A perturbed fractional p -Kirchhoff problem with critical nonlinearity

Abstract

We consider a quasilinear partial differential equation governed by the p-Kirchhoff fractional operator. By using variational methods, we prove several results concerning the existence of solutions and their stability properties with respect to some parameters.

Keywords

critical growth

1. Introduction

In this paper we are concerned in the study of the problem $\begin{matrix} (P_{a, b}^{λ}) & \{\begin{matrix} (a + b \int_{Q} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + p s}} d x d y) {(- Δ_{p})}^{s} u = | u |^{p_{s}^{*} - 2} u + λ g (x, u) & in Ω, \\ u = 0 & in R^{N} ∖ Ω, \end{matrix} \end{matrix}$ where $Ω \subset R^{N}$ is a bounded domain with Lipschitz boundary $\partial Ω$ , $Q = R^{2 N} ∖ O$ and $O = Ω^{c} \times Ω^{c}$ , a and b are strictly positive real numbers, $s \in (0, 1)$ . If $1 < p < 2$ we require $N > 2 p s$ while if $p > 2$ we suppose $N > p^{2} s$ . Here $p_{s}^{*} : = N p / (N - p s)$ denotes the critical exponent for the Sobolev embedding of $W^{s, p} (R^{N})$ into Lebesgue spaces. The p-fractional Laplacian can be defined up to a normalization constant as $\begin{matrix} {(- Δ_{p})}^{s} u (x) = 2 lim_{ϵ \to 0^{+}} \int_{R^{N} ∖ B_{ϵ} (0)} \frac{| u (x) - u (y) |^{p - 2} (u (x) - u (y))}{| x - y |^{N + p s}} d y . \end{matrix}$ Problem ( P a , b λ ) can be seen as a non local stationary generalized version of the classical Kirchhoff equation $\begin{matrix} (1.1) & ρ h \partial_{t t}^{2} u - (p_{0} + \frac{E h}{2 L} \int_{0}^{L} | \partial_{x} u |^{2} d x) \partial_{x x}^{2} u + δ \partial_{t} u + g (x, u) = 0 \end{matrix}$ for $t ⩾ 0$ and $0 < x < L$ , where $u = u (t, x)$ is the lateral displacement at time t and at position x, $E$ is the Young modulus, ρ is the mass density, h is the cross section area, L the length of the string, $p_{0}$ is the initial stress tension, δ the resistance modulus and g the external force. As pointed out by Murthy in [33], Kirchhoff in [21], attempting to generalize the well known d’Alembert equation of the vibrating string, introduced this model taking into account not only the transversal displacement. At a later time, the Kirchhoff equation found application in various fields. Indeed, Alves et al. in [1] emphasized that the solutions u of the Kirchhoff equation can also describe a process which depends on the average of itself such as the population density. Moreover, operators such as the one introduced by Kirchhoff also arise in phase transition phenomena, continuum mechanics, population dynamics, game theory, nonlinear optic and minimal surfaces. The interested reader can consult [7,12–14,26] and the references therein. The interest in generalizing this kind of problems to the fractional case is not only for mathematical purposes. In fact, Fiscella and Valdinoci in [18] constructed a model for the vibrating string in which the tension of the string is related to nonlocal measurements of the displacement of the string from its rest position. In recent years the fractional quasilinear Kirchhoff case has attracted the attention of many researchers. For instance, Franzina and Palatucci in [19] and Lindgren and Lindqvist in [23] studied some properties of the eigenvalues of ${(- Δ_{p})}^{s}$ . Furthermore, Brasco and Lindgren in [9], Di Castro et al. in [15] and Iannizzoto et al. in [20] obtained some results regarding the regularity of solutions involving the fractional p-Laplace operator. Also the attention to the fractional quasilinear case has grown considerably in the last years. We refer to [4,5] for results on existence, multiplicity and concentration of positive solutions for a singularly perturbed fractional p-Schrödinger equation by means of variational methods and the Lyusternik–Shnirel’man theory. Pucci et al. in [31] obtained a multiplicity result for the so called Kirchhoff–Schr̈odinger equation in $R^{N}$ where a potential was added to the Kirchhoff operator. Xiang et al. in [34] proved the existence of a nontrivial weak solution to a problem driven by a non local operator with a more general kernel than the one taken into consideration here. Moreover Xiang et al. in [35] proved the existence of a nontrivial solution for a problem with the fractional p-Laplace operator and a critical exponent. It is also worth mentioning [27] where the authors obtained the existence of a sequence of nontrivial solutions by using the symmetric mountain pass theorem under the assumption that the nonlinear term f satisfies a superlinear growth condition. We finally cite [6], where the authors investigate fractional p-Kirchhoff type problems in $R^{N}$ with subcritical, critical and supercritical growth.

