ ={}& \int _{\Omega \cap \{u:u< 1-\delta \}} \vert \nabla u \vert ^{2}\varphi ^{2}\,dx+2 \int _{\Omega \cap \{u:u< 1-\delta \}}\varphi \nabla u\cdot \nabla \varphi (u-1+\delta )\,dx \\ &{}+\beta \int _{\Omega \cap \{u:u< 1-\delta \}}u^{- \gamma}\bigl(\varphi ^{2}(u-1+ \delta )\bigr)\,dx. \end{aligned} \end{aligned}$$

By an application of integration by parts to the second term of (6.6), we get

$$\begin{aligned} \begin{aligned} &\int _{\Omega \cap \{u:u< 1-\delta \}} \vert \nabla u \vert ^{2}\varphi ^{2}\,dx \\ &\quad =-2 \int _{\Omega \cap \{u:u< 1-\delta \}}\varphi \nabla u\cdot \nabla \varphi (u-1+\delta )\,dx \\ &\qquad {}+\beta \int _{\Omega \cap \{u:u< 1-\delta \}}u^{- \gamma}\bigl(\varphi ^{2}(u-1+ \delta )\bigr)\,dx \\ &\quad \leq 4 \int _{\Omega}u^{2} \vert \nabla \varphi \vert ^{2}\,dx-\beta \int _{\Omega}u^{1- \gamma}\varphi ^{2}\,dx \\ &\quad \leq 4 \int _{\Omega}u^{2} \vert \nabla \varphi \vert ^{2}\,dx. \end{aligned} \end{aligned}$$

On passing to the limit \(\delta \rightarrow 0\), we conclude that \(u\in H_{\mathrm{loc}}^{1,2}(\Omega )\).

Furthermore, for nonnegative \(\zeta \in C_{0}^{\infty}(\Omega )\), we have

$$\begin{aligned} \begin{aligned} &- \int _{\Omega}\nabla \zeta \cdot \nabla u\,dx+\beta \int _{\Omega}u^{- \gamma}\zeta \,dx \\ &\quad = \biggl( \int _{\Omega \cap \{u:0< u< 1-2\delta \}}+ \int _{\Omega \cap \{u:1-2\delta < u< 1-\epsilon \}}+ \int _{\Omega \cap \{u:1-\delta < u< 1\}} \\ &\qquad {} + \int _{\Omega \cap \{u:u>1\}} \biggr) \\ & \qquad {}\times \biggl[\nabla \biggl(\zeta \max \biggl\{ \min \biggl\{ 2- \frac{1-u}{\delta},1 \biggr\} ,0 \biggr\} \biggr)\cdot \nabla u \\ &\qquad {} +\beta u^{-\gamma}\zeta \biggr]\,dx \\ &\quad \geq \int _{\Omega \cap \{u:1-2\delta < u< 1-\delta \}} \biggl[ \biggl(2- \frac{1-u}{\delta} \biggr)\nabla \zeta \cdot \nabla u+ \frac{\zeta}{\delta} \vert \nabla u \vert ^{2} \\ &\qquad {} +\beta u^{-\gamma}\zeta \biggr]\,dx. \end{aligned} \end{aligned}$$

On passing to the limit \(\delta \rightarrow 0\), we obtain \(\Delta (u-1)_{-}\geq 0\) in the distributional sense, and hence there exists a Radon measure μ (say) such that \(\mu =\Delta (u-1)_{-}\geq 0\). □

7 Proof of the main theorem

Finally, we are in a position to prove Theorem 1.1.

