Qualitative properties of solutions to generalized eigenvalue problems

Abdelhamid Gouasmia can be contacted at: gouasmia.abdelhamid@gmail.com

Conflict of interest: The author declares that there is no conflict of interest.

Purpose

This paper investigates the qualitative properties of solutions to a fractional (p,)-Laplace eigenvalue problem with nonhomogeneous terms. We aim to establish non-existence results for specific parameter ranges prove the existence of weak solutions using variational methods and analyze their regularity. Additionally, we refine the Diaz–Saa inequality to derive a uniqueness result in the case r = q. Our study extends existing results in the field of fractional and nonlocal operators, providing new insights into their mathematical structure and potential applications.

Design/methodology/approach

We analyze a fractional (p,)-Laplace eigenvalue problem using variational methods and functional analysis techniques. Non-existence results are established via test function arguments. Existence of solutions is proved using the mountain pass theorem and minimization techniques. Regularity properties are derived using Hölder estimates, and a refined Diaz–Saa inequality is employed to establish uniqueness in the case r = q. The fractional Sobolev embedding theorem and compactness arguments play a crucial role in our analysis. This approach extends existing methods for nonlocal and nonhomogeneous operators.

Findings

This study establishes the existence, non-existence, uniqueness and regularity properties of weak solutions to a fractional (p,)-Laplace eigenvalue problem. We prove non-existence for certain values of λ and obtain existence results using variational methods and the mountain pass theorem. Furthermore, we refine the Diaz–Saa inequality to establish uniqueness when r = q. Regularity results are derived using Hölder continuity estimates. These findings provide new insights into the qualitative behavior of solutions and extend existing results for fractional and nonhomogeneous operators.

Research limitations/implications

One limitation of this study is that the analysis is restricted to a bounded domain Ω with a smooth boundary. Extending the results to more general domains, including those with irregular boundaries, remains an open problem. Moreover, the uniqueness result established here applies only to the case r = q, and further investigation is required to address the case r = q. From an applied perspective, the fractional (p,)-Laplacian model could also be explored in practical scenarios, such as optimization and image processing, where nonlocal effects play a significant role.

Practical implications

The results of this study contribute to the mathematical foundation of fractional and nonhomogeneous operators, which have applications in various fields, including physics, engineering and image processing. The fractional (p,)-Laplace model can be utilized in problems involving nonlocal diffusion, anomalous transport, and phase transitions. Moreover, the uniqueness and regularity results provide a rigorous basis for numerical approximations and computational methods in the applied sciences. Future research may explore extensions to irregular domains and real-world applications where nonlocal effects play a crucial role.

Social implications

The study of fractional (p,)-Laplace equations has potential societal impact in areas where nonlocal phenomena play a key role, such as medical imaging, material science, and environmental modeling. Improved understanding of these operators can enhance techniques in image reconstruction, tumor growth prediction, and porous media analysis. Furthermore, the mathematical framework developed in this work can contribute to advancements in computational methods used in engineering and technology, ultimately leading to more efficient solutions in real-world applications that affect daily life.

Originality/value

This study provides novel contributions to the analysis of fractional (p,)-Laplace eigenvalue problems by extending classical techniques to a nonhomogeneous and nonlocal setting. The refinement of the Diaz–Saa inequality and the uniqueness result for r = q represent significant advancements in the field. Moreover, the combination of variational methods, compactness arguments and fractional Sobolev embeddings offers new insights into the existence and regularity of weak solutions. These results not only enhance theoretical understanding but also provide a foundation for future research in applied mathematics and related disciplines.

1. Introduction

The purpose of this paper is to study the existence, non-existence, uniqueness, and regularity of weak solutions to the following generalized eigenvalue problem involving fractional operators:

{(- Δ)}_{p}^{s_{1}} u + {(- Δ)}_{q}^{s_{2}} u = g (x, u) : = λ a_{r} (x) u^{r - 1}, u > 0 in Ω; u = 0, in R^{N} \ Ω, (P)

where r ∈ {p, q}, $a_{r} \in {(L^{\infty} (Ω))}^{+} \ {0}$ ⁠, and λ is a positive real parameter. The domain Ω is a bounded subset of $R^{N}$ with a boundary of class C^1,1, satisfying N > s₁p, 0 < s₂ ≤ s₁ < 1, and 1 < q < p < ∞. Here, ${(- Δ)}_{r}^{s}$ denotes the fractional r-Laplace operator, defined for s ∈ {s₁, s₂} and r ∈ {p, q} as

{(- Δ)}_{r}^{s} u (x) ≔ 2 P . V . \int_{R^{N}} \frac{| u (x) - u (y) |^{r - 2} (u (x) - u (y))}{| x - y |^{N + s r}} d y,

where P.V. denotes the Cauchy principal value. Nonlocal operators of this type have numerous applications in real-world problems, including optimization, finance, phase transitions, soft thin films, and image processing. Moreover, the fractional Laplacian serves as a mathematical model for describing certain jump Lévy processes in probability theory and porous media in physics, among other applications. For further discussions, see Refs. [1, 2, 3, 4] and the references therein.

1.1 State of the art

The non-homogeneous operator in problem (P), known as the fractional (p, q)-Laplacian, serves as the natural counterpart to the classical (p, q)-Laplace operator (−Δ_p − Δ_q). The latter arises in various fields, including biophysics, plasma physics, reaction-diffusion equations, and chemical reactions; see Refs. [5, 6, 7, 8] for further references.

In the local case (s₁ = s₂ = 1), problem (P) in its general form with a nonlinear term g has been extensively studied in the literature using various methods. In particular, Marano and Mosconi [7] provided a comprehensive survey on the existence and multiplicity of solutions to problem (P). More specifically, they investigated the existence and non-existence of solutions to generalized eigenvalue problems of the form:

g (x, u) = α {|u|}^{p - 2} u + β {|u|}^{q - 2} u,

where $(α, β) \in R^{2}$ ⁠; see Refs. [9, 10, 11] and the references therein for further details.

Furthermore, Tanaka [12] established the existence and non-existence of positive weak solutions to (P) in the special case:

g (x, u) = λ m r (x) u^{r - 1}, where r \in {p, q},

where m_r ∈ L^∞(Ω) and the Lebesgue measure of {x ∈ Ω∣m_r(x) > 0} is positive. Additionally, the uniqueness of a positive solution for this problem was established in Ref. [13] for r = q.

Returning to Ref. [7], the authors also compiled various results concerning the existence and multiplicity of solutions to problem (P), considering nonlinearities of the form:

g (x, u) = α {|u|}^{p - 2} u + β {|u|}^{q - 2} u + f (u),

where $f \in C^{1} (R)$ satisfies appropriate growth conditions at ±∞ and/or near zero. For related results, we refer the reader to Refs. [14, 15, 16, 17].

In the non-local framework, various authors have studied problem (P) through different analytical approaches. For instance, Goel [18] derived L^∞ estimates and interior Hölder regularity results (for p, q ≥ 2) for weak solutions of the system

\{\begin{aligned} {(- Δ)}_{p}^{s_{1}} u + β {(- Δ)}_{q}^{s_{2}} u & = λ a (x) {|u|}^{δ - 2} u + b (x) {|u|}^{r - 2} u & in Ω, \\ u & = 0 & in R^{N} \ Ω, \end{aligned}

where $1 < δ \leq q \leq p < r \leq p_{s_{1}}^{*}, 0 < s_{2} < s_{1} < 1,$ and N > ps₁. Here, λ and β are real parameters, while $a \in L^{\frac{r}{r - δ}} (Ω)$ and b ∈ L^∞(Ω) are sign-changing weight functions. Furthermore, Giacomoni et al. [19], [20] established global Hölder regularity for weak solutions of the problem

{(- Δ)}_{p}^{s_{1}} u + {(- Δ)}_{q}^{s_{2}} u = f in Ω,

under the assumptions 1 < q ≤ p < ∞, 0 < s₂ ≤ s₁ < 1, and $f \in L_{loc}^{γ} (Ω)$ with $γ > \frac{N}{p s_{1}}$ if N > ps₁ and γ ≥ 1 otherwise. In addition, they proved a Hopf-type maximum principle and a strong comparison principle. For further developments on the fractional (p, q)-Laplacian, see also [21, 22, 23]. We also highlight the recent contribution of Khieu and Nguyen [24], where the authors investigate the existence and non-existence of positive solutions to a related eigenvalue problem. Their analysis provides new insights into the spectral properties of the fractional (p, q)-Laplace operator with weights.

1.2 Objectives of this paper

The primary objectives of this work are to analyze the non-existence, existence, uniqueness, and Hölder regularity of solutions to problem (P). Specifically, by employing the variational method and the Mountain Pass Theorem, we establish the existence of a positive weak solution to (P) for r = q and r = p, respectively. Furthermore, the regularity estimates follow from the boundedness of weak solutions combined with the results obtained in Refs. [19, 20]. Additionally, a refined version of the Díaz-Saa inequality (see Lemma 2.4) ensures the uniqueness of weak solutions to (P) when r = q.