The aim of the present paper is to generalize to the fractional quasilinear case some results obtained by Appolloni et. al in [8] following the approach proposed in [17]. We point out that to the best of our knowledge these results we are going to prove are new even for the local case $s = 1$ . The main mathematical difficulty we have to face in order to study existence of solutions for problem ( P a , b λ ) is the presence of the term $| u |^{p_{s}^{*} - 2} u$ . Due to the lack of compactness of the embedding $W^{s, p} (Ω) ↪ L^{p_{s}^{*}} (Ω)$ , the energy functional associated to problem ( P a , b λ ) is not even weakly sequentially lower semicontinuous. Moreover, the validity of the Palais–Smale condition is not assured. In order to overcome these difficulties, we will invoke the concentration-compactness principle developed by Lions in [24] and [25] and generalized to the p-fractional case by Mosconi and Squassina in [29]. Helped by this result and choosing the quantity $a^{(N - 2 p s) / p s)} b$ adequately, we will show that the functional associated to the problem is weakly sequentially lower semicontinuous and satisfies the Palais–Smale condition at any level. In addition to that, while in the semilinear case $p = 2$ the minimizers for the best Sobolev constant are completely characterized, if $p \neq 2$ we can only rely on some asymptotic estimates at infinity. As regards studying the different levels of energy on which the solutions are, we will use a fiber type approach. Defining appropriately a map depending on a parameter we will identify a parameter ${\bar{λ}}_{0}^{s}$ that will play a crucial role to establish if the ground state is attained at a negative level. Since we will assume that the function g has a subcritical growth for the sake of simplicity at the beginning we will focus our attention to the auxiliary problem $\begin{matrix} (P_{a, b}) & \{\begin{matrix} (a + b \int_{R^{2 N}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + p s}} d x d y) {(- Δ_{p})}^{s} u = | u |^{p_{s}^{*} - 2} u & in Ω, \\ u = 0 & in R^{N} ∖ Ω . \end{matrix} \end{matrix}$ We denote by $‖ \cdot ‖_{p}$ the standard norm of the Lebesgue space $L^{p} (R^{N})$ , and we define the fractional Sobolev space $\begin{matrix} W^{s, p} (R^{N}) : = {u \in L^{p} (R^{N}) : \int_{R^{2 N}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + p s}} d x d y < \infty}, \end{matrix}$ endowed with the norm $‖ u ‖_{W^{s, p} (R^{N})} = ‖ u ‖_{p} + ‖ u ‖$ , where $\begin{matrix} ‖ u ‖^{p} = \int_{R^{2 N}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + p s}} d x d y . \end{matrix}$ We also set $\begin{matrix} X_{0}^{s, p} (Ω) : = {u \in W^{s, p} (R^{N}) : u = 0 a.e. in R^{N} ∖ Ω} . \end{matrix}$

Remark 1.1.
The norm $‖ \cdot ‖$ is equivalent to $‖ \cdot ‖_{W^{s, p} (R^{N})}$ in $X_{0}^{s, p} (Ω)$ .

We define the functional $I : X_{0}^{s, p} (Ω) \to R$ by $\begin{matrix} I (u) : = \frac{a}{p} ‖ u ‖^{p} + \frac{b}{2 p} ‖ u ‖^{2 p} - \frac{1}{p_{s}^{}} ‖ u ‖_{p_{s}^{}}^{p_{s}^{}}, \end{matrix}$ whose critical points are weak solutions to ( P a , b ). To see a complete summary of the notation used we refer the reader to the next section. Our paper is structured as follows. Section 2 is devoted to introducing the notation and to collect some preliminary lemmas. In Section 3 we give the proof of the main results for the auxiliary problem ( P a , b ). Finally in Section 4 we investigate the existence of solutions for problem ( P a , b λ ). To conclude the section we collect here the main results we are going to prove along the paper.
Theorem 1.2.
Let* $\begin{matrix} L (N, p, s) : = 2 p s \frac{{(N - 2 p s)}^{\frac{N - 2 p s}{p s}}}{N^{\frac{N - p s}{p s}} S_{s, p}^{\frac{N}{p s}}}, \end{matrix}$ where $S_{s, p}$ is the best Sobolev constant defined in ( 2.1 ) below. The functional $I$ is sequentially weakly lower semicontinuous on $X_{0}^{s, p} (Ω)$ if and only if $a^{\frac{N - 2 p s}{p s}} b ⩾ L (N, p, s)$ .
Theorem 1.3.
Define $\begin{matrix} PS (N, p, s) : = p s \frac{{(N - 2 p s)}^{\frac{N - 2 p s}{p s}}}{{(N - p s)}^{\frac{N - p s}{p s}} S_{s, p}^{\frac{N}{p s}}} . \end{matrix}$ If $a^{(N - 2 p s) / p s)} b > PS (N, p, s)$ , the functional $I$ satisfies the compactness Palais–Smale condition at any level $c \in R$ .
Remark 1.4.
We point out that $PS (N, p, s) ⩾ L (N, p, s)$ in our setting. Indeed, this inequality is equivalent to $\begin{matrix} {(\frac{N}{N - p s})}^{\frac{N - p s}{p s}} ⩾ 2, \end{matrix}$ or $\begin{matrix} {(1 + \frac{p s}{N - p s})}^{\frac{N - p s}{p s}} ⩾ 2 . \end{matrix}$ The generalized Bernouilli inequality $\begin{matrix} {(1 + x)}^{r} ⩾ 1 + r x, r ⩾ 1, x ⩾ - 1 \end{matrix}$ and the assumption that $N > 2 p s$ yield $\begin{matrix} {(1 + \frac{p s}{N - p s})}^{\frac{N - p s}{p s}} ⩾ 1 + \frac{N - p s}{p s} \cdot \frac{p s}{N - p s} = 2 . \end{matrix}$

