Nonhomogeneous nonlinear integral equations on bounded domains

1.   Introduction

This paper concerns the existence of positive solutions for the following integral equation:

where $u \in L^{p}\left(\Omega\right), f \in L^{p}\left(\Omega\right)$, $n > 2,$ $\frac{2n}{n+\alpha} = p_{\alpha}\leq p < 2,$ $1 < \alpha < n$ and $\Omega$ is a bounded domain in $\mathbb R^{n}$.

When we set $f(x) = 0$, Eq (1.1) simplifies to the subsequent integral equation:

Indeed, the existence of solutions for problem (1.2) is connected to the classic sharp Hardy-Littlewood-Sobolev (HLS) inequality:

Theorem A. Let $\alpha \in(0, n).$ The classical sharp HLS inequality (^{[15,16,19,20,21]}) states that

for all $f \in L^{p}\left(\mathbb{R}^{n}\right), g \in L^{t}\left(\mathbb{R}^{n}\right), 1 < p, t < \infty, 0 < \alpha < n$ and $1 / p+1 / t+(n-\alpha) / n = 2$. If $p = t = 2 n /(n+\alpha)$, then

holds for all $f, g \in L^{\frac{2 n}{n+\alpha}}\left(\mathbb{R}^{n}\right)$ where

And the equality holds if and only if

where $c_{2}$ is any constant, $c_{1}, c_{3}$ are positive constants and $x_{0} \in \mathbb{R}^{n}$. Clearly, inequality (1.3) is applicable to bounded domains as well. Motivated by this, Dou and Zhu in ^[11] recently explored the Euler-Lagrange equation for inequality (1.3) in a bounded domain, as per the following equation:

where $\Omega$ is a bounded domain in $\mathbb R^{n}$. Additionally, Dou and Zhu examined the subsequent general equation:

Using the compact embedding theorem along with a blowing-up and rescaling argument (as mentioned in Lemma 4.3 of ^[11]), they established the following theorem.

Theorem B. Assume $\alpha \in(1, n)$ and $\Omega$ is a smooth bounded domain.

(1) For $\frac{2 n}{n+\alpha} < p < 2$ (subcritical case), there is a positive solution $u \in C^{1}(\bar{\Omega})$ to Eq (1.5) for any given $\lambda \in \mathbb{R}$;

(2) For $p = \frac{2 n}{n+\alpha}$ (critical case) and $\lambda > 0,$ there is a positive solution $u \in C(\bar{\Omega})$ to Eq (1.5).

Dou and Zhu in ^[11] established the existence results for weak solutions to (1.5) when $\lambda > 0$ and $p = p_\alpha$. They considered the functional

Due to homogeneity, we know that the corresponding Euler-Lagrange equation for nonnegative extremal functions up to a constant multiplier is the integral equation (1.5) for $p = p_\alpha$. It should be noted that Eq (1.5) differs from Eq (1.1) due to the nonhomogeneous nature of Eq (1.1). Therefore, we cannot directly obtain the existence results for weak solutions to (1.1) using the approach of setting up extremal problems as done in ^[11]. Integral equations or systems of integral equations on the whole space, bounded domains or upper half space have been extensively studied previously as shown in ^{[6,7,8,9,10,12,13,14,17,18,23,24,25]} and the references therein.

In relation to the nonhomogeneous critical semilinear elliptic equation associated with Eq (1.1),

where $2^{*} = \frac{2 n}{n-2}$ is the critical Sobolev exponent, $n > 2,$ $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary. Tarantello ^[22] demonstrated that problem (1.6) possesses at least two solutions. The fundamental idea is to partition the Nehari manifold $\Lambda = \left\{u \in W_{0}^{1, 2}(\Omega); \langle I^{\prime}(u), u\rangle = 0\right\}$ into three disjoint subsets, namely $\Lambda^{+}, \Lambda^{-}$ and $\Lambda_{0}$ and to employ the Ekeland variational principle to obtain one solution in $\Lambda^{+}$ and another solution in $\Lambda^{-}$. The existence results for an elliptic problem of $(p, q)$-Laplacian type, involving a critical term, a power-type nonlinearity at the critical level with a subcritical term, nonnegative weights and a positive parameter $\lambda$ have been discussed in the literature, specifically in references ^[2,3], for the entire space $\mathbb{R}^N$.