Proof of Theorem 1.1

Choose \(\alpha >\lambda \) and a sequence \(\delta _{j}\rightarrow 0\) such that \(\delta _{j}<\delta _{0}(\alpha )\). For each j, Lemma 5.1 gives a minimizer \(u_{1}^{\delta}>0\) of \(E_{\delta _{j}}\) that obeys

$$\begin{aligned} E_{\delta _{j}}\bigl(u_{1}^{\delta _{j}}\bigr)&\leq m_{1}(\alpha )+2\alpha \delta _{j} c_{0} \vert \Omega \vert < 0. \end{aligned}$$

Further, by Lemma 5.2, we can guarantee the existence of the second critical point \(0< u_{2}^{\delta}\leq u_{1}^{\delta _{j}}\) such that

$$\begin{aligned} m_{2}(\alpha )&\leq E_{\delta _{j}} \bigl(u_{2}^{\delta _{j}}\bigr)\leq \frac{1}{2} \bigl\Vert u_{1}^{\delta _{j}} \bigr\Vert ^{2}+ \vert \Omega \vert . \end{aligned}$$

The next step is to show that \((u_{1}^{\delta _{j}})\), \((u_{2}^{\delta _{j}})\) are bounded in \(H_{0}^{1}(\Omega )\cap L^{\infty}(\Omega )\). We shall then apply Lemma 3.1.

Since \(H\geq 0\) and

$$ H_{\delta}\bigl(x,(t-1)_{+}\bigr)\leq c_{0}(t-1)_{+}+ \frac{c_{1}}{p}(t-1)_{+}^{p} \leq \biggl(c_{0}+ \frac{c_{1}}{p} \biggr) \vert t \vert ^{p} $$

for all t by \((f_{1})\), it follows that

$$\begin{aligned} \frac{1}{2} \bigl\Vert u_{1}^{\delta} \bigr\Vert ^{2}&\leq E_{\delta}\bigl(u_{1}^{\delta} \bigr)+ \alpha \biggl(c_{0}+\frac{c_{1}}{p} \biggr) \int _{\Omega}\bigl(u_{1}^{ \delta} \bigr)^{p}\,dx+\beta \int _{\Omega}\bigl(u_{1}^{\delta} \bigr)^{1-\gamma}\,dx. \end{aligned}$$

Since \(E_{\delta _{j}}(u_{1}^{\delta})<0\) by (7.1) and \(p<2\), we have that \((u_{1}^{\delta _{j}})\) is bounded in \(H_{0}^{1}(\Omega )\).

Since \(f_{\delta}(x,(t-1)_{+})=f_{\delta}(x,0)=0\) for any \(t\leq 1\) and

$$ f_{\delta}\bigl(x,(t-1)_{+}\bigr)\leq c_{0}+c_{1}(t-1)^{p-1} \leq (c_{0}+c_{1})t^{p-1} $$

whenever \(t>1\) by \((f_{1})\), we get

$$\begin{aligned} -\Delta u_{1}^{\delta _{j}}&=-\frac{1}{\delta _{j}}h \biggl( \frac{u_{1}^{\delta _{j}}-1}{\delta _{j}} \biggr)+\alpha f_{\delta _{j}}\bigl(x, \bigl(u_{1}^{ \delta _{j}}-1\bigr)_{+}\bigr)+\beta \bigl(u_{1}^{\delta _{j}}\bigr)^{-\gamma} \\ &\leq \alpha (c_{0}+c_{1}) \bigl(u_{1}^{\delta _{j}} \bigr)^{p-1}+\beta \bigl(u_{1}^{\delta _{j}} \bigr)^{- \gamma}. \end{aligned}$$

However, when \(u_{1}^{\delta _{j}}<1\),

$$\begin{aligned} -\Delta u_{1}^{\delta _{j}}&=\beta \bigl(u_{1}^{\delta _{j}}\bigr)^{-\gamma}, \end{aligned}$$

in which case \(u_{1}^{\delta _{j}}=u_{\beta}|_{\{u_{1}^{\delta _{j}}<1\}}\). □

The sublinearity of (7.5) together with the boundedness of \((u_{1}^{\delta _{j}})\) in \(H_{0}^{1}(\Omega )\) implies by the Moser iteration method that \((u_{1}^{\delta _{j}})\) in \(L^{\infty}(\Omega )\). By a similar argument, \((u_{2}^{\delta _{j}})\) is also bounded in \(L^{\infty}(\Omega )\) since \(0< u_{1}^{\delta _{j}}\leq u_{2}^{\delta _{j}}\) in Ω. On renaming the subsequence of \((\delta _{j})\), the sequences \((u_{1}^{\delta _{j}})\), \((u_{2}^{\delta _{j}})\) converge uniformly to a Lipschitz continuous functions, say \(u_{1},u_{2}\in H_{0}^{1}(\Omega )\cap C^{2}(\bar{\Omega}\setminus G(u))\) respectively, of problem (1.1) that satisfies

