Positive solutions for critical singular elliptic equations without Ambrosetti-Rabinowitz type conditions

1.   Introduction

Consider the Hardy-Sobolev's critical exponent problem

where $\Omega \subset \mathbb{R}^{N}$, $N\geq3$, $\lambda\in \mathbb{R}$, $0\leq s_1 < s_2 < 2$, $2^{*}(s) = \frac{2(N-s)}{N-2}$ for $0 \leq s \leq 2, 2 < p\leq 2^{*}(s_1), 2 < q \leq 2^{*}(s_2).$ We aim to study the existence of positive solutions of this problem when $q < 2^{*}(s_2)$.

In 1983, H. Brézis and L. Nirenberg^[1] studied Problem (1.1) with Sobolev critical exponent where $s_1 = s_2 = 0, \ p = 2, \ q = 2^*(s_2) = 2^*(0) = \frac{2N}{N-2}$, and $\lambda\in (-\lambda_1, 0)$. Here, $\lambda_1$ is the first eigenvalue of $-\Delta$ with zero Dirichlet boundary condition. In 2000, N. Ghoussoub and C. Yuan ^[2] investigated the Hardy-Sobolev critical exponent problem where $0 \in \Omega, \ s_1 = 0, \ s_2\not = 0$ and $\lambda < 0$. Instead of $0 \in \Omega$, N. Ghoussoub and X.S. Kang considered the Hardy-Sobolev critical exponent problem where $0 \in \partial\Omega$ in their work ^[3].

Then, in ^[4], Y. Li and C.-S. Lin studied Problem (1.1) with two Hardy-Sobolev critical exponents ($0\in \partial\Omega, \ p = 2^{*}(s_1) < 2^{*}(s_2) = q, \ \lambda\in \mathbb{R}$) and posed an open question: Does the problem have a positive solution when $\lambda > 0$ and $p > q = 2^{*}(s_2)$? For convenience, we refer to the two-critical Li–Lin's open problem with $2^{*}(s_2) = q < p = 2^{*}(s_1)$ and the one-critical Li–Lin's open problem with $2^{*}(s_2) = q < p < 2^{*}(s_1)$.

Our motivation is to address the open question. For more details on recent progress see ^{[5,6,7,8,9,10]} and the references therein. Specially, in 2015, G. Cerami, X. Zhong and W. Zou ^[7] obtained some existence results of positive solutions by the perturbation approach and the monotonicity trick. The related results are the following two theorems.

Theorem 1. (see ^[7], Theorem 1.5) Suppose that $\Omega \subset \mathbb{R}^{N}$ is a $C^{1}$ bounded domain such that $0 \in \partial\Omega$. Assume that $\partial \Omega$ is $C^{2}$ at $0$ and $H(0) < 0$. Let $0\leq s_1 < s_2 < 2$ and $2^{*}(s_2) < p\leq 2^{*}(s_1)$. Then there exists $\lambda_{0} > 0$ such that Problem (1.1) has a positive solution for all $\lambda \in (0, \lambda_{0})$.

Theorem 2. (see ^[7], Theorem 1.6) Suppose that $\Omega \subset \mathbb{R}^{N}$ is a $C^{1}$ bounded domain such that $0 \in \partial\Omega$. Assume that $\partial \Omega$ is $C^{2}$ at $0$ and $H(0) < 0$. Let $0\leq s_1 < s_2 < 2$ and $2^{*}(s_2) < p\leq 2^{*}(s_1)-\frac{2}{N-2}$. Then, for almost every $\lambda > 0$, Problem (1.1) has a positive solution.

Recently, in ^[10], we presented the first nonexistence result for the two-critical case of Li–Lin's open problem employing proof by contradiction, along with the Hölder inequality, the Hardy inequality, and the Young inequality. Furthermore, we obtained a second existence result for the Li–Lin's open problem with $2^{*}(s_1)\geq p > q = 2^{*}(s_2)$ based on Theorem 2. The main theorems are as follows.

Theorem 3. (see ^[10], Theorem 1.1) Suppose that $\Omega \subset \mathbb{R}^{N}$ is a domain. Assume that $0\leq s_1 < s_2 < 2$, $p = 2^{*}(s_1)$ and $q = 2^{*}(s_2)$. Then there exists $\lambda_{1} > 0$ such that Problem (1.1) has no nonzero solution for all $\lambda > \lambda_{1}$.

