Existence of solutions to mixed local and nonlocal anisotropic quasilinear singular elliptic equations

1.   Introduction

Our main purpose of this study is to investigate the existence of positive solutions to the following mixed local and nonlocal quasilinear singular elliptic equation

where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N}(N > 2)$, $\Delta_{\vec{p}}u$ is an anisotropic version of the $p$-Laplace operator, which is sometimes referred as the pseudo $p$-Laplace operator,

where $\vec{p} = (p_{1}, p_{2}, ..., p_{N})$, $p_{i}\geq2$ for all $i = 1, 2, ..., N$. The fractional $p$-Laplace operator $\left(-\Delta \right)_{p}^{s}$, $(s\in(0, 1), p\geq1)$ is defined by

where $\text{P.V.}$ denotes the Cauchy principal value.

Recently, there has been increasing attention focused on the study of elliptic operators that involve mixed local and nonlocal operators.These equations often arise spontaneously in the study of plasma physics and population dynamics^[1,2]. For some other related results of mixed local and nonlocal equation, see ^{[3,4,5,6,7,8]} and the references therein. In the nonlocal case $(0 < s < 1)$, Barrios et al. ^[9] investigated the existence and uniqueness results of positive solutions to the following problem with $p = 2$,

In the case $\delta > 0$, the existence of solutions to problem (1.2) obtained by the range of $\delta$ to the summability of $f$. In case $0 < \delta < 1$ and $1 < \delta$, Youssfi and Mahmoud ^[10] studied the existence of solutions to problem (1.2) with $p = 2$ under some suitable assumptions on the datum $f$. For further information, readers may refer to the related work ^[11,12,13] and references therein.

In the local case, Boccardo and Orsina ^[14] used the method of approximation to prove the existence of solutions to following the problem with $p = 2$,

They also studied the summability of these solutions when $\delta\in(0, \infty)$. Giacomoni and Schindler ^[15] employ variational methods proved the existence of solution to quasilinear elliptic problem for $p_{i} = p\in(1, \infty)$ with $\delta\in(0, 1)$. During the past few years, there has been a vast amount of literature devoted to studying the anisotropic operator, which has numerous applications in fluid dynamics and physical phenomena, (we refer readers to ^{[16,17,18,19]} and references therein). Miri ^[19] further extended some results of ^[14] to an anisotropic quasilinear singular elliptic problem with variable exponent $\delta(x)$, and obtained existence of a solution to this problem. Bal and Garain [20, Theorems 2.7 and Theorems 2.9] established existence and uniqueness of solutions to the following mixed singular problems

and

where $\Omega$ is a bounded smooth subset of $\mathbb{R}^{N}$, $N > 2$, $\delta > 0$, $\gamma > 1$, $f_{j}$, $g_{j}$ ($j = 1$, $2$) are nonnegative integrable functions,

When $g_{j} = 0$ ($j = 1, 2$), they obtained a solution to problems (1.4) and (1.5) associated with the following minimizing problems

and

Garain and Ukhlov [21, Theorems 2.13] proved the existence of solution to the following problem

It has been shown that problem (1.6) has a weak solution $u\in{W}_{\text{{0}}}^{1, p}(\Omega)$ when $\delta\in(0, 1]$ and $f\in L^{m}(\Omega)\setminus\{0\}$ with $m = (\frac{p^*}{1-\delta})^{\prime}$, where $p^* = \frac{Np}{N-p}$, while if $\delta\in(1, \infty)$ and $f\in L^{1}(\Omega)\setminus\{0\}$, then problem (1.6) has a weak solution $u\in{W}_{\text{{loc}}}^{1, p}(\Omega)$ with $u^{\frac{p+\delta-1}{p}}\in{W}_{\text{{0}}}^{1, p}(\Omega)$. Moreover, they proved that mixed Sobolev inequalities are both necessary and sufficient for the existence of weak solutions to such singular problems. For related results about mixed local and nonlocal elliptic operators see ^{[22,23,24,25,26,27,28,29,30]} and references therein.