2. Preliminaries and main results

2.1 Notation and function spaces

To state our main results, we introduce the notation and function spaces that will be used throughout the paper:

Let $Ω \subset R^{N}$ be an open bounded domain with a boundary of class C^1,1, where N ≥ sp, 0 < s < 1, and p > 1.
The norm in the Banach space L^p(Ω) is given by

‖ u ‖_{L^{p} (Ω)} ≔ {(\int_{Ω} | u |^{p} d x)}^{\frac{1}{p}} .

The fractional Sobolev space $W^{s, p} (R^{N})$ is defined as

W^{s, p} (R^{N}) ≔ \{u \in L^{p} (R^{N}) : \int_{R^{N}} \int_{R^{N}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + s p}} d x d y < \infty\},

endowed with the norm

‖ u ‖_{W^{s, p} (R^{N})} ≔ {(‖ u ‖_{L^{p} (R^{N})}^{p} + \int_{R^{N}} \int_{R^{N}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + s p}} d x d y)}^{\frac{1}{p}} .

Consider the space

X_{p, s} ≔ \{u : R^{N} \to R : u \in L^{p} (Ω), \int_{Q} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + s p}} d x d y < \infty\},

where $Q = R^{N} \ (Ω^{c} \times Ω^{c})$ ⁠, equipped with the norm

‖ u ‖_{X_{p, s}} ≔ {(‖ u ‖_{L^{p} (R^{N})}^{p} + \int_{Q} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + s p}} d x d y)}^{\frac{1}{p}} .

The space $W_{0}^{s, p} (Ω)$ is defined as

W_{0}^{s, p} (Ω) ≔ \{u \in W^{s, p} (R^{N}) : u = 0 a.e. in R^{N} \ Ω\} .

The norm in $W_{0}^{s, p} (Ω)$ is given by the Gagliardo seminorm:

‖ u ‖_{W_{0}^{s, p} (Ω)} ≔ {(\int_{R^{N}} \int_{R^{N}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + s p}} d x d y)}^{\frac{1}{p}} .

(2.1)

We can equivalently define $W_{0}^{s, p} (Ω)$ as the closure of $C_{0}^{\infty} (Ω)$ in X_p,s, where

C_{c}^{\infty} (Ω) ≔ \{φ : R^{N} \to R ∣ φ \in C^{\infty} (R^{N}) and supp φ \subset \subset Ω\} .

By employing the fractional Poincaré inequality (see, for instance, [ [25], Theorem 6.5]), there exists a positive constant c > 0 such that

c^{- 1} ‖ u ‖_{W^{s, p} (R^{N})} \leq ‖ u ‖_{W_{0}^{s, p} (Ω)} \leq c ‖ u ‖_{W^{s, p} (R^{N})}, for all u \in W_{0}^{s, p} (Ω) .

It is well known that $W_{0}^{s, p} (Ω)$ is a uniformly convex Banach space endowed with the norm (2.1). Consequently, by [ [26], Proposition 1.3], the fractional p-Laplacian operator is of type (S), meaning that every sequence $(u_{n}) \subset W_{0}^{s, p} (Ω)$ satisfying u_n ⇀ u in $W_{0}^{s, p} (Ω)$ and

⟨J^{'} (u_{n}), u_{n} - u⟩ \to 0,

admits a subsequence that converges strongly to u.

Here, the functional $J$ is defined by

J : W_{0}^{s, p} (Ω) \to R, J (u) = \frac{1}{p} \int_{R^{N}} \int_{R^{N}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + s p}} d x d y,

and for every $u, v \in W_{0}^{s, p} (Ω)$ ⁠, we have

⟨J^{'} (u), v⟩ = \int_{R^{N}} \int_{R^{N}} \frac{{|u (x) - u (y)|}^{p - 2} (u (x) - u (y)) (v (x) - v (y))}{| x - y |^{N + s p}} d x d y .

It is well known that $W_{0}^{s, p} (Ω)$ is continuously embedded in L^t(Ω) for $1 \leq t \leq p_{s}^{*}$ and compactly embedded for $1 \leq t < p_{s}^{*}$ ⁠, where $p_{s}^{*} ≔ \frac{N p}{N - s p}$ (see [ [25], Theorem 6.5] for further details).
Next, we denote by d(x) the distance from a point $x \in \bar{Ω}$ to the boundary ∂Ω, where $\bar{Ω} = Ω \cup \partial Ω$ denotes the closure of Ω, i.e.,

d (x) ≔ dist (x, \partial Ω) = \inf_{y \in \partial Ω} | x - y | .

For $α \in (0,1]$ ⁠, we consider the Hölder space:

C^{0, α} (\bar{Ω}) ≔ \{u \in C (\bar{Ω}) ∣ \sup_{x, y \in \bar{Ω}, x \neq y} \frac{|u (x) - u (y)|}{{|x - y|}^{α}} < \infty\},

endowed with the Banach norm:

‖ u ‖_{C^{0, α} (\bar{Ω})} = ‖ u ‖_{L^{\infty} (Ω)} + \sup_{x, y \in \bar{Ω}, x \neq y} \frac{|u (x) - u (y)|}{{|x - y|}^{α}} .

For 1 < r < ∞ and a given function m_r ∈ L¹(Ω), we denote by ϕ_1,s,r(m_r) the positive, normalized eigenfunction $({‖ϕ_{1, s, r} (m_{r})‖}_{L^{\infty} (Ω)} = 1)$ of ${(- Δ)}_{r}^{s}$ with weight m_r in $W_{0}^{s, r} (Ω)$ associated with the first eigenvalue λ_1,s,r(m_r), i.e., satisfying

{(- Δ)}_{r}^{s} ϕ_{1, s, r} (m_{r}) = λ_{1, s, r} (m_{r}) m_{r} ϕ_{1, s, r} {(m_{r})}^{r - 1}, ϕ_{1, s, r} (m_{r}) = 0, in R^{N} \ Ω .

Furthermore, it is known that $ϕ_{1, s, r} (m_{r}) \in C^{0, α} (\bar{Ω})$ for some $α \in (0, s]$ (see [[27], Theorem 1.1]).

Now, we recall the embedding of $W_{0}^{s_{1}, p} (Ω)$ into $W_{0}^{s_{2}, q} (Ω)$ in the following lemma.

Lemma 2.1.

([ [18], Lemma 2.1]). Let 1 < q ≤ p < ∞ and 0 < s₂ < s₁ < 1. Then, there exists a constant $C = C (|Ω|, N, p, q, s_{1}, s_{2}) > 0$ such that

{‖u‖}_{W_{0}^{s_{2}, q} (Ω)} \leq C {‖u‖}_{W_{0}^{s_{1}, p} (Ω)}

for all $u \in W_{0}^{s_{1}, p} (Ω)$ ⁠.

Remark 2.2.

The embedding in Lemma 2.1 does not hold when s₁ = s₂ with p ≠ q, as demonstrated by a counterexample in [[28], Theorem 1.1]. Therefore, we consider the space $W ≔ W_{0}^{s_{1}, p} (Ω)$ for the case 0 < s₂ < s₁ < 1. In the case s = s₁ = s₂, we define

W ≔ W_{0}^{s, p} (Ω) \cap W_{0}^{s, q} (Ω),

endowed with the norm

{‖\cdot‖}_{W} ≔ {‖\cdot‖}_{W_{0}^{s, p} (Ω)} + {‖\cdot‖}_{W_{0}^{s, q} (Ω)} .

Since $W_{0}^{s, r} (Ω)$ is a separable reflexive Banach space for s ∈ {s₁, s₂} and r ∈ {p, q}, it follows that W is also a separable reflexive Banach space.

For future reference, we state the following lemma.

Lemma 2.3.

For 1 < q ≤ p and d > 0, define

W_{d} ≔ \{u \in W : {‖u‖}_{W_{0}^{s_{1}, p} (Ω)}^{p} + {‖u‖}_{W_{0}^{s_{2}, q} (Ω)}^{p} \leq d {‖u‖}_{L^{p} (Ω)}^{p}\} .

Then, for each d > 0, there exists a constant C = C(d) > 0 such that

{‖u‖}_{W} \leq C {‖u‖}_{L^{q} (Ω)}, for all u \in W_{d} .

proof. The proof proceeds by contradiction. Suppose that there exist some d > 0 and a sequence ${\{u_{n}\}}_{n} \subset W_{d}$ satisfying

{‖u_{n}‖}_{W_{0}^{s_{1}, p} (Ω)} + {‖u_{n}‖}_{W_{0}^{s_{2}, q} (Ω)} > n {‖u_{n}‖}_{L^{q} (Ω)} .

(2.2)

Define

v_{n} = \frac{u_{n}}{{‖u_{n}‖}_{L^{p} (Ω)}} .

First, we observe that the sequence ${\{v_{n}\}}_{n}$ is bounded in W. Indeed, using the inequality

a^{t} + b^{t} \geq C_{t} {(a + b)}^{t}, \forall a, b \geq 0, \forall t > 1,

and the fact that u_n ∈ W_d, we obtain

C_{p} {‖v_{n}‖}_{W}^{p} \leq {‖v_{n}‖}_{W_{0}^{s_{1}, p} (Ω)}^{p} + {‖v_{n}‖}_{W_{0}^{s_{2}, q} (Ω)}^{p} \leq d .