Theorem 4. (see ^[10], Theorem 1.4) Suppose that $\Omega \subset \mathbb{R}^{N}$ is a $C^{1}$ bounded domain such that $0 \in \partial\Omega$. Assume that $\partial \Omega$ is $C^{2}$at $0$ and $H(0) < 0$. Let $0\leq s_1 < s_2 < 2$, $q = 2^{*}(s_2)$, $2^{*}(s_1)-\frac{N(s_2-s_1)}{(N-2)(N+1-s_2)} < p\leq 2^{*}(s_1)$ and $2^{*}(s_1)-\frac{s_2-s_1}{N-2}\leq p$. Let $\lambda_{*} = \sup \{\lambda \in \mathbb{R}\ |$ Problem (1.1) has a positive solution}. Then $\lambda_{*} > 0$ and Problem (1.1) has at least a positive solution for all $\lambda \in (0, \lambda_{*})$.

Clearly, the present results only deal with special cases of Li–Lin's open problem and are far from giving a full solution. The main difficulty of this problem is that it's impossible to obtain the boundedness of the $(PS)$ sequences for the energy functional.

A natural question is: What will happen if we exchange the critical property of $p$ and $q$ in the one-critical Li–Lin's open problem, that is, $p = 2^{*}(s_1), 2 < q < 2^{*}(s_2)$? This question is the same as replacing $q = 2^{*}(s_2)$ with $q < 2^{*}(s_2)$ in the two-critical Li–Lin's open problem. In fact, obtaining the boundedness of the $(PS)$ sequences for the energy functional is still a challenge. However, we have proved a new inequality to overcome this difficulty.

In this paper, we study more general questions, that is, $2 < p\leq 2^{*}(s_1), 2 < q < 2^{*}(s_2)$. It is noteworthy that for small $\lambda$, we can obtain two positive solutions, while for large $\lambda$, there are no positive solutions. The main results are the following theorems.

Theorem 1.1. Assume that $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with $0 \in \partial\Omega$, $\lambda > 0$, $0\leq s_1 < s_2 < 2$, $2 < p\leq 2^{*}(s_1)$ and $2 < q < \frac{2-s_2}{2-s_1}p+\frac{2s_2-2s_1}{2-s_1}$. Then there exists $\lambda_0 > 0$ such that Problem (1.1) has at least two positive solutions for all $\lambda \in (0, \lambda_0)$.

Corollary 1.2. Assume that $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with $0 \in \partial\Omega$, $\lambda > 0, 0\leq s_1 < s_2 < 2$, $p = 2^{*}(s_1)$ and $2 < q < 2^{*}(s_2)$. Then there exists $\lambda_0 > 0$ such that Problem (1.1) has at least two positive solutions for all $\lambda \in (0, \lambda_0)$.

Theorem 1.3. Suppose that $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with $0 \in \partial\Omega$, $\lambda > 0$, $0\leq s_1 < s_2 < 2$, $2 < p\leq 2^{*}(s_1)$ and $2 < q < \frac{N+2-2s_2}{N+2-2s_1}p+\frac{2s_2-2s_1}{N+2-2s_1}$. Then there exists $\lambda_{*} > 0$ such that Problem (1.1) has no positive solution for all $\lambda > \lambda_{*}$, and has at least one positive solution for all $\lambda \in (0, \lambda_{*}]$.

Corollary 1.4. Suppose that $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with $0 \in \partial\Omega$, $\lambda > 0$, $0\leq s_1 < s_2 < 2$, $p = 2^{*}(s_1)$ and $2 < q < 2^{*}(s_2)$. Then there exists $\lambda_{*} > 0$ such that Problem (1.1) has no positive solution for all $\lambda > \lambda_{*}$, and has at least one positive solution for all $\lambda \in (0, \lambda_{*}]$.

Remark 1.5. This question is related to the subcritical approximations of the two-critical Li–Lin's open problem.

Remark 1.6. Theorem 1.3 represents a global bifurcation for positive solutions. Moreover, its proof is carried out using the variational method combined with the method of sub-supersolutions.

2.   Preliminaries

We introduce the work space

with scalar product and norm given by

It is well-known that the solutions of problem (1.1) are precisely the critical points of the energy functional $I_\lambda:H^1_0{(\Omega)}\rightarrow \mathbb{R}$ defined by

It is easy to see that $I_\lambda$ is well-defined and $I_\lambda\in C^1(H^1_0{(\Omega)}, \mathbb{R})$. Then, for any $u, v\in H^1_0{(\Omega)}$,

where $I'_\lambda(u)$ is the Gâteaux derivative of $I_\lambda(u)$.