Motivated by the results of the above cited papers, especially ^[20,21], the our purpose of this study is to establish the existence of solutions to problem (1.1) according to the range of the power exponent $\delta$ and to the summability of datum $f(x)$. The main results as follows:

Theorem 1.1. Let $0 < \delta < 1$ and $1 < \bar{p} < N$. Suppose that $f > 0$, $f\in L^{m}(\Omega)$ with $m\geq 1$. Then there exists a weak solution $u$ to problem (1.1) such that

(i) $u\in {{L}^{\infty}(\Omega)}$ if $m > \frac{N\bar{p}}{N{\bar{p}}-(N-\bar{p}){p_{N}}}$, where $\bar{p}$ satisfies

(ii) $u\in {{L}^{t}(\Omega)}$ if $\bar{m} < m < \frac{N\bar{p}}{N{\bar{p}}-(N-\bar{p}){p_{N}}}$, where

(iii) $u\in{W}_{0}^{1, q}(\Omega)$ if $1\leq m < \bar{m}$, where

Remark 1.2. Notice that when $p_{i} = 2$, the range of corresponding $m$ values is exactly the summability of solutions obtained in ^[14].

When $p_{i} = p$, then problem (1.1) reduces to problem (1.6). Therefore

(ⅰ) If $f\in L^{m}(\Omega)$ with $\frac{Np}{Np-(1-\delta)(N-p)} < m < \frac{N}{p}$, then the solutions $u$ to problem (1.6) satisfies $u\in {{L}^{t}(\Omega)}$ with

(ⅱ) If $f\in L^{m}(\Omega)$ with $1\leq m < \frac{Np}{Np-(1-\delta)(N-p)}$, then the solutions $u$ to problem (1.6) satisfies $u\in{W}_{\text{{0}}}^{1, q}(\Omega)$ with $q = \frac{mN(1-\delta-p)}{m(1-\delta)-N}$.

Theorem 1.3. Suppose that $\delta = 1$ and $1 < \bar{p} < N$, $f > 0$, $f\in L^{m}(\Omega)$ with $m > 1$. Then there exists a weak solution $u$ to problem (1.1) such that

(i) $u\in {{L}^{\infty}(\Omega)}$ if $m > \frac{N\bar{p}}{N{\bar{p}}-(N-\bar{p}){p_{N}}}$.

(ii) $u\in {{L}^{t}(\Omega)}$ if $1\leq m < \frac{N\bar{p}}{N{\bar{p}}-(N-\bar{p}){p_{N}}}$, where

Theorem 1.4. Let $\delta > 1$ and $1 < \bar{p} < N.$ Suppose that $f > 0$, $f\in L^{m}(\Omega)$ with $m > 1$. Then there exists a weak solution $u$ to problem (1.1) such that

(i) $u\in {{L}^{\infty}(\Omega)}$ if $m > \frac{N\bar{p}}{N{\bar{p}}-(N-\bar{p}){p_{N}}}$.

(ii) $u\in {{L}^{t}(\Omega)}$ if $1\leq m < \frac{N\bar{p}}{N{\bar{p}}-(N-\bar{p}){p_{N}}}$, where

The order of the article is organized as follows: In Section 2, we provide basic notations and algebraic inequalities needed in this paper, as well as some definitions and useful lemmas. In Section 3, we present the proofs of Theorem 1.1, Theorem 1.3 and Theorem 1.4.

2.   Preliminaries and auxiliary results

2.1.   Preliminaries

In this article, we will use the following notations:

For any $v$, we denote by $v^{+} = \max\{v, 0\}$, $v^{-} = \max\{-v, 0\}$. For $p > 1$, we denote by $p^{\prime} = \frac{p}{p-1}$ to mean the conjugate exponent of $p$.