There exists a notable distinction between integral and differential equations. For instance, consider

where $\Omega$ is a bounded domain and $c(n, \alpha)$ is a constant, dependent only on $n, \alpha$. Given $f = u^{\frac{n-2}{n+2}}$ and $\alpha = 2$, it can be observed that $u$ must fulfill

but not conversely, as seen in ^[11]. Furthermore, the difference between $W^{1, 2}(\Omega)$ and $L^{p}(\Omega)(1 < p < 2)$ arises challenges when attempting to treat integral equations in the same manner as differential equations. For instance, the Brezis-Lieb lemma ^[4] cannot be applied in $L^{p}(\Omega)(1 < p < 2)$ because almost-everywhere convergence of sequences cannot be inferred from weak convergence of sequences in $L^{p}(\Omega)(1 < p < 2)$. This fact complicates our attempts to prove the existence of the solution to Eq (1.1) using the variational method to handle Eq (1.6).

Inspired by the work described above, our study differs from previous works on integral equations which primarily focused on the homogeneous case in that we instead handle the nonhomogeneous case. Therefore, we consider the existence of positive solutions for Eq (1.1) for $p_{\alpha}\leq p < 2$. A function $u \in L^{p}(\Omega)$ is said to be a solution of $(1.1)$ if $u$ satisfies

Consider functionals $\mathrm{I}$ : $L^{p}(\Omega)\rightarrow \mathbb R$:

Let

For $u, w \in L^{p}(\Omega),$ due to HLS inequality and Hölder inequality, we have

This implies that $I \in C^{1}\left(L^{p}(\Omega), \mathbb{R}\right)$.

We first investigate the critical problem, leading to the following existence result, which is the principal outcome of this paper.

Theorem 1.1. Assume that $f(x)$ is a non-negative function satisfying the following conditions:

($A_{1}$) For small enough $\epsilon$, $\|f\|_{p_{\alpha}} < \min\{C(n, p_{\alpha}, \alpha, \Omega)N^{\frac{1}{p_{\alpha}-2}}_{\alpha}, \epsilon^{\frac{n+\alpha}{2}}\}$;

($A_{2}$) $f(x)\in C^{0}(B_{\delta}(x_{\ast}))\bigcap L^{p_{\alpha}+\delta}(\Omega)$, $f(x_{\ast}) > 0$ where $B_{\delta}(x_{\ast})\subseteq\Omega$ for some $x_{\ast}\in\Omega$ and $\delta > 0$ is small enough.

Then, problem (1.1) has at least one positive solution $u \in L^{p_{\alpha}}\left(\Omega\right)$, $1 < \alpha < (\sqrt{2}-1)n, n > 2$ and $\Omega$ is a bounded domain in $\mathbb R^{n}$.

Next, we examine the existence result for Eq (1.1) in the subcritical case.

Theorem 1.2. Let $f(x)\in L^{p}(\Omega)$, $f(x)\neq 0$, $\|f\|_{p} < C(n, p, \alpha, \Omega)N^{\frac{1}{p-2}}_{\alpha}$. Then problem (1.1) has at least two positive solutions $u_{0}, u_{1} \in L^{p}\left(\Omega\right)$, $\frac{2n}{n+\alpha} < p < 2,$ $1 < \alpha < n, \ n > 2$ and $\Omega$ is a bounded domain in $\mathbb R^{n}$.

Remark 1.1. In what follows, we proceed with the proof of these theorems. For the critical case, we employ the Ekeland variational principle (see ^[22]) and a blow-up argument and a rescaling argument to find a weak solution of (1.1). In the process of proving the main theorem (Theorem 1.1), we encounter difficulties similar to those in ^[22]. In ^[22], the following core lemma is required to be proved:

Lemma 1.1. For $f \neq 0, \ n > 2, \ p = \frac{2n}{n-2}$,

is achieved, where $c_{n}$ is a constant that only depends on $n$.

