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 [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:

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

(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 [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:

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

(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 n̂ 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:

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

(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 [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 $\delta >0$

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

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

• 2. $0< u_{2}\leq u_{1}$ 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. $u_{1}$ 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

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

(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

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

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

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

• (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 [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

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

(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 [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

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

(3.11)

as $j\rightarrow \infty$. Here, n̂ 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)

 □

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

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

(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 [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

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

(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 [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,

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

(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.