Definition 2.1. Let $p > 1$, $\Omega\subset\mathbb{R}^{N}$ with $N > 2$. The fractional Sobolev space $\mathcal{W}^{s, p}(\Omega)$ is defined by

with

The space $\mathcal{W}^{s, p}(\mathbb{R}^{N})$ and $\mathcal{W}_{\text {loc}}^{s, p}(\Omega)$ are defined analogously. The space $\mathcal{W}_{0}^{s, p}(\Omega)$ is defined as

Both $\mathcal{W}^{s, p}(\Omega)$ and $\mathcal{W}_{0}^{s, p}(\Omega)$ are reflexive Banach spaces ^[31].

Recall that the Lebesgue space $L^{p_{i}}(E)$ is defined as the space of $p_{i}$-integrable functions $u:E\rightarrow \mathbb{R}$ with the finite norm

where $p_{i}\in(1, +\infty)$ for all $i = 1, 2, ..., N$.

The anisotropic Sobolev space is defined as follows:

and

endowed with the norm

Definition 2.2. A function $\mathcal{J}$: ${\mathcal{W}}_{0}^{1, p_{i}}\rightarrow \mathbb{R}$ is defined to be weakly lower semi-continuous if

for any sequence $u_{n}$ approaching $u\in {\mathcal{W}}_{0}^{1, p_{i}}$ in the weak topology on ${\mathcal{W}}_{0}^{1, p_{i}}$.

The zero Dirichlet boundary condition in this paper is defined as follows:

Definition 2.3. We say that $u\leq0$ in $\mathbb{R}^{N}\backslash\Omega$ if $u = 0$ in $\mathbb{R}^{N}\backslash\Omega$ and for any $\epsilon > 0$, we have

We say that $u = 0$ on $\mathbb{R}^{N}\backslash\Omega$, if $u$ is nonnegative and $u\leq0$ in $\mathbb{R}^{N}\backslash\Omega$.

The definition of weak solution in this paper is defined as

Definition 2.4. A positive function $u\in{W}_{\text{loc}}^{1, p_{i}}(\Omega)\cap L^{p_{i}-1}(\mathbb{R}^{N})$ is a weak solution to problem (1.1) if

for every $\phi\in C_{c}^{1}(\Omega)$, we have that

where

and

Lemma 2.5. [19, Theorem 1.2] There exists a positive constant $C$, such that for every $u\in{W}^{1, p_{i}}(\Omega)$, we have

where

and

Lemma 2.6. [32, Lemma 2.1] Let $1 < p_{i} < \infty.$ Then for $\xi$, $\eta\in\mathbb{R}^{N}$, there exists a constant $C = C(p_{i}) > 0$ such that

Lemma 2.7. [33, Lemma 2.1] Let $f$: $\mathbb{R}^{+}\rightarrow \mathbb{R}^+$ be a non-increasing function such that

where $M > 0$, $\alpha > 1$ and $\beta > 0$. Then $\psi(d) = 0$, where $d^{\beta} = C\psi(0)^{\alpha-1}2^{\frac{\alpha\beta}{(\alpha-1)}}$.

2.2.   Auxiliary results

For $n\in\mathbb{N}$, $f(x)\in L^{1}(\Omega)$ and $f(x) > 0$, let $f_{n}(x): = \min\{f(x), n\}$ and we consider the following approximated problem

First, we consider the following useful result.

Lemma 2.8. Let $g(x)\in L^{\infty}(\Omega)$, $g(x)\geq0$. Then the following elliptic problem

has a unique positive weak solution $u\in \mathcal{W}_{0}^{1, p_{i}}(\Omega)$.

Proof. ${\bf Existence}$: Define the energy functional $\mathcal{J}: \mathcal{W}_{0}^{1, p_{i}}(\Omega)\rightarrow \mathbb{R}$ as

where

and

(ⅰ) By the Sobolev embedding theorem and $g\in L^{\infty}(\Omega)$, we have

which implies the $\mathcal{J}$ is coercive.