Thus, up to a subsequence, we have v_n ⇀ v in W and v_n → v in L^r(Ω) for $1 \leq r < p_{s_{1}}^{*}$ ⁠. Moreover, since ${‖v_{n}‖}_{L^{p} (Ω)} = 1$ ⁠, it follows that ${‖v‖}_{L^{p} (Ω)} = 1$ ⁠, implying that v ≠ 0. From (2.2), we deduce

{‖v_{n}‖}_{L^{q} (Ω)} < \frac{{‖v_{n}‖}_{W}}{n} \leq \frac{d^{\frac{1}{p}}}{C_{p}^{\frac{1}{p}} n} \to 0 as n \to \infty .

Consequently, we obtain ${‖v‖}_{L^{q} (Ω)} = 0$ ⁠, which contradicts the fact that v ≠ 0. This completes the proof. □

Using [[29], Lemma 1.8], we extend the well-known Diaz-Saa inequality to the fractional and nonhomogeneous setting as follows.

Lemma 2.4.

(Diaz-Saa inequality). Let 1 < p < ∞ and 0 < s₁, s₂ < 1. Then, for 1 < q ≤ p, the following inequality holds in the sense of distributions:

\int_{Ω} (\frac{{(- Δ)}_{p}^{s_{1}} u + {(- Δ)}_{q}^{s_{2}} u}{u^{q - 1}} - \frac{{(- Δ)}_{p}^{s_{1}} v + {(- Δ)}_{q}^{s_{2}} v}{v^{q - 1}}) (u^{q} - v^{q}) d x \geq 0,

(2.3)

for any u, v ∈ W that are positive in Ω and satisfy $\frac{u}{v}, \frac{v}{u} \in L^{\infty} (Ω)$ ⁠. Moreover, if equality holds in (2.3), then the following statements hold:

$\frac{u}{v} \equiv const > 0$ a.e. in Ω.
If p ≠ q, then u ≡ v a.e. in Ω.

Proof. It is straightforward to verify that (2.3) admits the following distributional interpretation:

\begin{aligned} \int_{R^{N}} \int_{R^{N}} \frac{{|u (x) - u (y)|}^{p - 2} (u (x) - u (y))}{{|x - y|}^{N + s_{1} p}} [\frac{u {(x)}^{q} - v {(x)}^{q}}{u {(x)}^{q - 1}} - \frac{u {(y)}^{q} - v {(y)}^{q}}{u {(y)}^{q - 1}}] d x d y \\ + \int_{R^{N}} \int_{R^{N}} \frac{{|v (x) - v (y)|}^{p - 2} (v (x) - v (y))}{{|x - y|}^{N + s_{1} p}} [\frac{v {(x)}^{q} - u {(x)}^{q}}{v {(x)}^{q - 1}} - \frac{v {(y)}^{q} - u {(y)}^{q}}{v {(y)}^{q - 1}}] d x d y \\ + \int_{R^{N}} \int_{R^{N}} \frac{{|u (x) - u (y)|}^{q - 2} (u (x) - u (y))}{{|x - y|}^{N + s_{2} q}} [\frac{u {(x)}^{q} - v {(x)}^{q}}{u {(x)}^{q - 1}} - \frac{u {(y)}^{q} - v {(y)}^{q}}{u {(y)}^{q - 1}}] d x d y \\ + \int_{R^{N}} \int_{R^{N}} \frac{{|v (x) - v (y)|}^{q - 2} (v (x) - v (y))}{{|x - y|}^{N + s_{2} q}} [\frac{v {(x)}^{q} - u {(x)}^{q}}{v {(x)}^{q - 1}} - \frac{v {(y)}^{q} - u {(y)}^{q}}{v {(y)}^{q - 1}}] d x d y \geq 0, \end{aligned}

(2.4)

for any u, v ∈ W that are positive in Ω and satisfy u/v, v/u ∈ L^∞(Ω). All integrals are understood in the Lebesgue sense. To guarantee their well-posedness, we observe the following:

There exists a constant M > 0 such that

0 \leq \frac{u^{q}}{v^{q - 1}} = {(\frac{u}{v})}^{q - 1} u \leq M u, 0 \leq \frac{v^{q}}{u^{q - 1}} = {(\frac{v}{u})}^{q - 1} v \leq M v .

Hence, we conclude that

\frac{u^{q}}{v^{q - 1}} \in L^{p} (R^{N}), \frac{v^{q}}{u^{q - 1}} \in L^{p} (R^{N}),

both vanishing in $R^{N} \ Ω$ ⁠.

By applying Lagrange's Mean Value Theorem, one obtains

\begin{array}{l} |\frac{u^{q} (x)}{v^{q - 1} (x)} - \frac{u^{q} (y)}{v^{q - 1} (y)}| & \leq \frac{|u^{q} (x) - u^{q} (y)|}{v^{q - 1} (x)} + u^{q} (y) \frac{|v^{q - 1} (x) - v^{q - 1} (y)|}{v^{q - 1} (x) v^{q - 1} (y)} \\ \leq \frac{q \max {u^{q - 1} (x), u^{q - 1} (y)}}{v^{q - 1} (x)} |u (x) - u (y)| \\ + \frac{u^{q} (y) (q - 1) \max {v^{q - 2} (x), v^{q - 2} (y)}}{v^{q - 1} (x) v^{q - 1} (y)} |v (x) - v (y)| \\ \leq C (q, {‖\frac{u}{v}‖}_{L^{\infty} (Ω)}) (|u (x) - u (y)| + |v (x) - v (y)|) . \end{array}

Consequently, we deduce that u^q/v^q−1, v^q/u^q−1 ∈ W. On the other hand, by using [[29], Lemma 1.8], we infer that

\begin{array}{l} {|u (x) - u (y)|}^{p - 2} (u (x) - u (y)) [\frac{u {(x)}^{q} - v {(x)}^{q}}{u {(x)}^{q - 1}} - \frac{u {(y)}^{q} - v {(y)}^{q}}{u {(y)}^{q - 1}}] \\ + {|v (x) - v (y)|}^{p - 2} (v (x) - v (y)) [\frac{v {(x)}^{q} - u {(x)}^{q}}{v {(x)}^{q - 1}} - \frac{v {(y)}^{q} - u {(y)}^{q}}{v {(y)}^{q - 1}}] \geq 0, \end{array}

and similarly,

\begin{array}{l} {|u (x) - u (y)|}^{q - 2} (u (x) - u (y)) [\frac{u {(x)}^{q} - v {(x)}^{q}}{u {(x)}^{q - 1}} - \frac{u {(y)}^{q} - v {(y)}^{q}}{u {(y)}^{q - 1}}] \\ + {|v (x) - v (y)|}^{q - 2} (v (x) - v (y)) [\frac{v {(x)}^{q} - u {(x)}^{q}}{v {(x)}^{q - 1}} - \frac{v {(y)}^{q} - u {(y)}^{q}}{v {(y)}^{q - 1}}] \geq 0 . \end{array}

Multiplying the above inequalities by $| x - y |^{- (N + s_{1} p)}$ and $| x - y |^{- (N + s_{2} q)}$ ⁠, respectively, integrating over $R^{N} \times R^{N}$ ⁠, and then adding the resulting expressions, we arrive at (2.4). □

In this paper, we apply the Mountain Pass Theorem to demonstrate the existence of a solution in the case r = p. To this end, we first recall the classical statement of the Mountain Pass Theorem.

Theorem 2.1.

(Mountain Pass Theorem [1]). Let X be a real Banach space, and let $K \in C^{1} (X, R)$ be a functional satisfying the Palais–Smale condition. Assume that:

$K (0) = 0$ ;
there exist constants ρ, α > 0 such that $K (u) \geq α$ whenever ‖u‖ = ρ;
there exists e ∈ X with ‖e‖ > ρ such that $K (e) < 0$ ⁠.

Then $K$ possesses a critical value c ≥ α, which can be characterized by

c = \inf_{γ \in Γ} \max_{t \in [0,1]} K (γ (t)),

where

Γ ≔ \{γ \in C ([0,1], X) | γ (0) = 0, γ (1) = e\} .

2.2 Statements of main results

We define the notion of weak solutions to (P) as follows:

Definition 2.5.

A non-negative function u ∈ W is called a weak solution to (P) if, for any φ ∈ mathbfW, we have:

\begin{align} \int_{R^{N}} \int_{R^{N}} & \frac{{|u (x) - u (y)|}^{p - 2} (u (x) - u (y)) (φ (x) - φ (y))}{| x - y |^{N + s_{1} p}} d x d y \\ + \int_{R^{N}} \int_{R^{N}} \frac{{|u (x) - u (y)|}^{q - 2} (u (x) - u (y)) (φ (x) - φ (y))}{| x - y |^{N + s_{2} q}} d x d y \\ = λ \int_{Ω} a_{r} (x) u^{r - 1} φ d x . \end{align}

(2.6)

Moreover, if u satisfies u > 0 in Ω, we refer to u as a positive weak solution.

The following remark provides an important regularity property of weak solutions to fractional and non-homogeneous equations, which will be used repeatedly in the sequel:

Remark 2.6.