Proposition 2.1. Suppose that $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with $0 \in \partial\Omega$, $0\leq s_1 < s_2 < 2$, $2 < p\leq 2^{*}(s_1)$ and $2 < q < \frac{2-s_2}{2-s_1}p+\frac{2s_2-2s_1}{2-s_1}$. Then there exist three positive constants ${\gamma_1}, {\gamma_2}, {\gamma_3}$ with ${\gamma_1}+ {\gamma_2} +{\gamma_3} = 1$ such that

for every $u\in H^1_0(\Omega)$.

Proof Let

Then we have ${\gamma_1}, {\gamma_2}, {\gamma_3} > 0$ and

It follows from the Hölder inequality that

for every $u\in H^1_0(\Omega)$. This completes the proof of the proposition. □

Lemma 2.2. Suppose that $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with $0 \in \partial\Omega$, $\lambda > 0$, $0\leq s_1 < s_2 < 2$, $2 < p\leq 2^{*}(s_1)$ and $2 < q < \frac{2-s_2}{2-s_1}p+\frac{2s_2-2s_1}{2-s_1}$. Then there exists $\lambda_{1} > 0$ such that Problem (1.1) has no nonzero solution for every $\lambda > \lambda_{1}$.

Proof Suppose that $u$ is a nonzero solution of Problem (1.1). Then one has $I'_\lambda(u) = 0$. Hence, $\langle I'_\lambda(u), u\rangle = 0$, that is,

which implies that

for some constant $C > 0$ according to the Hardy-Sobolev inequality. It follows that

By Proposition 2.1 and the Hardy inequality (see ^[11], Theorem 4.1), we have

where $C_N = \left(\frac{2}{N-2}\right)^2$. It follows from the Young inequality that

which implies that

by (2.1). Let $\lambda_1 = {\gamma_1} (C_N^{\gamma_2} |\Omega|^{\gamma_3} C^\frac{2{\gamma_3}}{q-2})^\frac{1}{{\gamma_1}}$, then Problem (1.1) has no nonzero solution for every $\lambda > \lambda_{1}$. □

Lemma 2.3. Suppose that $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with $0 \in \partial\Omega$, $0\leq s_1 < s_2 < 2$, $2 < p\leq 2^{*}(s_1)$ and $2 < q < \frac{2-s_2}{2-s_1}p+\frac{2s_2-2s_1}{2-s_1}$. Then the functional $I_{\lambda}$ is coercive, i.e., $I_{\lambda}(u)\to+\infty$ as $\|u\|\to\infty$. Moreover, the functional $I_{\lambda}$ satisfies the $(PS)_c$ condition for every $c\in\mathbb{R}$. Specifically, if $I_{\lambda}(u_n)\to c$ and $I_{\lambda}'(u_n)\to0$ as $n\to\infty$, then $\{u_n\}$ has a convergent subsequence.

Proof By Proposition 2.1, the Hardy inequality and the Young inequality, we have

where $C_N = \left(\frac{2}{N-2}\right)^2$ and $A = C_N(p\lambda^{-1})^\frac{{\gamma_1}}{{\gamma_2}}|\Omega|^\frac{{\gamma_3}}{{\gamma_2}}$. It follows that

on $H^1_0(\Omega)$, which implies that the functional $I_\lambda$ is coercive due to $0 < \frac{{\gamma_2}}{1-{\gamma_1}} < 1$.

Suppose that $\{ u_n\}$ is a $(PS)_c$ sequence for some $c\in \mathbb{R}^{N}$, that is, $I_\lambda(u_n)\rightarrow c$ and $I'_\lambda(u_n)\rightarrow0$. Then $\{ u_n\}$ is bounded in $H^1_0{(\Omega)}$ by the coercivity of $I_\lambda$. Up to a subsequence, there is $u_0\in H^1_0{(\Omega)}$ such that, as $n\rightarrow \infty$,

By the definition of $I'_\lambda$ we have

which implies that $u_n\rightarrow u_0$ in $H^1_0{(\Omega)}$ as $n\rightarrow \infty$ by

and

where $C_1$ and $C_2$ are some positive constants. □

3.   Proof of Theorems 1.1 and 1.3

3.1.   Proof of Theorem 1.1

Proof At the beginning, we define

Our goal is to prove that $0 < \lambda_{0} < +\infty$. We choose $u_0 \in H^1_0(\Omega)\setminus \{0\}$. Since $q > 2$, there exists $t_0 > 0$ such that

According to the definition of $\lambda_{0}$, one has $\lambda_{0} > 0$.