(ⅱ) $\mathcal{J}(v)$ is weakly lower semi-continuous in $\mathcal{W}_{0}^{1, p_{i}}(\Omega)$.

It easy to see that $\mathcal{J}_{1}(u)$ is differentiable, according to [34, Lemma 3.4], we know that $\mathcal{J}_{i}(u), i = 1, 2$ are weakly lower semi-continuous. On the other hand, we know that $\mathcal{J}_{3}(u)$ is a bounded linear functional. Thus $\mathcal{J}_{3}(u)$ is continuous. Therefore, $\mathcal{J}$ is weakly lower semi-continuous and

Combining the above properties of $\mathcal{J}$, we know that there exists a minimizer $u\in \mathcal{W}_{0}^{1, p_{i}}(\Omega)$ and which is also a critical point of $\mathcal{J}$, which also is the solution to Eq (2.6).

${\bf Uniqueness}$: Let $u_{1}$, $u_{2}\in \mathcal{W}_{0}^{1, p_{i}}(\Omega)$ be two solutions to problem (2.6). Thus, for any $\varphi\in \mathcal{W}_{0}^{1, p_{i}}(\Omega)$, we have

and

Choosing $\varphi = u_{1}-u_{2}$ and then subtracting (2.7) and (2.8), we obtain

Using Lemma 2.6, we get the first term of the left hand side of (2.9) is nonnegative. On the other hand, by the monotonicity of the function $f(t) = t^{p-1} (p > 1)$, we have

Consequently,

Therefore, $u_{1}(x)-u_{2}(x) = C$ for all $x\in\mathbb{R}^{N}$. Note that $u_{1}-u_{2} = 0$ on $\mathbb{R}^{N}\backslash\Omega$ since $u_{i}(x) = 0$ for $x\in\mathbb{R}^{N}\backslash\Omega$. Thus $u_{1}(x)\equiv u_{2}(x)$, which implies that the solution of (2.6) is unique.

${\bf Boundedness}$: For any $k > 1$, decompose $\mathbb{R}^{N}$ as $\mathbb{R}^{N} = A_{k}\cup A^{c}_{k}$, where

Taking $G_{k}(u): = (u-k)^{+} = \max\{u-k, 0\}$ as a test function in (2.6), we have

Obviously,

Therefore, combining (2.13) and (2.12) with Sobolev embedding theorem, we have

Therefore

For every $1\leq k < h$ we know that $A(h)\subset A(k)$ and $u(x)-k\geq(h-k)$ in $A(h)$, we get

Hence, we have

Obviously,

Hence, using Lemma 2.7 we obtain

${\bf Positivity}$: First, taking $u_{-}(x): = \min\{u(x), 0\}$ as a test function in (2.6) and using $g\geq0$, we have

where $\mathcal{K}u(x, y) = |u(x)-u(y)|^{p-2}(u(x)-u(y))$. Rewrite

Denote,

Therefore,

Obviously,

Using (2.16) in (2.15) we obtain

Therefore, $u_{-} = C$ for all $x\in\mathbb{R}^{N}$. Note that $u_{-} = 0$ on $\mathbb{R}^{N}\backslash\Omega$ since $u_{-}: = \min\{u, 0\}$. Thus $u\geq0$ in $\Omega$.