Similarly, we aim to show that for $p = p_{\alpha},$

is achieved in this paper. It's important to note that the Brezis-Lieb lemma ^[4] plays a crucial role in proving Lemma 1.1 through the variational method. However, since $1 < p_{\alpha} < 2,$ the Brezis-Lieb lemma ^[4] does not hold in $L^{p_{\alpha}}(\Omega)$. Consequently, the proof method of Lemma 1.1 fails to prove that $Q_{p_{\alpha}}(\Omega)$ is achieved. To solve the problem, we use a blow-up argument and a rescaling argument in this paper. First, for $\frac{2 n}{n+\alpha} < p < 2$, we can show $Q_p(\Omega)$ is achieved at a point $u_p$. For $p = p_\alpha$, we will show $\lim _{p \rightarrow p_\alpha}\left\|u_p\right\|_{L^{\infty}(\Omega)} \leq C$, by a blow-up argument and a rescaling argument. Thus, $u_p \rightarrow u_*$ as $p \rightarrow p_\alpha$ in $C(\bar{\Omega})$. Once $Q_p(\Omega)$ is achieved, we can prove that problem (1.1) has at least one positive solution by Ekeland variational principle.

The structure of this paper is as follows: In Section 2, we provide preliminary results. In Section 3, we prove Theorems 1.1 and 1.2.

Throughout this paper, we utilize the symbols c and C to represent various positive constants, the value of which may change from one line to another.

2.   Preliminaries

To obtain the proof of the main theorems, several preliminary are needed. Let

Lemma 2.1. Let $f \neq 0$ satisfy ($A_{1}$). For every $u\in L^{p}\left(\Omega\right), p_{\alpha}\leq p < 2, u \neq 0,$ there exists unique $t^{+} = t^{+}(u) > 0$ such that $t^{+} u \in \Lambda^{-}.$ In particular:

and I $\left(t^{+} u\right) = \max _{t \geq t_{\max }} I(t u).$ Moreover, if $\int_{\Omega} \int_{\Omega} \frac{u(x) f(y)}{|x-y|^{n-\alpha}} d x d y > 0,$ then there exists a unique $t^{-} = t^{-}(u) > 0$ such that $t^{-}(u) \in \Lambda^{+}.$ In particular,

$I\left(t^{-} u\right) \leq I(t u), \forall t \in\left[0, t^{+}\right]$.

Proof. Let $\varphi(t) = t^{p-1}\|u\|_{p}^{p}-\mathrm{t}\int_{\Omega} \int_{\Omega} \frac{u(x) u(y)}{|x-y|^{n-\alpha}} d x d y.$ Easy computations show that $\varphi$ is concave and achieves its maximum at

Also

that is

Thus, if $\int_{\Omega} \int_{\Omega} \frac{u(x) f(y)}{|x-y|^{n-\alpha}} d x d y \leq 0,$ then there exists a unique $t^{+} > t_{\max }$ such that: $\varphi\left(t^{+}\right) = \int_{\Omega} \int_{\Omega} \frac{u(x) f(y)}{|x-y|^{n-\alpha}} d x d y \quad$ and $\quad \varphi^{\prime}\left(t^{+}\right) < 0. \quad$ Equivalently $\quad t^{+} u \in \Lambda^{-} \quad$ and $\mathrm{I}\left(t^{+} u\right) \geq \mathrm{I}(t u), \ \forall t \geq t_{\max }.$ In case $\int_{\Omega} \int_{\Omega} \frac{u(x) f(y)}{|x-y|^{n-\alpha}} d x d y > 0,$ by assumption ($A_{1}$) we have that necessarily,

Therefore, in this case, we have unique $0 < t^{-} < t_{\max } < t^{+}$ such that

and

Equivalently $t^{+} u \in \Lambda^{-}$ and $t^{-} u \in \Lambda^{+}$. □

Let

we show

Lemma 2.2. Assume that $f(x)$ is a non-negative function satisfying ($A_{2}$).