$$ -\Delta u=\alpha \chi _{\{u>1\}}f\bigl(x,(u-1)_{+}\bigr)+\beta u^{-\gamma} $$

classically in the region \(\Omega \setminus G(u)\), the free boundary condition in the generalized sense and furthermore continuously vanishes on Ω. We also have that

$$\begin{aligned} E(u_{1})\leq \lim \inf E_{\delta _{j}} \bigl(u_{1}^{\delta _{j}}\bigr)\leq \lim \sup E_{\delta _{j}} \bigl(u_{1}^{\delta _{j}}\bigr)\leq E(u_{1})+ \bigl\vert \{u_{1}:u_{1}=1 \} \bigr\vert \end{aligned}$$


$$\begin{aligned} E(u_{2})\leq \lim \inf E_{\delta _{j}} \bigl(u_{2}^{\delta _{j}}\bigr)\leq \lim \sup E_{\delta _{j}} \bigl(u_{2}^{\delta _{j}}\bigr)\leq E(u_{2})+ \bigl\vert \{u_{2}:u_{2}=1 \} \bigr\vert . \end{aligned}$$

Using (7.6) in combination with (7.1) and (4.10) yields

$$ E(u_{1})\leq \lim \sup E_{\delta _{j}}\bigl(u_{1}^{\delta _{j}} \bigr)\leq m_{1}( \alpha )\leq E(u_{1}). $$


$$\begin{aligned} E(u_{1})&=m_{1}(\alpha )< - \vert \Omega \vert . \end{aligned}$$

Similarly, combining (7.7) with (7.2) yields

$$ 0< m_{2}(\alpha )\leq \lim \inf E_{\delta _{j}}\bigl(u_{2}^{\delta _{j}} \bigr) \leq E(u_{2})+ \bigl\vert \{u_{2}:u_{2}=1\} \bigr\vert . $$


$$\begin{aligned} E(u_{2})>- \bigl\vert \{u_{2}:u_{2}=1 \} \bigr\vert \geq - \vert \Omega \vert . \end{aligned}$$

So, from (7.8) and (7.9) we can conclude that \(u_{1}\), \(u_{2}\) are distinct and nontrivial solutions of problem (1.1). Here \(u_{1}\) is a minimizer, whereas \(u_{2}\) is not. Also, since \(u_{2}^{\delta _{j}}\leq u_{1}^{\delta _{j}}\) for each j, we have \(u_{2}\leq u_{1}\). Since \(u_{2}\) is a nontrivial solution, it follows that \(0< u_{2}\leq u_{1}\) and the sets \(\{u_{1}:u_{1}<1\}\subset \{u_{2}:u_{2}<1\}\) are connected if Ω is connected. Moreover, the sets \(\{u_{2}:u_{2}>1\}\subset \{u_{1}:u_{1}>1\}\) are nonempty.

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The first author (DC) has received funding from the National Board for Higher Mathematics (NBHM), Department of Atomic Energy (DAE) India, [02011/47/2021/NBHM(R.P.)/R&D II/2615]. The second author (DDR) has received funding from the Slovenian Research Agency grants P1-0292, J1-4031, J1-4001, N1-0278, N1-0114, and N1-0083.

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The authors DC and DDR have contributed equally to the study of the problem and have written the main manuscript text. All authors reviewed the manuscript.

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Choudhuri, D., Repovš, D.D. An elliptic problem of the Prandtl–Batchelor type with a singularity. Bound Value Probl 2023, 63 (2023).

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