Second, assume that there exists a point $x_0\in\Omega$ such that $u(x_0) = \inf_{x\in\Omega} u(x) = 0$, thus

This is a contradiction since $g(x_0)\geq0$. Hence, $u > 0$ in $\Omega$. □

Lemma 2.9. For any $n\in\mathbb{N}$, there exists a unique positive solution $u_{n}\in \mathcal{W}_{0}^{1, p_{i}}(\Omega)\cap L^{\infty}(\Omega)$ to problem (2.5). Moreover, The sequence $\{u_{n}\}$ is increasing with respect to $n$ and

Proof. ${\bf Step 1.}$ ${\bf (Existence)}$ Let $n\in\mathbb{N}$. By Lemma 2.8, for every $u\in \mathcal{W}_{0}^{1, p_{i}}(\Omega)\cap L^{\infty}(\Omega)$, there exists a unique $v\in \mathcal{W}_{0}^{1, p_{i}}(\Omega)\cap L^{\infty}(\Omega)$ such that

Define the operator $\mathcal{T}: u\mapsto v = \mathcal{T}(u)$, where $v$ is the unique solution to (2.18). Choosing $v$ as a test function in (2.18), using Sobolev imbedding theorem, we obtain

Thus

which implies that the ball with radius $R$ in $\mathcal{W}_{0}^{1, p_{i}}(\Omega)$ remains unchanged under ${\mathcal{T}}$.

Now, we have to prove the continuity and compactness of $\mathcal{T}$, which is an operator from $\mathcal{W}_{0}^{1, p_{i}}(\Omega)$ to $\mathcal{W}_{0}^{1, p_{i}}(\Omega).$

(ⅰ) Continuity of $\mathcal{T}$: In order to do this, we have to show that $\lim_{k\rightarrow \infty}\|v_{k}-v\|_{\mathcal{W}_{0}^{1, p_{i}}(\Omega)} = 0$ if $\lim_{k\rightarrow \infty}\|u_{k}-u\|_{\mathcal{W}_{0}^{1, p_{i}}(\Omega)} = 0$, where $v_{k} = \mathcal{T}(u_{k})$ and $v = \mathcal{T}(u)$.

Choosing $\bar{v}_{k}(x) = {v}_{k}(x)-{v}(x)$ as a test function of the equations of $v_{k}$ and $v$ respectively, using (2.19), we get

Using Hölder and Sobolev inequalities we infer that

By (2.19) and (2.20), using [10, Lemma 2.2], we find

since the pointwise convergence of ${u}_{k}\rightarrow {u}$ in $\mathcal{W}_{0}^{1, p_{i}}(\Omega)$. we get

Therefore, in the case $p_{i}\geq2$, the operator $\mathcal{T}$ is continuous from $\mathcal{W}_{0}^{1, p_{i}}(\Omega)$ to $\mathcal{W}_{0}^{1, p_{i}}(\Omega)$.

(ⅱ) Compactness of ${\mathcal{T}}$: To achieve this, we have to show that, for some $v\in{W}_{0}^{1, p_{i}}(\Omega)$, it holds

Let ${u_{k}}$ be a bounded sequence in $\mathcal{W}_{0}^{1, p_{i}}(\Omega)$ and $v_{k}: = \mathcal{T}(u_{k})$. Then we have

According to (2.18), we have $\|\mathcal{T}(u_{k})\|_{\mathcal{W}_{0}^{1, p_{i}}(\Omega)}\leq C$. Therefore there exists a subsequence, still denoted by $\{u_{k}\}$, such that

For any $\varphi\in\mathcal{W}_{0}^{1, p_{i}}(\Omega)$,

Now, we show that as $k\rightarrow \infty$, (2.22) converges to

By the dominated convergence theorem, we have

and

Therefore, for every $\varphi\in\mathcal{C}_{c}^{1}(\Omega)$, we have

Since $\varphi\in\mathcal{C}_{c}^{1}(\Omega)$ and $v_{k}$ is uniformly bounded in $\mathcal{W}_{0}^{1, p_{i}}(\Omega)$,

by the pointwise convergence of $v_{k}(x)$ to $v(x)$

Since

we get that the (2.22) converges to the (2.23). Similarly, combining (2.21) and (2.18), we have

Therefore, the operator $\mathcal{T}$ is continuous from $\mathcal{W}_{0}^{1, p_{i}}(\Omega)$ to $\mathcal{W}_{0}^{1, p_{i}}(\Omega)$. Then, Schauder fixed point theorem implies the existence of a fixed points $u_{n}$ such that $u_{n} = \mathcal{ T}(u_{n})$, which is a weak solution to approximated problem (2.5).