Let u₀ ∈ W ∩ L^∞(Ω) be a nontrivial weak solution to problem (P). Then, Theorem 2.3 in Ref. [19], Corollary 2.4, and Remark 2.3 in Ref. [20] establish the Hölder continuity of u₀, ensuring that $u_{0} \in C^{0, α} (\bar{Ω})$ for some $α \in (0, s_{1})$ ⁠. Furthermore, by [[19], Theorem 2.5], it follows that u₀ > 0 in Ω. Applying Hopf's Lemma [[19], Proposition 2.6], we deduce that

u_{0} \geq k d^{s_{1} + ϵ_{0}} (x)

for some k = k(ϵ₀) > 0 and any ϵ₀ > 0. Moreover, by [[19], Proposition 3.11], for all $σ \in (0, s_{1})$ , there exists a constant K = K(σ) > 0 such that

u_{0} \leq K d^{σ} (x) in Ω .

We now establish a non-existence result for positive weak solutions to problem (P):

Theorem 2.7.

Let r = p or q, with s = s₁ or s₂, respectively. If λ ≤ λ_1,s,r(a_r), then problem (P) admits no nontrivial solutions.

Next, we state the results concerning existence, uniqueness, and regularity:

Theorem 2.8.

Let 0 < s₂ ≤ s₁ < 1 and 1 < q < p < ∞. Then, we have the following:

If λ > λ_1,s,r(a_r), where r = p (or q) with s = s₁ (or s₂, respectively), then problem (P) admits at least one positive solution u. Moreover, $u \in C^{0, α} (\bar{Ω})$ for some α ∈ (0, s₁), and for any σ ∈ (0, s₁) and σ′ > s₁, there exists a positive constant c = c(σ, σ′) > 0 such that:

c^{- 1} d^{σ^{'}} \leq u \leq c d^{σ} in Ω .

Furthermore, in the case s = s₂ and r = q, the solution is unique.

Theorem 2.9.

We define the following nonlocal Rayleigh quotient:

{\underline{λ}}_{s, s^{*}, r, r^{*}} (a) ≔ \inf_{u \in W} \{\frac{\int_{R^{N}} \int_{R^{N}} \frac{{|u (x) - u (y)|}^{r}}{| x - y |^{N + s r}} d x d y + \frac{r}{r^{*}} \int_{R^{N}} \int_{R^{N}} \frac{{|u (x) - u (y)|}^{r^{*}}}{| x - y |^{N + s^{*} r^{*}}} d x d y}{\int_{Ω} a_{r} (x) u^{r} d x}\} .

Here, r = p (or q) with s = s₁ (or s₂) if r* = q (or p) with s = s₂ (or s₁, respectively). Then, we have

{\underline{λ}}_{s, s^{*}, r, r^{*}} (a_{r}) = λ_{1, s, r} (a_{r}) .

Moreover, the infimum in the definition of ${\underline{λ}}_{s, s^{*}, r, r^{*}} (a_{r})$ is not attained.

2.3 Outline of the paper

The rest of the paper is organized as follows. In Section 3, we provide the proof of the non-existence result (Theorem 2.7), as well as the existence, uniqueness, and Hölder regularity of positive solutions to problem (P) (Theorem 2.8). The proof of Theorem 2.8 is divided into two main cases. First, using a variational method, we establish the existence of weak solutions in the sense of Definition 2.5 for r = q. Next, we employ the mountain pass theorem to demonstrate the existence of weak solutions when r = p. Moreover, by applying Lemma 2.4, we also obtain the uniqueness result for r = q (Section 3).

3. Proof of the main results

In this section, we first establish the non-existence of positive weak solutions for problem (P).

Proof of Theorem 2.7. Assume, by contradiction, that u ∈ W is a nontrivial solution of (P) and that λ ≤ λ_1,s,r(a_r). Choosing u as a test function in (2.6) and using the definition of λ_1,s,r(a_r), we obtain

λ_{1, s, r} (a_{r}) \leq \frac{{‖u‖}_{W_{0}^{s, r} (Ω)}^{r}}{\int_{Ω} a_{r} (x) u^{r} d x} < λ = \frac{{‖u‖}_{W_{0}^{s_{1}, p} (Ω)}^{p} + {‖u‖}_{W_{0}^{s_{2}, q} (Ω)}^{q}}{\int_{Ω} a_{r} (x) u^{r} d x} .

This contradiction completes the proof. □

Next, we establish the existence results and qualitative properties of solutions to problem (P).

Proof of Theorem 2.8. We distinguish two cases:

Case 1: s = s₂ and r = q. We divide the proof into three steps.

Step 1: Existence of a weak solution.

Consider the energy functional $J$ corresponding to (P), defined on W by (with $u^{+} = \max \{u, 0\}$ ⁠):

J (u) = \frac{1}{p} \int_{R^{N}} \int_{R^{N}} \frac{{|u (x) - u (y)|}^{p}}{{|x - y|}^{N + s_{1} p}} d x d y + \frac{1}{q} \int_{R^{N}} \int_{R^{N}} \frac{{|u (x) - u (y)|}^{q}}{{|x - y|}^{N + s_{2} q}} d x d y - \frac{λ}{q} \int_{Ω} a_{q} (x) {(u^{+})}^{q} d x .

It is straightforward to verify that $J$ is well-defined on W. Moreover, $J$ is weakly lower semi-continuous on W. Indeed, let (u_n) ⊂ W converge weakly to some u in W as n → ∞. Then, we have:

{‖u‖}_{W} \leq \lim_{n \to \infty} \inf {‖u_{n}‖}_{W} .

On the other hand, the fractional Sobolev embedding theorem [ [25], Theorem 6.5] implies that, up to a subsequence, u_n → u in L^t(Ω) for every $1 \leq t < p_{s_{1}}^{*}$ ⁠, and u_n(x) → u(x) for a.e. $x \in R^{N}$ ⁠. By the Dominated Convergence Theorem, we obtain:

\lim_{n \to \infty} \int_{Ω} a (x) {(u_{n}^{+})}^{q} d x = \int_{Ω} a (x) {(u^{+})}^{q} d x .

This establishes the weak lower semi-continuity of $J$ ⁠. Finally, $J$ is also coercive on W. Indeed, for every u ∈ W, using Hölder's inequality and the Sobolev embedding theorem, we obtain:

\begin{array}{l} J (u) & \geq \frac{1}{p} {‖u‖}_{W_{0}^{s_{1}, p} (Ω)}^{p} + \frac{1}{q} {‖u‖}_{W_{0}^{s_{2}, q} (Ω)}^{q} - \frac{λ}{q} {‖a_{q}‖}_{L^{\infty} (Ω)} {‖u‖}_{L^{q} (Ω)}^{q} \\ \geq \frac{1}{p} {‖u‖}_{W_{0}^{s_{1}, p} (Ω)}^{p} - {|Ω|}^{1 - \frac{q}{p}} \frac{λ}{q} {‖a_{q}‖}_{L^{\infty} (Ω)} {‖u‖}_{L^{p} (Ω)}^{q} \\ \geq {‖u‖}_{W_{0}^{s_{1}, p} (Ω)}^{p} [\frac{1}{p} - {|Ω|}^{1 - \frac{q}{p}} \frac{λ}{q λ_{1, s_{1}, p}^{\frac{q}{p}} (1)} {‖a_{q}‖}_{L^{\infty} (Ω)} {‖u‖}_{W_{0}^{s_{1}, p} (Ω)}^{q - p}] . \end{array}

Since 1 < q < p, we conclude that $J (u) \to + \infty$ as ${‖u‖}_{W} \to + \infty$ ⁠. Hence, $J$ admits a global minimizer, denoted by u₀. Noting that, with the notation t = t⁺ − t⁻, we have:

\begin{array}{l} J (u_{0}) & = J (u_{0}^{+}) \\ + \frac{1}{p} \int_{R^{N}} \int_{R^{N}} \frac{{|(u_{0}^{-}) (x) - (u_{0}^{-}) (y)|}^{p}}{{|x - y|}^{N + s_{1} p}} d x d y \\ + \frac{1}{q} \int_{R^{N}} \int_{R^{N}} \frac{{|(u_{0}^{-}) (x) - (u_{0}^{-}) (y)|}^{q}}{{|x - y|}^{N + s_{2} q}} d x d y \\ + \frac{2}{p} \int_{R^{N}} \int_{R^{N}} \frac{{|(u_{0}^{+}) (x) - (u_{0}^{-}) (y)|}^{p}}{{|x - y|}^{N + s_{1} p}} d x d y \\ + \frac{2}{q} \int_{R^{N}} \int_{R^{N}} \frac{{|(u_{0}^{+}) (x) - (u_{0}^{-}) (y)|}^{q}}{{|x - y|}^{N + s_{2} q}} d x d y \\ \geq J (u_{0}^{+}) . \end{array}

Therefore, u₀ ≥ 0. To show that u₀ ≢ 0 in Ω, we find a suitable function u ∈ W such that $J (u) < 0 = J (0)$ ⁠. To this end, for any t > 0:

J (t ϕ_{1, s_{2}, q} (a)) \leq t^{q} [\frac{t^{p - q}}{p} {‖ϕ_{1, s_{2}, q} (a_{q})‖}_{W_{0}^{s_{1}, p} (Ω)}^{p} + \frac{λ_{1, s_{2}, q} (a_{q}) - λ}{q λ_{1, s_{2}, q} (a_{q})} {‖ϕ_{1, s_{2}, q} (a_{q})‖}_{W_{0}^{s_{2}, q} (Ω)}^{q}] .