Then, $Q_{p}(\Omega) < \frac{1}{c_{n, \alpha}(N_{\alpha})^{\frac{p-1}{2-p}}}, where\ p = p_{\alpha}.$

Proof. Similar to the proof of Lemma 4.1 of ^[11], let $x_{*} \in \Omega.$ For small positive $\epsilon$ and a fixed $R > 0$ so that $B_{R}\left(x_{*}\right) \subset \Omega,$ we define

where

Obviously, $\tilde{u}_{\epsilon} \in L^{p_{\alpha}}\left(\mathbb{R}^{n}\right).$ Thus, similar to the proof of Proposition 2.1 of ^[11] we have

where

By ($A_{2}$), we have $f(x)\in C^{0}(B_{\delta}(x_{\ast})), B_{\delta}(x_{\ast})\subseteq\Omega$ for some point $x_{\ast}$ within $\Omega$ and a positive real number $\delta$. Subsequently, we can select $\delta_{1}$ such that $0 < \delta_{1} < \delta$ thereby ensuring $f(x) > C$ for every $x$ in the ball $B_{\delta_{1}}(x_{\ast})$ where C is a constant independent of $x$. Choose $\epsilon < R$ so that $|\epsilon \eta| < \delta_{1}$ if $\eta\in B_{1}(0)$. Set

For $I_{3},$ we have

So, for $1 < \alpha < (\sqrt{2}-1)n$ and small enough $\epsilon,$ we get

□

Notation: For any function $u(x)$ defined on $\Omega$, we always use

Lemma 2.3. (Lemma 3.1 of ^[11]) Let $p > p_{\alpha}$ and $p^{'} = \frac{p}{p-1}$ be its conjugate. There exists a positive constant $C(n, \alpha, \Omega) > 0$ such that

holds for any $u\in L^{p}(\Omega)$. Moreover, for $\alpha > 1$ operator $I_{\alpha, \Omega}:L^{p}(\Omega)\hookrightarrow L^{p^{'}}(\Omega)$ is a compact embedding.

Lemma 2.4. Assume $f(x)$ is a non-negative function satisfying ($A_{1}$), ($A_{2}$). Then,

is achieved and $Q_{p}(\Omega) > 0$, where $p = p_{\alpha}.$

Proof. In order to establish the conclusion, we need to prove that

is achieved, where $p = p_{\alpha}.$ For this purpose, for $2 > p > p_{\alpha},$ we wil show that the infinum is attained by a positive function $u_{p}.$ To do this, all we have to do is show

is achieved. By Lemma 2.3, we have

where $p^{'} = \frac{p}{p-1}$. Together with the HLS inequality this implies:

Select a minimizing positive sequence $\left\{u_{j}\right\}_{j = 1}^{\infty}$ such that $\left\|I_{\alpha, \Omega} u_{j}\right\|_{L^{p^{\prime}}(\Omega)} = 1$. Thus, $\left\{u_{j}\right\}$ is bounded in $L^{p}(\Omega)$. It follows that there exists a subsequence $\{u_{j}\}$(still denoted as $\{u_{j}\}$) and $u_{*} \in L^{p}(\Omega)$ such that

By Lemma 2.3, we get

Then,

Therefore, $u_{*}$ is a minimizer. Thus, we have

Also, by considering $\frac{u}{\left\|u\right\|_p}$, we have

is achieved, where $2 > p > p_{\alpha}.$ Thus, for $2 > p > p_{\alpha},$ the infinum is attained by a positive function $u_{p},$ which satisfies the following equation with subcritical exponent

where $\left\|u_{p}\right\|_{p} = 1.$ We claim that $u_{p} \in C(\bar{\Omega})$ and $Q_{ p} \rightarrow Q_{p_{\alpha}}$ for $p \rightarrow p_{\alpha}$. First, we prove that $u_{p} \in C(\bar{\Omega})$. According to Eq (2.3), by writing $g(x) = u^{p-1}(x)$, we can obtain a weak positive solution $g(x) \in L^{p^{\prime}}(\Omega)$ to