${\bf Step 2.}$ ${\bf (Monotonicity)}$ Since $u_{n}$ and $u_{n+1}$ are positive solutions to problem (2.8), for any $\varphi\in \mathcal{W}_{0}^{1, p_{i}}(\Omega)$, we have

and

Taking $\varphi = \left(u_{n}(x)-u_{n+1}(x)\right)^{+}$ as test function in (2.24) and (2.25), we get

and

Since $f_{n}(x)\leq f_{n+1}(x)$ for $x\in\Omega,$ we have

Subtracting (2.26) with (2.27) and using the (2.28), we obtain

Following the argument in the proof of [35, Lemma 9], we obtain

Therefore, applying (2.30) in (2.29) we get

Using Lemma 2.6 we obtain

Note that $u_{n}(x) = u_{n+1}(x) = 0$ on $\mathbb{R}^{N}\backslash\Omega$ thus $C = 0$, which implies that $u_{n+1}(x)\geq u_{n}(x)$ in $\Omega$.

${\bf Step 3.}$ ${\bf (Uniform\; Positivity)}$ Let $u_{1}$ solves (2.6). By Lemma 2.8, for every $K\Subset\Omega$, there exists a constant $C_{K} > 0$ such that $u_{1}\geq C_{K} > 0$ in $K$. Again, since the monotonicity of $u_{n}$, we have $u_{n}\geq u_{1}$ in $K$. Therefore, for any $K\Subset\Omega$,

□

3.   Proof of main results

In order to prove the existence of positive solution to (1.1), we use the sequence of solutions $u_{n}$ of problem (2.5). Then we need a priori estimates on $u_{n}$.

3.1.   Auxiliary lemma

Lemma 3.1. Let $0 < \delta < 1$ and $1 < \bar{p} < N$. Suppose that $f > 0$, $f\in L^{m}(\Omega)$ with

Then, the sequence solutions $u_{n}$ to the approximate problem (2.5) such that

$(i)$ $u_{n}\in {{L}^{\infty}(\Omega)}$ if $m > \frac{N\bar{p}}{N{\bar{p}}-{p_{N}}(N-\bar{p})}$.

$(ii)$ $u_{n}\in {{L}^{t}(\Omega)}$, where $t = \frac{m(1-\delta-p_{i})N\bar{p}}{N\bar{p}(m-1)-p_{i}m(N-\bar{p})}$ if

Proof. $(i)$ Let $A_{k} = \left\{x\in\Omega: u_{n}(x)\geq k\right\}$. Choosing $G_{k}(u): = \left(u-k\right)^{+}\in \mathcal{W}_{0}^{1, p_{i}}(\Omega)$ as a test function in (2.5), we get

Foe any $k > 1$, by (2.13) we know that,

By Hölder inequality, Sobolev embedding theorem, $f_{n}(x)\leq f(x)$ and (3.1), we have

Hence

Let $h > k\geq1$, we know that $A_{h}\subset A_{k}$ and $G_{k}(u_{n})\geq h-k$ for in $\Omega$, we have that

Therefore

Note that

if $m > \frac{N\bar{p}}{N{\bar{p}}-{p_{N}}(N-\bar{p})}$. Hence, apply Lemma 2.7 with

there exists $k_{0}$ such that $\psi(k)\equiv0$ for all $k\geq k_{0}$. Thus,

(ⅱ) Choose $u_{n}^{p_{i}(\gamma-1)+1}$ ($\gamma > 1$) as test function in (2.17), we have