Since $λ > λ_{1, s_{2}, q} (a_{q})$ ⁠, for sufficiently small t > 0, we obtain $J (t ϕ_{1, s_{2}, q} (a_{q})) < 0$ ⁠. Since $J (0) = 0$ ⁠, we deduce that u₀ ≢ 0. By the Gâteaux differentiability of $J$ ⁠, u₀ satisfies (2.6), meaning that u₀ is a weak solution to (P).

Step 2: Regularity and Positivity of Weak Solutions

Firstly, we claim that all weak solutions to the problem (P) belong to L^∞(Ω). To this end, we follow the approach in [[30], Theorem 3.2]. Specifically, let u₀ ∈ W be a weak solution to (P). We define

v_{0} = \frac{u_{0}}{ρ {‖u_{0}‖}_{L^{q} (Ω)}}, where ρ = \max \{1, {‖u_{0}‖}_{L^{q} (Ω)}^{- 1}\} .

Since v₀ ∈ W and ${‖v_{0}‖}_{L^{q} (Ω)} = ρ^{- 1}$ ⁠, we now introduce the sequence of functions {w_k} defined as follows:

\{\begin{cases} w_{k} (x) ≔ {(v_{0} (x) - (1 - 2^{- k}))}^{+}, & k \geq 1, \\ w_{0} (x) = {(v_{0} (x))}^{+} . \end{cases}

We first state the following fundamental properties of w_k(x):

w_{k} \in W, and w_{k} = 0 a.e. in R^{N} \ Ω .

Additionally, we have the following inequalities:

\{\begin{cases} 0 \leq w_{k + 1} (x) \leq w_{k} (x), & a.e. in R^{N}, \\ v_{0} (x) < (2^{k + 1} - 1) w_{k} (x), & for x \in {w_{k + 1} > 0} . \end{cases}

(3.1)

Moreover, the inclusion

{w_{k + 1} > 0} \subseteq {w_{k} > 2^{- (k + 1)}} holds for all k \in N .

(3.2)

Now, we define the sequence

V_{k} ≔ ‖ w_{k} ‖_{L^{q} (Ω)}^{q} .

Claim. V_k → 0 as k → ∞.

Indeed, since 1 < q < p, we note that $ρ {‖u_{0}‖}_{L^{q} (Ω)} \geq 1$ ⁠. Furthermore, using the inequality

{|x^{+} - y^{+}|}^{l} \leq {|x - y|}^{l - 2} (x^{+} - y^{+}) (x - y), for x, y \in R, l > 1,

we obtain

\begin{array}{l} ‖ w_{k + 1} ‖_{W_{0}^{s_{2}, q} (Ω)}^{q} = \int_{R^{N}} \int_{R^{N}} \frac{| w_{k + 1} (x) - w_{k + 1} (y) |^{q}}{| x - y |^{N + s_{2} q}} d x d y \\ \leq \int_{R^{N}} \int_{R^{N}} \frac{| w_{k + 1} (x) - w_{k + 1} (y) |^{p}}{| x - y |^{N + s_{1} p}} d x d y + \int_{R^{N}} \int_{R^{N}} \frac{| w_{k + 1} (x) - w_{k + 1} (y) |^{q}}{| x - y |^{N + s_{2} q}} d x d y \\ \leq {(ρ {‖u_{0}‖}_{L^{q} (Ω)})}^{1 - p} \int_{R^{N}} \int_{R^{N}} \frac{| u_{0} (x) - u_{0} (y) |^{p - 2} (w_{k + 1} (x) - w_{k + 1} (y)) (u_{0} (x) - u_{0} (y))}{| x - y |^{N + s_{1} p}} d x d y \\ + {(ρ {‖u_{0}‖}_{L^{q} (Ω)})}^{1 - q} \int_{R^{N}} \int_{R^{N}} \frac{| u_{0} (x) - u_{0} (y) |^{q - 2} (w_{k + 1} (x) - w_{k + 1} (y)) (u_{0} (x) - u_{0} (y))}{| x - y |^{N + s_{2} q}} d x d y \\ \leq {(ρ {‖u_{0}‖}_{L^{q} (Ω)})}^{1 - q} {‖a_{q}‖}_{L^{\infty} (Ω)} \int_{{w_{k + 1} > 0}} u_{0}^{q - 1} w_{k + 1} d x \\ = {‖a_{q}‖}_{L^{\infty} (Ω)} \int_{{w_{k + 1} > 0}} v_{0}^{q - 1} w_{k + 1} d x . \end{array}

By combining this with (3.1), we obtain

‖ w_{k + 1} ‖_{W_{0}^{s_{2}, q} (Ω)}^{q} \leq C_{1} {(2^{k + 1} + 1)}^{q - 1} V_{k},

(3.3)

where C₁ > 0 is a constant. On the other hand, applying Hölder's inequality and the fractional Sobolev embedding theorem [[25], Theorem 6.5], we deduce that

V_{k + 1} = \int_{{w_{k + 1} > 0}} w_{k + 1}^{q} d x \leq C_{2} ‖ w_{k + 1} ‖_{W_{0}^{s_{2}, q} (Ω)}^{q} {|\{w_{k + 1} > 0\}|}^{1 - \frac{q}{q_{s_{2}}^{*}}}

(3.4)

where C₂ > 0 is a constant. Now, using (3.2), we obtain

V_{k} = \int_{Ω} w_{k}^{q} d x \geq \int_{{w_{k + 1} > 0}} w_{k}^{q} d x \geq 2^{- (k + 1) q} | {w_{k + 1} > 0} | .

Thus, inequality (3.4) can be rewritten as:

V_{k + 1} \leq C^{k} V_{k}^{1 + α}, for all k \in N,

(3.5)

for a suitable constant C > 1 and $α = \frac{s_{2} q}{N}$ ⁠. This implies that

V_{k} \leq \frac{η^{k}}{ρ^{q}}, for all k \in N,

(3.6)

where

η = C^{- \frac{1}{α}}, ρ = \max \{1, {‖u_{0}‖}_{L^{q} (Ω)}^{- 1}, C^{\frac{1}{q α^{2}}}\} .

By induction, we verify the following:

Base case: Clearly, $V_{0} = {‖v_{0}^{+}‖}_{L^{q} (Ω)}^{q} \leq {‖v_{0}‖}_{L^{q} (Ω)}^{q} = \frac{1}{ρ^{q}}$ ⁠.
Inductive step: Assuming (3.6) holds for some $k \in N$ ⁠, using (3.5), we obtain

V_{k + 1} \leq C^{k} V_{k}^{1 + α} \leq \frac{η^{k + 1}}{ρ^{q}} .

Since η ∈ (0, 1), it follows that

\lim_{k \to \infty} V_{k} = 0 .

(3.7)

Since w_k converges to ${(v_{0} - 1)}^{+}$ almost everywhere in $R^{N}$ ⁠, from (3.7) we conclude that w_k → 0 almost everywhere in Ω. Consequently, v₀ ≤ 1 almost everywhere in Ω, which implies

{‖u_{0}‖}_{L^{\infty} (Ω)} \leq ρ {‖u_{0}‖}_{L^{q} (Ω)} .

Thus, we deduce that u₀ ∈ L^∞(Ω). Furthermore, by Remark 2.6, we conclude that $u_{0} \in C^{0, α} (\bar{Ω})$ for some $α \in (0, s_{1})$ ⁠, and for any ϵ₀ > 0, there exists a constant K = K(ϵ₀) > 0 such that

K^{- 1} d^{s_{1} + ϵ_{0}} \leq u_{0} \leq K d^{s_{1} - ϵ_{0}} in Ω .

Step 3: Uniqueness of the Weak Solution.

We apply Lemma 2.4 (the Díaz–Saa inequality) to establish the uniqueness of the positive weak solution to problem (P). To this end, let v ∈ W be a weak positive solution of (P). For any ϵ > 0, define

u_{ϵ} = u_{0} + ϵ, v_{ϵ} = v + ϵ,

and introduce the functions

Φ = \frac{u_{ϵ}^{q} - v_{ϵ}^{q}}{u_{ϵ}^{q - 1}}, Ψ = \frac{v_{ϵ}^{q} - u_{ϵ}^{q}}{v_{ϵ}^{q - 1}} .

