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

## Abstract

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}
(1.1)

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}
(1.2)

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

The case when $$f(x,t)=1$$, $$\beta =0$$ 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 , Elcrat and Miller , Acker , and Jerison and Perera . We drew our motivation for studying the present problem in this paper from perera . The problem studied by Perera  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  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 , Friedman and Liu , and Temam ).

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  from 1991 that opened a new door for the researchers in elliptic and parabolic PDEs. The problem considered in  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}
(1.3)

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  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. . Jerison and Perera  obtained a mountain pass solution of this problem for the superlinear subcritical case. Yang and Perera  addressed the problem for the critical case. Recently, Choudhuri and Repovš  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. .

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}
(1.4)

$$\text{for all} \psi \in C_{0}^{1}(\Omega ,\mathbb{R}^{N})$$ 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 , 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}
(1.5)

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}
(1.6)

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}
(1.7)

The existence of $$u_{\beta}$$ can be guaranteed by Lazer and McKenna . 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 . 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}
(1.8)

Define

\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}
(1.9)

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 ). 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}
(3.1)

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}
(3.2)

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}
(3.3)

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}
(3.4)

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}
(3.5)

By Lazer and McKenna , 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}
(3.6)

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}
(3.7)

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}
(3.8)

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}
(3.9)

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}
(3.10)

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 , 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}
(3.11)

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}
(3.12)

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}
(3.13)

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}
(3.14)

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}
(3.15)

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}
(3.16)

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}
(4.1)

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}
(4.2)

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}
(4.3)

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}
(4.4)

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}
(4.5)

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}
(4.6)

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}
(4.7)

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}
(4.8)

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}
(4.9)

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}
(4.10)

Set

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

## 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}
(5.1)

### Proof

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}
(5.2)

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, $$\emptyset \neq \{u_{2}^{\delta}:u_{2}^{\delta}>1\}\subset \{u_{1}^{ \delta}:u_{1}^{\delta}>1\}$$.

### Proof

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}
(5.3)

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}
(5.4)

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

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

and

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

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}
(5.5)
\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}
(5.6)

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}
(5.7)

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}
(5.8)

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}
(5.9)

□

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}
(6.1)

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

### Proof

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}
(6.2)

By the Schauder estimates, we have $$u\in C^{2,a}(\Omega )$$, and by Lazer and McKenna  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}
(6.3)

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

$$\varphi (x)= \textstyle\begin{cases} 1& \text{if } x\in V, \\ 0< \varphi < 1& \text{if } x\in W\setminus V, \\ 0& \text{if } x\in \Omega \setminus W. \end{cases}$$
(6.4)

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}
(6.5)

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 $$\beta \in (0,\beta _{*})$$.

### Proof

We follow the proof due to Alt and Caffarelli . 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}
(6.6)

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}
(6.7)

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}
(6.8)

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}
(7.1)

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}
(7.2)

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}
(7.3)

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}
(7.4)

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

\begin{aligned} -\Delta u_{1}^{\delta _{j}}&=\beta \bigl(u_{1}^{\delta _{j}}\bigr)^{-\gamma}, \end{aligned}
(7.5)

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}
(7.6)

and

\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}
(7.7)

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

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

Therefore,

\begin{aligned} E(u_{1})&=m_{1}(\alpha )< - \vert \Omega \vert . \end{aligned}
(7.8)

Similarly, combining (7.7) with (7.2) yields

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

Thus,

\begin{aligned} E(u_{2})>- \bigl\vert \{u_{2}:u_{2}=1 \} \bigr\vert \geq - \vert \Omega \vert . \end{aligned}
(7.9)

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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## Funding

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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Correspondence to Dušan D. Repovš.

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