According to [10, Lemma 2.2], we have

Combining (3.4) and (3.3), and using Hölder inequality, we get

By Sobolev inequality,

Therefore,

Now we choose $\gamma$ such that

that is

Since $\gamma > 1$, we know

Thus $\frac{p_{N}}{\bar{p}^*} > \frac{1}{m^{\prime}}$ gives

Therefore, $u_{n}$ is uniformly bounded in ${{L}^{t}(\Omega)}$ with $t = \gamma\bar{p}^*$. □

Lemma 3.2. Let $0 < \delta < 1$ and $1 < \bar{p} < N$. Suppose that $f > 0$ and $f\in L^{m}(\Omega)$ with

Then, the sequence solutions $\{u_{n}\}$ to the approximate problem (2.5) are uniformly bounded in $\mathcal{W}_{\mathit{\text{{0}}}}^{1, q}(\Omega)$ with

Proof. Similar to above taking $u_{n}^{p_{i}(\gamma-1)+1}$ as test function in (2.5) with $\frac{\delta+p_{i}-1}{p_{i}}\leq\gamma < 1$. However, this option is not acceptable, since the gradient of such a test function will be singular where $u_n(x) = 0$. Hence, for $n$ fixed, choose $(u_{n}+\varepsilon)^{p_{i}(\gamma-1)+1}-\varepsilon^{p_{i}(\gamma-1)+1}$ ($0 < \varepsilon < \frac{1}{n}$) as test function in (2.5), we get

By $f_{n}(x)\leq f(x)$ and $\varepsilon < \frac{1}{n}$, we have

By Sobolev inequality,

Hence, by (3.6) and (3.7), we get

Let $\varepsilon\rightarrow 0$, we have

If $m = 1$, we choose $\gamma = \frac{p_{i}+\delta-1}{p_{i}}$ in the (3.8), so that $u_{n}\in{{L}^{\frac{(\delta+p_{i}-1)N\bar{p}}{{p_{i}}(N-\bar{p})}}(\Omega)}$.

If $m > 1$, from the proof of Lemma 3.1, we get that $u_{n}(x)\in{{L}^{t}(\Omega)}$ with

Since $\gamma < 1$, by (3.6), we have

We can apply Hölder inequality (since $q < p_{i}$),

Choice $\gamma$ and $q$ such that

Therefore, $u_{n}\in \mathcal{W}_{\text{{0}}}^{1, q}(\Omega)$ with

□

Lemma 3.3. Suppose that $\delta = 1$ and $1 < \bar{p} < N$, $f > 0$, $f\in L^{m}(\Omega)$ with $m > 1$. Then there exists a weak solution $u_{n}$ to problem (2.5) such that

(i) If $m > \frac{N\bar{p}}{N{\bar{p}}-{p_{N}}(N-\bar{p})}$, Then $u_{n}\in {{L}^{\infty}(\Omega)}$;

(ii) If $1\leq m < \frac{N\bar{p}}{N{\bar{p}}-{p_{N}}(N-\bar{p})}$, Then $u_{n}\in{{L}^{t}(\Omega)}$, where

Proof. The proof of $(i)$ is identical to that of Lemma 3.1, so we will omit it.

As for $(ii)$, observe that if $m = 1$, then $t = \frac{N\bar{p}}{N-\bar{p}} = \bar{p}^*$. If $m > 1$, similar to Lemma 3.1, Choosing $u_{n}^{p_{i}(\gamma-1)+1}$ as test function in (2.5), we know that there is

Choose $\gamma$ such that

Obviously

Since $\gamma > 1$, we arrive at $1 < m$. Thus $\frac{p_{N}}{\bar{p}^*} > \frac{1}{m^{\prime}}$ being

so that $u_{n}\in {L}^{t}(\Omega)$ with $t = \gamma\bar{p}^*$. □

Simple modifications to the proof of Lemma 3.1 enable us to demonstrate Lemma 3.4.