Using the inequalities established in the proof of Lemma 2.4, it follows that Φ and Ψ belong to W. Then, applying (2.6), we obtain

\begin{array}{l} \int_{R^{N}} \int_{R^{N}} & \frac{{|u_{0} (x) - u_{0} (y)|}^{p - 2} (u_{0} (x) - u_{0} (y)) (Φ (x) - Φ (y))}{| x - y |^{N + s_{1} p}} d x d y \\ + \int_{R^{N}} \int_{R^{N}} \frac{{|u_{0} (x) - u_{0} (y)|}^{q - 2} (u_{0} (x) - u_{0} (y)) (Φ (x) - Φ (y))}{| x - y |^{N + s_{2} q}} d x d y \\ = λ \int_{Ω} a_{q} (x) u_{0}^{q - 1} Φ d x . \end{array}

Similarly, we have

\begin{array}{l} \int_{R^{N}} \int_{R^{N}} & \frac{{|v (x) - v (y)|}^{p - 2} (v (x) - v (y)) (Ψ (x) - Ψ (y))}{| x - y |^{N + s_{1} p}} d x d y \\ + \int_{R^{N}} \int_{R^{N}} \frac{{|v (x) - v (y)|}^{q - 2} (v (x) - v (y)) (Ψ (x) - Ψ (y))}{| x - y |^{N + s_{2} q}} d x d y \\ = λ \int_{Ω} a_{q} (x) v^{q - 1} Ψ d x . \end{array}

Summing the above identities, we obtain

\begin{align} \int_{R^{N}} & \int_{R^{N}} \frac{{|u_{0} (x) - u_{0} (y)|}^{p - 2} (u_{0} (x) - u_{0} (y)) (Φ (x) - Φ (y))}{| x - y |^{N + s_{1} p}} d x d y \\ + \int_{R^{N}} \int_{R^{N}} \frac{{|u_{0} (x) - u_{0} (y)|}^{q - 2} (u_{0} (x) - u_{0} (y)) (Φ (x) - Φ (y))}{| x - y |^{N + s_{2} q}} d x d y \\ + \int_{R^{N}} \int_{R^{N}} \frac{{|v (x) - v (y)|}^{p - 2} (v (x) - v (y)) (Ψ (x) - Ψ (y))}{| x - y |^{N + s_{1} p}} d x d y \\ + \int_{R^{N}} \int_{R^{N}} \frac{{|v (x) - v (y)|}^{q - 2} (v (x) - v (y)) (Ψ (x) - Ψ (y))}{| x - y |^{N + s_{2} q}} d x d y \\ = λ \int_{Ω} a_{q} (x) (\frac{u_{0}^{q - 1}}{u_{ϵ}^{q - 1}} - \frac{v^{q - 1}}{v_{ϵ}^{q - 1}}) (u_{ϵ}^{q} - v_{ϵ}^{q}) d x . \end{align}

(3.8)

Following the proof of [[22], Theorem 2.10], we now pass to the limit in the right-hand side of (3.8). Specifically, we obtain

{(\frac{u_{ϵ}}{v_{ϵ}})}^{q} \leq 2^{q - 1} [{(\frac{u_{0}}{v})}^{q} + 1] \in L^{1} (Ω) .

(3.9)

Using Hölder's inequality, the fractional Hardy inequality, and the boundary behavior of u₀, v (see Remark 2.6), we obtain

\int_{Ω} {(\frac{u_{0}}{v})}^{q} d x \leq C \int_{Ω} {(\frac{u_{0}}{d^{s_{1} + ϵ_{0}} (x)})}^{q} d x \leq C {(\int_{Ω} \frac{1}{d^{\frac{p q}{p - q} ϵ_{0}} (x)})}^{\frac{p - q}{p}} {‖u_{0}‖}_{W}^{q} < \infty

for sufficiently small ϵ₀, where C = C(ϵ₀) > 0. Similarly, we find that for small enough ϵ₀

{(\frac{v_{ϵ}}{u_{ϵ}})}^{q} \leq 2^{q - 1} [{(\frac{v}{u_{0}})}^{q} + 1] \in L^{1} (Ω) .

Taking the limit as ϵ → 0 in (3.8), and using Fatou's lemma along with the dominated convergence theorem, we obtain

\int_{Ω} (\frac{{(- Δ)}_{p}^{s_{1}} u_{0} + {(- Δ)}_{q}^{s_{2}} u_{0}}{u_{0}^{q - 1}} - \frac{{(- Δ)}_{p}^{s_{1}} v + {(- Δ)}_{q}^{s_{2}} v}{v^{q - 1}}) (u^{q} - v^{q}) d x = 0 .

Applying Lemma 2.4, we conclude that u₀ = v since 1 < q < p.

(2) Case 2: s = s₁ and r = p. In this case, we employ the Mountain Pass Theorem to establish the existence of a solution. To this end, we consider the energy functional $K$ associated with (P), defined on W as follows:

K (u) = \frac{1}{p} \int_{R^{N}} \int_{R^{N}} \frac{{|u (x) - u (y)|}^{p}}{{|x - y|}^{N + s_{1} p}} d x d y + \frac{1}{q} \int_{R^{N}} \int_{R^{N}} \frac{{|u (x) - u (y)|}^{q}}{{|x - y|}^{N + s_{2} q}} d x d y - \frac{λ}{p} \int_{Ω} a_{p} (x) {(u^{+})}^{p} d x .

The proof is structured in two steps.

Step 1: Verification of the Palais–Smale condition for $K$ at any level $c \in R$ ⁠.

Following the approach in Lemma 8 of [12], we provide a detailed proof for completeness. Let ${(u_{k})}_{k \in N} \subset W$ be a Palais–Smale sequence for $K$ at level $c \in R,$ satisfying:

K (u_{k}) = c + o_{k} (1) and K^{'} (u_{k}) = o_{k} (1) .

Thus, for any φ ∈ W, we obtain

\begin{align} \int_{R^{N}} & \int_{R^{N}} \frac{{|u_{k} (x) - u_{k} (y)|}^{p - 2} (u_{k} (x) - u_{k} (y)) (φ (x) - φ (y))}{| x - y |^{N + s_{1} p}} d x d y \\ + \int_{R^{N}} \int_{R^{N}} \frac{{|u_{k} (x) - u_{k} (y)|}^{q - 2} (u_{k} (x) - u_{k} (y)) (φ (x) - φ (y))}{| x - y |^{N + s_{2} q}} d x d y \\ - λ \int_{Ω} a_{p} (x) {(u_{k}^{+})}^{p - 1} φ d x = o_{k} (1) . \end{align}

(3.10)

Claim The sequence ${(u_{k})}_{k \in N}$ is bounded in W.

To establish this, we proceed by contradiction. Suppose that

{‖u_{k}‖}_{W} \to \infty, as k \to \infty .

Defining

w_{k} ≔ \frac{u_{k}}{{‖u_{k}‖}_{W}},

we infer the existence of a subsequence such that w_k ⇀ w in W and w_k → w in L^r(Ω) for. $1 \leq r < p_{s_{1}}^{*} .$

First, we show that w ≥ 0 in Ω.

To verify this, we use ${(u_{k})}^{-}$ as a test function in (3.10) and apply the inequality

{|x^{-} - y^{-}|}^{l} \leq {|x - y|}^{l - 2} (x^{-} - y^{-}) (x - y), for x, y \in R, l > 1 .

This yields

{‖{(u_{k})}^{-}‖}_{W_{0}^{s_{1}, p} (Ω)}^{p} + {‖{(u_{k})}^{-}‖}_{W_{0}^{s_{2}, q} (Ω)}^{q} = ⟨K^{'} (u_{k}), {(u_{k})}^{-}⟩ \leq o_{k} (1) {‖{(u_{k})}^{-}‖}_{W},

(3.11)

where $⟨\cdot, \cdot⟩$ denotes the duality pairing. Since 1 < q < p, it follows that ${‖{(w_{k})}^{-}‖}_{W} \to 0$ as k → ∞. To justify this, it suffices to establish that ${\{{(u_{k})}^{-}\}}_{k}$ is bounded in W. Assuming the contrary, we consider the following three alternatives:

Alternative 1: ${‖{(u_{k})}^{-}‖}_{W_{0}^{s_{1}, p} (Ω)} \to + \infty$ and ${‖{(u_{k})}^{-}‖}_{W_{0}^{s_{2}, q} (Ω)} \to + \infty .$

Then, for k large enough, we have ${‖{(u_{k})}^{-}‖}_{W_{0}^{s_{1}, p} (Ω)} > 1$ ⁠. By use the following inequality

a^{q} + b^{q} \geq C_{q} {(a + b)}^{q}, \forall a, b \geq 0, \forall q > 1,

and (3.11), we obtain:

C_{q} {‖{(u_{k})}^{-}‖}_{W}^{q - 1} \leq o_{k} (1)

and this gives a contradiction.

Alternative 2: ${‖{(u_{k})}^{-}‖}_{W_{0}^{s_{1}, p} (Ω)} \to + \infty$ and ${‖{(u_{k})}^{-}‖}_{W_{0}^{s_{2}, q} (Ω)}$ is bounded.

From (3.11) we have

1 \leq o_{k} (1) ({‖{(u_{k})}^{-}‖}_{W_{0}^{s_{1}, p} (Ω)}^{1 - p} + \frac{{‖{(u_{k})}^{-}‖}_{W_{0}^{s_{2}, q} (Ω)}}{{‖{(u_{k})}^{-}‖}_{W_{0}^{s_{1}, p} (Ω)}^{p}}) .

Since p > 1 and passing to the limit as k → 0, we obtain that 1 ≤ 0 and this is a contradiction.

Alternative 3: ${‖{(u_{k})}^{-}‖}_{W_{0}^{s_{2}, q} (Ω)} \to + \infty$ and ${‖{(u_{k})}^{-}‖}_{W_{0}^{s_{1}, p} (Ω)}$ is bounded.

Symmetrically to Alternative 2.

Then, we obtain:

{‖{(w_{k})}^{-}‖}_{W} = \frac{{‖{(u_{k})}^{-}‖}_{W}}{{‖u_{k}‖}_{W}} \to 0 as k \to \infty .