for $p^{\prime} < \frac{2 n}{n-\alpha} = q_\alpha$. By (2.4) and HLS inequality, we have

for $1 / s = 1 / t-\alpha / n$. By employing a similar method as in Lemma 3.3 of ^[11], we can use the above inequality in an iterative process to obtain $g \in C(\bar{\Omega})$. Therefore, we can conclude that $u_p \in C(\bar{\Omega})$. Using a similar method as in Lemma 2.3 of ^[5], we apply Proposition 2.1 in ^[11] and the Hölder inequality to find a minimizing sequence of $Q_{p_{\alpha}}$ from the minimizer $u_p$. Consequently, we can establish that $Q_p \rightarrow Q_{p_{\alpha}}$ as $p \rightarrow p_{\alpha}$.

Next, we need to show $\lim _{p \rightarrow p_{\alpha}}\left\|u_{p}\right\|_{L^{\infty}(\Omega)} \leq C.$ We prove this by contradiction. Suppose not. Let $u_{p}\left(x_{p}\right) = \max _{\bar{\Omega}} u_{p}(x).$ Then $u_{p}\left(x_{p}\right) \rightarrow \infty$ as $p \rightarrow p_{\alpha}.$ Let $\mu_{p} = u_{p}^{p-2+\frac{p}{2-p}}\left(x_{p}\right)$ and $\Omega_{\mu} = \frac{\Omega-x_{p}}{\mu_{p}}: = \left\{z | z = \frac{x-x_{p}}{\mu_{p}} \text { for } x \in \Omega\right\}$. We define $g_{p}(z) = \mu_{p}^{\frac{-p^{2}+4+p}{p-2}} u_{p}\left(\mu_{p} z+x_{p}\right)$ for $z \in \Omega_{\mu}$. Then, $g_{p}$ satisfies

and $g_{p}(0) = 1, g_{p}(z) \in(0, 1]$.

For $p$ close to $p_{\alpha}$ with $1 < \alpha < n$, we have $(1-p)((p-2+\frac{p}{p-1})n)(\frac{p-1}{2-p}+1) < 0$.

Additionally, let $\Omega_{R}^{c} = \Omega \backslash \bar{B}_{R \mu_{p}}\left(x_{p}\right).$ For $p$ close to $p_{\alpha},$ we know $\alpha < n / p.$ We can observe that for any fixed $\left|x-x_{p}\right| < C \mu_{p},$ as $R$ being chosen large enough

Thus,

as $p \rightarrow p_{\alpha}$ and $R \rightarrow \infty.$ As $p \rightarrow p_{\alpha},$ there are two cases:

Case 1. $\Omega_{\mu} \rightarrow \mathbb{R}^{n},$ and $u_{p}(z) \rightarrow g(z)$ point-wise in $\mathbb{R}^{n}$ where $g(z)$ satisfies from estimates $(2.5) \text { and }(2.6):$

Also, direct computation yields

Thus $\int_{\mathbb{R}^{n}} g^{p_{\alpha}} d z \leq 1.$ Combining this with (2.7) and Lemma 2.2, we have

This is a contradiction.

Case 2. $\Omega_{\mu} \rightarrow \mathbb{R}_{T}^{n}: = \left\{\left(z_{1}, z_{2}, \cdots, z_{n}\right) | z_{n} > -T\right\}$ for some $T > 0, g_{q}(z) \rightarrow g(z)$ pointwise in $\mathbb{R}_{T}^{n}, \text { where } g(z)$ satisfies from estimates $(2.5) \text { and }(2.6):$

Similarly, we know $\int_{\mathbb{R}^{n}} g^{p_{\alpha}} d z \leq 1.$ Combining this with (2.7), $A_{2}$ and Lemma 2.2, we have

This is a contradiction.