Lemma 3.4. Suppose that $\delta > 1$ and $1 < \bar{p} < N$, $f > 0$, $f\in L^{m}(\Omega)$ with $m > 1$. Then there exists a weak solution $u_{n}$ to problem (2.5) such that

(i) If $m > \frac{N\bar{p}}{N{\bar{p}}-{p_{N}}(N-\bar{p})}$, then $u_{n}\in {{L}^{\infty}(\Omega)}$.

(ii) If $1\leq m < \frac{N\bar{p}}{N{\bar{p}}-{p_{N}}(N-\bar{p})}$, then $u_{n}\in{{L}^{t}(\Omega)}$ with

Proof. The proof of $(i)$ is identical to that given in Lemma 3.1, so we omit it.

For $(ii)$, by [21, Lemma 3.7], we known, if $m = 1$, the sequence $u_{n}^{\frac{\delta+p_{i}-1}{p_{i}}}$ is uniformly bounded in $\mathcal{W}_{0}^{1, p_{i}}(\Omega)$, This also gives $u_{n}$ is bounded in $\mathcal{W}_{\text {loc}}^{1, p_{i}}(\Omega)$.

If $1 < m < \frac{N\bar{p}}{N{\bar{p}}-{p_{N}}(N-\bar{p})}$, similar to Lemma 3.1, taking $u_{n}^{p_{i}(\gamma-1)+1}$ as test function in (2.5), this time with $\gamma > 1$ since $\gamma > {\frac{\delta+p_{i}-1}{p_{i}}}$, we have

Choosing $\gamma$ in such a way that

since $\gamma > {\frac{\delta+p_{i}-1}{p_{i}}}$ gives $m > 1$, and by $\frac{p_{N}}{\bar{p}^*} > \frac{1}{m^{\prime}}$ being

Therefore, $u_{n}$ is uniformly bounded in ${{L}^{t}(\Omega)}$ as well. □

3.2.   Proof of main theorem

In this section, we give the proof of Theorem 1.1 by the approximate method.

Proof of Theorem 1.1. Let $f\in L^{m}(\Omega)$. By Lemma 3.2 and 3.1, we know that the solutions $u_{n}$ to problem (2.5) are bounded in $\mathcal{W}_{0}^{1, p_{i}}(\Omega)$. Then, the pointwise limit $u$ in $\mathcal{W}_{0}^{1, p_{i}}(\Omega)\cap{L}^{p_{i}-1}(\Omega)$. For any $\varphi\in\mathcal{W}_{0}^{1, p_{i}}(\Omega)$,

Then, for any $\varphi\in C_{c}^{1}(\Omega)$, we get

Since $\left\{u_{n}\right\}$ is uniformly bounded in $\mathcal{W}_{0}^{1, p_{i}}(\Omega)$,

By point-wise convergence of $u_{n}(x)$ to $u(x)$

Then, we have

By Lemma 2.9, for any $K\Subset\Omega$, $u_{n}(x)\geq C_{K} > 0$ with $\operatorname{supp}(\varphi) = C_{K} > 0$. Therefore, for any $\varphi\in C_{c}^{1}(\Omega)$ such that

We conclude that

Finally, passing to the limit in (3.9), we conclude that

for all $\varphi\in C_{c}^{1}(\Omega)$, which shows that $u$ is a solution to problem (1.1) and $u\in \mathcal{W}_{\text{{0}}}^{1, q}(\Omega)$. □

Proof of Theorem 1.3 and Theorem 1.4. The proof of Theorem 1.3 and 1.4 are similar, here we omit the details. □

4.   Conclusions

The manuscript establishes the existence of solutions to mixed local and nonlocal anisotropic quasilinear singular elliptic eqautions. The interplay between the integrability and the singularity power is investigated. This results generalizes and complements the existing results.

Use of AI tools declaration

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

Acknowledgments

This works was partially supported by Fundamental Research Funds for the Central Universities (No. 31920220041) and Innovation Team Project of Northwest Minzu University (No. 1110130131).

Conflict of interest

The authors declare no competing interests.