This implies that ${(w_{k})}^{+} \to w,$ which yields w ≥ 0 a.e. in Ω.

Secondly, we can infer that w ≢ 0 in Ω.

Indeed, by taking $ϕ = \frac{w_{k} - w}{{‖u_{k}‖}_{W}^{p - 1}}$ as test function in (3.10), we obtain

\begin{array}{l} \int_{R^{N}} \int_{R^{N}} \frac{{|w_{k} (x) - w_{k} (y)|}^{p - 2} (w_{k} (x) - w_{k} (y)) ((w_{k} - w) (x) - (w_{k} - w) (y))}{| x - y |^{N + s_{1} p}} d x d y \\ + \frac{1}{{‖u_{k}‖}_{W}^{p - q}} \int_{R^{N}} \int_{R^{N}} \frac{{|w_{k} (x) - w_{k} (y)|}^{q - 2} (w_{k} (x) - w_{k} (y)) ((w_{k} - w) (x) - (w_{k} - w) (y))}{| x - y |^{N + s_{2} q}} d x d y \\ - λ \int_{Ω} a_{p} (x) w_{k}^{p - 1} (w_{k} - w) d x = o_{k} (1) . \end{array}

On the other hand, by using the Hölder's inequality, we have:

\begin{array}{l} |\int_{R^{N}} \int_{R^{N}} \frac{{|w_{k} (x) - w_{k} (y)|}^{q - 2} (w_{k} (x) - w_{k} (y)) ((w_{k} - w) (x) - (w_{k} - w) (y))}{| x - y |^{N + s_{2} q}} d x d y| \\ \leq {‖w_{k}‖}_{W}^{q - 1} {‖w_{k} - w‖}_{W} < c_{1} \end{array}

and

|\int_{Ω} a_{p} (x) w_{k}^{p - 1} (w_{k} - w) d x| \leq c_{2} {‖a_{p}‖}_{L^{\infty} (Ω)} {‖w_{k}‖}_{W}^{p - 1} {‖w_{k} - w‖}_{L^{p} (Ω)}

where c₁, c₂ > 0 are constants.

By ${‖w_{k}‖}_{W} \to \infty$ and since w_k is bounded in W, w_k → w in L^p(Ω) and passing to the limit as k → ∞, we deduce that:

\int_{R^{N}} \int_{R^{N}} \frac{{|w_{k} (x) - w_{k} (y)|}^{p - 2} (w_{k} (x) - w_{k} (y)) ((w_{k} - w) (x) - (w_{k} - w) (y))}{| x - y |^{N + s_{1} p}} d x d y = o_{k} (1) .

Then w_k → w in $W_{0}^{s_{1}, p} (Ω)$ (by using (S) property of fractional p − Laplace operator on $W_{0}^{s_{1}, p} (Ω)$ ⁠). Since ${‖w_{k}‖}_{W} = 1,$ we deduce w ≢ 0 in Ω.

Thirdly, we show that w is an eigenfunction of ${(- Δ)}_{p}^{s_{1}}$ with weight a_p in $W_{0}^{s_{1}, p} (Ω) .$

From (3.10), we have

\begin{array}{l} \int_{R^{N}} \int_{R^{N}} & \frac{{|w_{k} (x) - w_{k} (y)|}^{p - 2} (w_{k} (x) - w_{k} (y)) (φ (x) - φ (y))}{| x - y |^{N + s_{1} p}} d x d y \\ + \frac{1}{{‖u_{k}‖}_{W}^{p - q}} \int_{R^{N}} \int_{R^{N}} \frac{{|w_{k} (x) - w_{k} (y)|}^{q - 2} (w_{k} (x) - w_{k} (y)) (φ (x) - φ (y))}{| x - y |^{N + s_{2} q}} d x d y \\ - λ \int_{Ω} a_{p} (x) w_{k}^{p - 1} φ d x = \frac{o_{n} (1)}{{‖u_{k}‖}_{W}^{p - 1}} . \end{array}

Passing to the limit as k → ∞, we deduce that w is a non-negative and non-trivial solution of the following problem:

{(- Δ)}_{p}^{s_{1}} w = λ a_{p} (x) w^{p - 1}, w = 0, in R^{N} \ Ω .

By the Moser iteration process as in Step 2 of the proof of Case 1, we obtain that w ∈ L^∞(Ω). On the other hand, from Remark 2.6, we infer that w > 0 in Ω. Then, from [ [30], Theorem 4.1] we deduce that $λ_{1, s_{1}, p} (a_{p}) = λ > λ_{1, s_{1}, p} (a_{p}),$ and this gives a contradiction. Consequently, ${(u_{k})}_{k \in N}$ is bounded in W. Then there exists a sub-sequence, such that u_k ⇀ u in W and u_k → u in L^p(Ω).

Now, by taking φ = u_k − u as test function in (3.10), we obtain

\begin{array}{l} \int_{R^{N}} & \int_{R^{N}} \frac{{|u_{k} (x) - u_{k} (y)|}^{p - 2} (u_{k} (x) - u_{k} (y)) (φ (x) - φ (y))}{| x - y |^{N + s_{1} p}} d x d y \\ + \int_{R^{N}} \int_{R^{N}} \frac{{|u_{k} (x) - u_{k} (y)|}^{q - 2} (u_{k} (x) - u_{k} (y)) (φ (x) - φ (y))}{| x - y |^{N + s_{2} q}} d x d y \\ - λ \int_{Ω} a_{p} (x) u_{k}^{p - 1} (u_{n} - u) d x = o_{k} (1) . \end{array}

Using the Hölder's inequality, we have

|\int_{Ω} a_{p} (x) u_{k}^{p - 1} φ d x| \leq c {‖a_{p}‖}_{L^{\infty} (Ω)} {‖u_{k}‖}_{W}^{p - 1} {‖u_{k} - u‖}_{L^{p} (Ω)} \to 0 as k \to \infty

where c > 0 is a constant. Then,

\begin{align} \int_{R^{N}} \int_{R^{N}} \frac{{|u_{k} (x) - u_{k} (y)|}^{p - 2} (u_{k} (x) - u_{k} (y)) (φ (x) - φ (y))}{| x - y |^{N + s_{1} p}} d x d y \\ + \int_{R^{N}} \int_{R^{N}} \frac{{|u_{k} (x) - u_{k} (y)|}^{q - 2} (u_{k} (x) - u_{k} (y)) (φ (x) - φ (y))}{| x - y |^{N + s_{2} q}} d x d y = o_{k} (1) . \end{align}

(3.12)

On the other hand, we have

\begin{align} \int_{R^{N}} \int_{R^{N}} \frac{{|u (x) - u (y)|}^{p - 2} (u (x) - u (y)) (φ (x) - φ (y))}{| x - y |^{N + s_{1} p}} d x d y \\ + \int_{R^{N}} \int_{R^{N}} \frac{{|u (x) - u (y)|}^{q - 2} (u (x) - u (y)) (φ (x) - φ (y))}{| x - y |^{N + s_{2} q}} d x d y = o_{k} (1) . \end{align}

(3.13)

Subtracting (3.12)-(3.13) and using inequalities in [ [31], Section 10], yields

\int_{R^{N}} \int_{R^{N}} \frac{{|u_{k} (x) - u_{k} (y)|}^{p - 2} (u_{k} (x) - u_{k} (y)) ((u_{k} - u) (x) - (u_{k} - u) (y))}{| x - y |^{N + s_{1} p}} d x d y = o_{k} (1)

and

\int_{R^{N}} \int_{R^{N}} \frac{{|u_{k} (x) - u_{k} (y)|}^{q - 2} (u_{k} (x) - u_{k} (y)) ((u_{k} - u) (x) - (u_{k} - u) (y))}{| x - y |^{N + s_{2} q}} d x d y = o_{k} (1) .

Then by using again (S) property of ${(- Δ)}_{p}^{s_{1}}$ and ${(- Δ)}_{q}^{s_{2}}$ on $W_{0}^{s_{1}, p} (Ω)$ and $W_{0}^{s_{2}, q} (Ω)$ ⁠, respectively, we infer that u_k → u in W.

Step 2: The functional $K$ possesses the Mountain Pass geometry.