Now we prove $\lambda_{0} < +\infty$. On the one hand, by the Hardy-Sobolev inequality, since $2 < q$, there is a positive constant $r_0$ such that

for all $u \in H^1_0(\Omega)\setminus \{0\}$ with $||u|| \leq r_0$. On the other hand, for $||u|| > r_0$, we have

by Proposition 2.1 and the Hardy inequality. Moreover, it follows from the Young inequality that

Therefore,

for $||u|| > r_0$. By the definition of $\lambda_{0}$, we can see that $\lambda_{0} < +\infty$.

In conclusion, we have $0 < \lambda_{0} < +\infty$.

Next, we define

and want to prove that $-\infty < m_\lambda < 0$ for every $\lambda \in (0, \lambda_{0})$. By the definition of $\lambda_{0}$, there exists $u_\lambda \in H^1_0(\Omega)\setminus \{0\}$ such that

which implies that

Thus, $m_\lambda < 0$. Furthermore, it follows from Lemma 2.3 and the boundedness of functional $I_\lambda$ that $m_\lambda > -\infty$. Therefore, we have proven that $-\infty < m_\lambda < 0.$

Now we prove the existence of a positive solution, which is the local minimum point of the functional $I_\lambda$. Note that the functional $I_\lambda$ is weakly lower semi-continuous since

is convex and continuous, and

is weakly continuous. By the least action principle (see Theorem 1.1 in ^[12]), $I_\lambda$ has a minimum point $w_\lambda$ such that $I_\lambda(w _\lambda) = m_\lambda$. Due to $m_\lambda < 0$, one has $w_\lambda\not = 0$. Since $I_\lambda(|w_\lambda|) = I_\lambda(w_\lambda) = m_\lambda$, we may assume that $w_\lambda\geq 0$. Hence, $w_\lambda$ is a nonzero nonnegative solution of Problem (1.1). It follows from the strong maximum principle (see ^[13]) that $w_\lambda$ is a positive solution of Problem (1.1).

Finally, we consider another positive solution which is the mountain pass point of the functional $I_\lambda$. By the Hardy-Sobolev inequality, we have

for all $u\in H^1_0(\Omega)$ and some constant $C > 0$. It follows that

on $H^1_0(\Omega)$, which implies that $I_\lambda(u)\geq\frac{1}{4}\rho^2$ for $\|u\| = \rho$ with $0 < \rho < \min\{\|w_\lambda\|, (4C)^{-\frac{1}{q-2}}\}$. Note that $I_\lambda(w _\lambda) = m_\lambda < 0$. Hence, the functional $I_\lambda$ has a mountain pass geometry structure. Then we define the minimax value

where

According to Lemma 2.3 and the mountain pass lemma (see ^[14]), $I_\lambda$ has a mountain pass point $v_\lambda$ such that $I_\lambda(v_\lambda) = c_\lambda$. Since $c_\lambda > 0$, we know that $v_\lambda\not = 0$. Also, because $I_\lambda(|v_\lambda|) = I_\lambda(v_\lambda) = c_\lambda$, we can assume that $v_\lambda\geq 0$. Consequently, $v_\lambda$ is a nonzero nonnegative solution of Problem (1.1). By the strong maximum principle, $v_\lambda$ is a positive solution of Problem (1.1). Moreover, since $I_\lambda(v_\lambda) = c_\lambda > 0 > m_\lambda = I_\lambda(w_\lambda)$, we have $v_\lambda\not = w_\lambda$.

In conclusion, for all $\lambda \in (0, \lambda_0)$, Problem (1.1) has at least two positive solutions $v_\lambda$ and $w_\lambda$. □

3.2.   Proof of Theorem 1.3

In this subsection, we will prove Theorem 1.3 using the method of sub-supersolutions and the variational method. Now, we recall the sub-supersolution method in ^[15].

Definition 3.1 (see ^[15], P2430, Definition 1.1) A function u is an $L^{1}-solution$ of

where $\Omega \subset \mathbb{R}^{N}$ is a smooth bounded domain and $f:\Omega \times \mathbb{R} \rightarrow \mathbb{R}$ is a Carath$\acute{e}$odory function, if

$(i)\ u \in {L}^{1}(\Omega);$

$(ii)\ f(\cdotp, u)\rho_0 \in {L}^{1}(\Omega);$

$(iii)$

Here, $\rho_0(x) = d(x, \partial\Omega), \ \forall x \in \Omega, \ and \ C_0^2(\overline{\Omega}) = \{ \varphi \in C^2(\overline{\Omega}); \ \varphi = 0\ on\ \partial \Omega \}$.