Let $u_{p} > 0$ be solutions to (2.3) for $p \in\left(p_{\alpha}, 2\right)$ which are also the minimizers of the energy $Q_{p}(\Omega).$ Then, $\left\|u_{p}\right\|_{L^{\infty}(\bar{\Omega})} \leq C,$ which yields that $u_{p}$ is uniformly bounded and equi-continuous due to Eq (2.3). Thus, $u_{p} \rightarrow u_{*}$ as $p \rightarrow p_{\alpha}$ in $C(\bar{\Omega}),$ and $u_{*}$ is the energy minimizer for $Q_{p_{\alpha}}(\Omega).$ □

Lemma 2.5. Let $f$ be a non-negative function satisfying ($A_{1}$), ($A_{2}$). For every $u \in \Lambda, u \neq 0$, $p = p_{\alpha}$ we have

(i.e., $\Lambda_{0} = \{0\}$).

Proof. By contradiction, assume that for some $u \in \Lambda$ with $u \neq 0$, we have

Thus, we have

Using the HLS inequality and the condition (2.8), we have

and from (2.9) we obtain:

which leads to a contradiction. □

As a consequence of Lemma 2.5 we have:

Lemma 2.6. Let $f(x)$ be a non-negative function satisfying ($A_{1}$), ($A_{2}$). Given $u \in \Lambda, u \neq 0$, $p = p_{\alpha},$ there exist $\varepsilon > 0$ and a differentiable function $t = t(w) > 0, \ w \in L^{p}(\Omega), \ \|w\| < \varepsilon$ satisfying the following properties:

and

Proof. Define $F: \mathbb{R} \times L^{p}(\Omega) \rightarrow \mathbb{R}$ as follows:

Since $F(1, 0) = 0$ and $\mathrm{F}_{t}(1, 0) = (p-1)\|u\|_{p}^{p}-\int_{\Omega} \int_{\Omega} \frac{u(x) u(y)}{|x-y|^{n-\alpha}} d x d y \neq 0$ (by Lemma 2.5), we can apply the implicit function theorem at the point $(1, 0)$ and obtain the desired result. □

3.   Proofs of Theorems 1.1 and 1.2

Proof of Theorem 1.1. Let us denote

We will first show that $I$ is bounded from below in $\Lambda$. For $u \in \Lambda$, we have:

Thus,

In particular, we have

To obtain an upper bound for $c_{0},$ let $v \in L^{p}(\Omega)$ be a positive solutions for $u^{p-1} = \int_{\Omega} \frac{ f(y)}{|x-y|^{n-\alpha}} d y.$ So, for $f \neq 0$

Set $t_{0} = t^{-}(v) > 0$ as defined by Lemma 2.1. It follows that $t_{0} v \in \Lambda^{+}$ and

This implies

It is clear that Ekeland's variational principle (see ^[1], Corollary 5.3.2) holds for the minimization problem (3.1). This principle provides a minimizing sequence $\left\{u_{m}\right\} \subset \Lambda$ with the following properties: (ⅰ) $\mathrm{I}\left(u_{m}\right) < c_{0}+\frac{1}{m},$ (ⅱ) $\mathrm{I}(w) \geq \mathrm{I}\left(u_{m}\right)-\frac{1}{m}\left\|\left(w-u_{m}\right)\right\|_{p}, \forall w \in \Lambda.$ By taking $m$ large, from (3.2) we have

Thus, it follows that

Therefore, we have $u_{m} \neq 0.$ By applying HLS inequality, $u_{m} \neq 0$ and (3.3), we obtain

Using HLS inequality and (3.4), we have

Applying (3.5) and (3.6), we obtain

Our goal is to show that $\left\|\mathrm{I}^{\prime}\left(u_{m}\right)\right\|_{p} \rightarrow 0$ as $m \rightarrow +\infty$. Hence, let us assume $\left\|I^{\prime}\left(u_{m}\right)\right\|_{p} > 0$ for $m$ large (otherwise we are done). Applying Lemma 2.6 with $u = u_{m}$ and $w = \delta \frac{I^{\prime}\left(u_{m}\right)}{\left\|I^{\prime}\left(u_{m}\right)\right\|_{p}}, \ \delta > 0$ small, we find $t_{m}(\delta): = t\left[\delta \frac{\mathrm{I}^{\prime}\left(u_{m}\right)}{\left\|\mathrm{I}^{\prime}\left(u_{m}\right)\right\|_{p}}\right]$ such that