Now, we show that the functional $K$ satisfies the following two conditions:

There exist α, ρ such that $K (u) \geq α$ for ${‖u‖}_{W} = ρ .$ Indeed, let u ∈ W where ${‖u‖}_{W} = ρ \in (0,1) .$

Then, by taking into account that ${‖u‖}_{W_{0}^{s_{2}, q} (Ω)} < 1$ and using 1 < q < p, we obtain that ${‖u‖}_{W_{0}^{s_{2}, q} (Ω)}^{p} \leq {‖u‖}_{W_{0}^{s_{2}, q} (Ω)}^{q} .$ Now, we distinguish two cases:

Case 1: For u ∉ W_d with $d = 2 λ {‖a_{p}‖}_{L^{\infty} (Ω)}$ (see Lemma 2.3), by using the following inequality:

a^{p} + b^{p} \geq C_{p} {(a + b)}^{p}, \forall a, b \geq 0, \forall p > 1,

we obtain:

\begin{array}{l} K (u) & \geq \frac{1}{p} ({‖u‖}_{W_{0}^{s_{1}, p} (Ω)}^{p} + {‖u‖}_{W_{0}^{s_{2}, q} (Ω)}^{p}) - \frac{λ {‖a_{p}‖}_{L^{\infty} (Ω)}}{p} {‖u^{+}‖}_{L^{p} (Ω)}^{p} \\ \geq \frac{C_{p}}{2 p} {‖u‖}_{W}^{p} + \frac{d}{2 p} {‖u‖}_{L^{p} (Ω)}^{p} - \frac{λ {‖a_{p}‖}_{L^{\infty} (Ω)}}{p} {‖u^{+}‖}_{L^{p} (Ω)}^{p} \geq \frac{C_{p}}{2 p} {‖u‖}_{W}^{p} (see Lemma 2.3) . \end{array}

Case 2: For u ∈ W_d, by using definition of $λ_{1, s_{1}, p} (a_{p})$ and $λ_{1, s_{2}, q} (1)$ together with $λ > λ_{1, s_{1}, p} (a_{p}),$ we get

\begin{array}{l} K (u) & \geq \frac{1}{p} (1 - \frac{λ}{λ_{1, s_{1}, p} (a_{p})}) {‖u‖}_{W_{0}^{s_{1}, p} (Ω)}^{p} + \frac{1}{q} {‖u‖}_{W_{0}^{s_{2}, q} (Ω)}^{q} \\ \geq \frac{1}{p} (1 - \frac{λ}{λ_{1, s_{1}, p} (a_{p})}) {‖u‖}_{W}^{p} + \frac{λ_{1, s_{2}, q} (1)}{q C {(d)}^{q}} {‖u‖}_{W}^{q} (see Lemma 2.3) . \end{array}

Since 1 < q < p, we can find α > 0 such that $K (u) \geq α$ for ${‖u‖}_{W} = ρ$ small enough.

There exists ϕ ∈ W such that

‖ ϕ ‖_{W} > ρ and K (ϕ) < 0 .

Indeed, for any t > 0, we have

\begin{array}{l} K (t ϕ_{1, s_{1}, p} (a_{p})) & = \frac{t^{p}}{p} {‖ϕ_{1, s_{1}, p} (a_{p})‖}_{W_{0}^{s_{1}, p} (Ω)}^{p} + \frac{t^{q}}{q} {‖ϕ_{1, s_{1}, p} (a_{p})‖}_{W_{0}^{s_{2}, q} (Ω)}^{q} - \frac{λ t^{p}}{p} {‖ϕ_{1, s_{2}, q} (a_{p})‖}_{L^{p} (Ω)}^{p} \\ = t^{p} [\frac{t^{q - p}}{q} {‖ϕ_{1, s_{1}, p} (a_{p})‖}_{W_{0}^{s_{2}, q} (Ω)}^{q} + \frac{λ_{1, s_{1}, p} (a_{p}) - λ}{p} {‖ϕ_{1, s_{2}, q} (a_{p})‖}_{L^{p} (Ω)}^{p}] . \end{array}

Using the assumptions p > q and $λ > λ_{1, s_{1}, p} (a_{p})$ ⁠, it follows that

K (t ϕ_{1, s_{1}, p} (a_{p})) \to - \infty, while ‖ t ϕ_{1, s_{1}, p} (a_{p}) ‖_{W} \to + \infty as t \to + \infty .

Consequently, there exists t₀ sufficiently large such that

‖ t_{0} ϕ_{1, s_{1}, p} (a_{p}) ‖_{W} > ρ and K (t_{0} ϕ_{1, s_{1}, p} (a_{p})) < 0 .

Therefore, the functional $K$ satisfies the mountain pass geometry. By invoking the Mountain Pass Theorem (see Theorem 2.1), we deduce that for every $λ > λ_{1, s_{1}, p} (a_{p})$ ⁠, the level c is a critical value of $K$ associated with a critical point u₀ ∈ W. Specifically,

K (u_{0}) = c, K^{'} (u_{0}) = 0,

where

c ≔ \inf_{γ \in Σ} \max_{t \in [0,1]} K (γ (t)),

and

Σ ≔ \{γ \in C ([0,1], W) : γ (0) = 0, γ (1) = t_{0} ϕ_{1, s_{1}, p} (a_{p})\} .

Since $c \geq α > 0 = K (0) > K (t_{0} ϕ_{1, s_{1}, p} (a_{p})),$ we deduce that u₀ ≢ 0. Moreover, by an argument similar to that used in Step 2 of the proof of Case 1, it follows that u₀ ∈ L^∞(Ω). In addition, invoking Remark 2.6, we infer that $u_{0} \in C^{0, α} (\bar{Ω}),$ for some α ∈ (0, s₁). Furthermore, for any ϵ₀ > 0, there exists a constant K = K(ϵ₀) > 0 such that

K^{- 1} d^{s_{1} + ϵ_{0}} \leq u_{0} \leq K d^{s_{1} - ϵ_{0}} in Ω .

□

Proof of Theorem 2.9. We begin by observing that

\begin{align} {\underline{λ}}_{s, s *, r, r^{*}} (a_{r}) & = \inf_{u \in W} \{\frac{\int_{R^{N}} \int_{R^{N}} \frac{| u (x) - u (y) |^{r}}{| x - y |^{N + s r}} d x d y + \frac{r}{r^{*}} \int_{R^{N}} \int_{R^{N}} \frac{| u (x) - u (y) |^{r *}}{| x - y |^{N + s^{*} r^{*}}} d x d y}{\int_{Ω} a_{r} (x) u^{r} d x}\} \\ \geq λ_{1, s, r} (a_{r}) > - \infty . \end{align}

(3.14)

Thus, ${\underline{λ}}_{s, s *, r, r^{*}} (a_{r})$ is well-defined.

Next, following the approach in [ [12], Proposition 4], let t > 0 and set v = tϕ_1,s,r(a_r). By the definition of ${\underline{λ}}_{s, s *, r, r^{*}} (a_{r})$ and λ_1,s,r(a_r), we obtain

{\underline{λ}}_{s, s *, r, r^{*}} (a_{r}) \leq \frac{\int_{R^{N}} \int_{R^{N}} \frac{| v (x) - v (y) |^{r}}{| x - y |^{N + s r}} d x d y + \frac{r}{r^{*}} \int_{R^{N}} \int_{R^{N}} \frac{| v (x) - v (y) |^{r *}}{| x - y |^{N + s^{*} r^{*}}} d x d y}{\int_{Ω} a_{r} (x) v^{r} (x) d x} = λ_{1, s, r} (a_{r}) + \frac{r t^{r^{*} - r}}{r^{*}} \frac{\int_{R^{N}} \int_{R^{N}} \frac{| ϕ_{1, s, r} (a_{r}) (x) - ϕ_{1, s, r} (a_{r}) (y) |^{r *}}{| x - y |^{N + s^{*} r^{*}}} d x d y}{\int_{Ω} a_{r} (x) ϕ_{1, s, r} {(a_{r})}^{r} d x} .

(3.15)

Now, we analyze two cases:

Case 1: If r = p with s = s₁ and r* = q with s* = s₂, then since 1 < q < p, taking the limit as t → +∞ in (3.15), we deduce

{\underline{λ}}_{s_{1}, s_{2}, p, q} (a_{p}) \leq λ_{1, s_{1}, p} (a_{p}) .

Case 2: If r = q with s = s₂ and r* = p with s* = s₁, then taking the limit as t → 0 in (3.15) and using again the fact that 1 < q < p, we obtain

{\underline{λ}}_{s_{2}, s_{1}, q, p} (a_{q}) \leq λ_{1, s_{2}, q} (a_{q}) .

From (3.14), it follows that

{\underline{λ}}_{s, s^{*}, r, r^{*}} (a_{r}) = λ_{1, s, r} (a_{r}) .

To conclude, suppose by contradiction that there exists u₀ ∈ W such that

\begin{array}{l} {\underline{λ}}_{s, s *, r, r^{*}} (a_{r}) & = \frac{\int_{R^{N}} \int_{R^{N}} \frac{| u_{0} (x) - u_{0} (y) |^{r}}{| x - y |^{N + s r}} d x d y + \frac{r}{r^{*}} \int_{R^{N}} \int_{R^{N}} \frac{| u_{0} (x) - u_{0} (y) |^{r *}}{| x - y |^{N + s^{*} r^{*}}} d x d y}{\int_{Ω} a_{r} (x) u_{0}^{r} d x} \\ \geq λ_{1, s, r} (a_{r}) + \frac{\frac{r}{r^{*}} \int_{R^{N}} \int_{R^{N}} \frac{| u_{0} (x) - u_{0} (y) |^{r *}}{| x - y |^{N + s^{*} r^{*}}} d x d y}{\int_{Ω} a_{r} (x) u_{0}^{r} d x} > λ_{1, s, r} (a_{r}) = {\underline{λ}}_{s, s *, r, r^{*}} (a_{r}), \end{array}

which is a contradiction. □

References

Ambrosetti

Rabinowitz

Dual variational methods in critical point theory and applications

J Funct Anal

1973

;

349

. doi:

Qualitative properties of solutions to generalized eigenvalue problems

1. Introduction

1.1 State of the art

1.2 Objectives of this paper

2. Preliminaries and main results

2.1 Notation and function spaces

2.2 Statements of main results

2.3 Outline of the paper

3. Proof of the main results

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