We also consider $L^1 - sub solutions$ and $L^1 -super solutions$ in analogy with this definition. For instance, $u$ is an $L^1 - sub solution$ of Problem (3.1) if $u$ satisfies (ⅰ)-(ⅲ) with "$\leq$" instead of "$=$" in (3.2). We will systematically omit the term "$L^1$" and simply say that $u$ is a solution of Problem (3.1), which means that $u$ satisfies (3.2); a similar convention applies to subsolutions and supersolutions.

Lemma 3.2. (see ^[15], P2436, Corollary 5.3) Let $v_1, v_2\in L^1(\Omega)$ be a sub and a supersolution of Problem (3.1), respectively. Assume that $v_1 \leq v_2$ a.e. and

Then, Problem (3.1) has a solution $u\in H^1_0(\Omega)$ such that $v_1 \leq u \leq v_2$ a.e.

Proof of Theorem 1.3 We define

From Theorem 1.1 and Lemma 2.2, we obtain $\lambda_{*}\in (0, +\infty)$. Hence, Problem (1.1) has no positive solution for all $\lambda > \lambda_{*}$.

By the definition of $\lambda_{*}$, for every $\lambda \in (0, \lambda_{*})$, there exists $\mu \in (\lambda, \lambda_{*})$ such that Problem (1.1) with $\lambda = \mu$ has a positive solution $u_{\mu}$.

Let $v_1 = u_{\mu}$, $v_2(x)\stackrel{\vartriangle }{ = }M|x|^{-\alpha}$, $\alpha\stackrel{\vartriangle }{ = }\frac{s_2-s_1}{p-q}$ and

We have $\alpha\leq N-2$, which follows from $q < \frac{N+2-2s_2}{N+2-2s_1}p+\frac{2s_2-2s_1}{N+2-2s_1}$.

In a way similar to the proof in ^[10], one obtains that $v_1$ is a subsolution of Problem (1.1), $v_2$ is a supersolution of Problem (1.1) and $v_2(x)\geq v_1(x)$ for a.e. $x\in \overline{\Omega}$.

By $q < \frac{N+2-2s_2}{N+2-2s_1}p+\frac{2s_2-2s_1}{N+2-2s_1}$, we have $[\alpha(p-1)+s_1]\frac{2N}{N+2} = [\alpha(q-1)+s_2]\frac{2N}{N+2} < N,$ which implies that (3.3) holds. Hence, Problem (1.1) has at least one positive solution for all $\lambda \in (0, \lambda_{*})$ by Lemma 3.2.

Next, we consider the case $\lambda = \lambda_{*}$. For an integer $n > \frac{1}{\lambda_{*}}$, there exists a positive $u_n\in H^1_0(\Omega)$ such that

For any $v\in H^1_0(\Omega)$, we have

for some positive constant $C$. So $\|u_n\|\leq C$, and $I_{\lambda_*}(u_n)$ is bounded. Without loss of generality, assume that $I_{\lambda_*}(u_n)\to c$ as $n\to\infty$ for some $c\in \mathbb{R}$. From (3.4), we get $I'_{\lambda_*}(u_n) = 1/n |x|^{-s_{1}}|u_n|^{p-2}u_n\to 0$ as $n\to\infty$. Thus, $(u_n)$ is a $(PS)_c$ sequence of $I_{\lambda_*}$. According to Lemma 2.3, $(u_n)$ has a convergent subsequence. We can assume that $u_n\to u_0$ as $n\to\infty$ for some $u_0\in H^1_0(\Omega)$. Moreover, $u_0$ is a nonnegative nonzero solution of Problem (1.1) with $\lambda = \lambda_{*}$. By the strong maximum principle, $u_0$ is a positive solution of Problem (1.1) with $\lambda = \lambda_{*}$, which completes our proof. □

Author contributions

Zhi-Yun Tang: Conceptualization, Writing–Original Draft, Writing–Review & Editing; Xianhua Tang: Supervision, Writing–Review & Editing.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This study was funded by the National Natural Science Foundation of China(No. 12371181) and the Fundamental Research Funds for the Central Universities of Central South University (No. 2024ZZTS0441).

Conflict of interest

The authors declare there is no conflict of interest.