Using condition (ii) we have

Dividing by $\delta > 0$ and passing to the limit as $\delta \rightarrow 0$ we derive

where we set $t_{m}^{\prime}(0) = \langle t^{\prime}(0), \frac{\mathrm{I}^{\prime}\left(u_{m}\right)}{\left\|\mathrm{I}^{\prime}\left(u_{m}\right)\right\|_{p}}\rangle$. Thus, from (3.7) we conclude that

for a suitable positive constant $\mathrm{C}$. We do this once we show that $\left|t_{m}^{\prime}(0)\right|$ is bounded uniformly on $m.$ From (2.10) and the estimate (3.7) we get

$\mathrm{C}_{1} > 0$ suitable constant. Hence, we need to show that $\mid(p-1)\|u_{m}\|_{p}^{p}-\int_{\Omega} \int_{\Omega} \frac{u_{m}(x) u_{m}(y)}{|x-y|^{n-\alpha}} d x d y\mid$ is bounded away from zero.

On the contrary, suppose that for a subsequence which we still call $u_{m}$ we have

Using the estimates (3.7) and (3.8), we obtain

and

Furthermore, combining (3.8) with the fact that $u_{m} \in \Lambda$ we also have

This, together with (3.9) and Lemma 2.4 implies

which is clearly impossible. Therefore,

Let $u_{0} \in L^{p}(\Omega)$ be the weak limit in $L^{p}(\Omega)$ of (a subsequence of) $u_{m}.$ From (3.7) we derive that

and from (3.10) we have

i.e., $u_{0}$ is a weak solution for (1.1). In particular, $u_{0} \in \Lambda$. Therefore,

Consequently $u_{m} \rightarrow u_{0}$ strongly in $L^{p}(\Omega)$ and $I\left(u_{0}\right) = c_{0} = \inf _{\Lambda} I$. Also, from Lemma 2.1 and (3.10), we can conclude that $u_{0} \in \Lambda^{+}$. Finally, since $f\geq0$ we can easily deduce that $u_{0}\geq0$ from ^[22]. Therefore, for $p = p_{\alpha}$, the problem (1.1) has a positive solution $u_{0} \in \Lambda^{+}$.

Proof of Theorem 1.2. Let $\frac{2 n}{n+\alpha} < p < 2$(subcritical case) and

Similar to the proof of Theorem 1.1, we can show that there is a solution $u_{0}\in\Lambda^{+}$ to Eq (1.1) using compactness imbedding theorem (see Lemma 2.3). Analogously to the proof of the first solution, one can show that the Ekeland's variational principle gives a sequence $\left\{u_{m}\right\} \subset \Lambda^{-}$ satisfying:

Furthermore, by the compactness imbedding theorem (Lemma 2.3) it can be proved that the functional $I$ satisfies the usual $(P S)_{c_{1}}$ condition for the subcritical equation. For $\frac{2 n}{n+\alpha} < p < 2$, there is another solution $u_{1}\in\Lambda^{-}$ to Eq (1.1). We can also deduce that $u_{1}\geq0$ from $f\geq0$ (see ^[22]).

4.   Conclusions

In this paper, we demonstrate that under suitable assumptions on $f$, the integral equation admits a positive solution in $L^{\frac{2n}{n+\alpha}}\left(\Omega\right)$. Our approach combines the Ekeland variational principle, a blow-up argument, and a rescaling argument. Additionally, we establish the existence of multiple solutions for this equation in the subcritical case. In the next section, we will investigate the existence of multiple solutions in the critical case.

Use of AI tools declaration

The author declares that he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author would like to express his sincere gratitude to Prof. Qiuyi Dai and Prof. Xiaohui Yu for their invaluable guidance throughout this research. Their expertise and insightful comments have greatly contributed to the quality of this work.

Conflict of interest

The author declares there is no conflicts of interest.