We next prove an existence result for ground states of problem ( P a , b λ ).
Theorem 1.5.
Let $a, b \in R^{+}$ such that $a^{(N - 2 p s) / p s} b ⩾ L (N, p, s)$ , and set $\begin{matrix} ι_{λ}^{s} : = inf {I^{λ} (u) : u \in X_{0}^{s, p} (Ω) ∖ {0}} for any λ > 0 . \end{matrix}$ There exists ${\overline{λ}}_{0}^{s} ⩾ 0$ such that for any $λ > {\overline{λ}}_{0}^{s}$ there exists $u_{λ}^{s} \in X_{0}^{s, p} (Ω) ∖ {0}$ satisfying $I^{λ} (u_{λ}^{s}) = ι_{λ}^{s} < 0$ .
Theorem 1.6.
Let $λ = {\overline{λ}}_{0}^{s}$ . The following statements hold:
if $a^{(N - 2 p s) / p s} b > L (N, p, s)$ then there exists $u_{λ}^{s} \in X_{0}^{s, p} (Ω) ∖ {0}$ such that $ι_{{\overline{λ}}_{0}^{s}}^{s} = I^{{\overline{λ}}_{0}^{s}} = 0$ ;

if $a^{(N - 2 p s) / p s} b = L (N, p, s)$ , then $u = 0$ in the only minimizer for $ι_{{\overline{λ}}_{0}^{s}}^{s}$ .

The following theorem states a sort of stability when the quantity $a^{(N - 2 p s) / p s} b$ converges to $L (N, p, s)$ .
Theorem 1.7.
Let ${(a_{k})}_{k}$ , ${(b_{k})}_{k}$ be sequences of real positive numbers such that $a_{k} \to a$ , $b_{k} \to b$ and $a_{k}^{(N - 2 p s) / p s} b_{k} ↘ L (N, p, s)$ . Setting $λ_{k} : = {\overline{λ}}_{0}^{s} (a_{k}, b_{k})$ we have that $λ_{k} \to 0$ as $k \to \infty$ . Furthermore, if ${(u_{k})}_{k} \subset X_{0}^{s, p} (Ω) ∖ {0}$ satisfies $λ_{k} = λ_{0}^{s} (u_{k})$ then $u_{k} ⇀ 0$ and $\begin{matrix} \frac{‖ u_{k} ‖_{p_{s}^{}}^{p}}{‖ u_{k} ‖^{p}} \to S_{s, p} . \end{matrix}$

Next statement describes the situations for mountains pass solutions.
Theorem 1.8.
If $λ ⩾ {\overline{λ}}_{0}^{s}$ and* $a^{(N - 2 p s) / p s} b > PS (N, p, s)$ , then there exists a $v_{λ}^{s} \in X_{0}^{s, p} (Ω) ∖ {0}$ such that $I^{λ} (v_{λ}^{s}) = c_{λ}^{s}$ and ${(I^{λ})}^{'} (v_{λ}^{s}) = 0$ where $\begin{matrix} c_{λ}^{s} : = inf_{h \in Γ_{λ}^{s}} max_{ζ \in [0, 1]} I^{λ} (h (ζ)) \end{matrix}$ and $\begin{matrix} Γ_{λ}^{s} : = {h \in C ([0, 1], X_{0}^{s, p} (Ω)) : h (0) = 0, h (1) = u_{{\overline{λ}}_{0}^{s}}^{s}} . \end{matrix}$

The last two Theorems analyse what happens to the set of solutions of ( P a , b λ ) when $λ < {\bar{λ}}_{0}^{s}$ . Theorem 1.9.
Assume $a^{(N - 2 p s) / p s)} b > PS (N, p, s)$ . There exist $δ > 0$ , $r > 0$ such that for any $λ \in ({\overline{λ}}_{0}^{s} - δ, {\overline{λ}}_{0}^{s})$ the value $\begin{matrix} {\hat{ι}}_{λ}^{s} : = inf {I^{λ} (u) : u \in X_{0}^{s, p} (Ω), ‖ u ‖ ⩾ r} \end{matrix}$ is attained at a function $w_{λ}^{s} \in X_{0}^{s, p} (Ω)$ satisfying $‖ w_{λ}^{s} ‖ > r$ .
Theorem 1.10.
Suppose $a^{(N - 2 p s) / p s)} b > PS (N, p, s)$ . For any $λ \in ({\overline{λ}}_{0}^{s} - δ, {\overline{λ}}_{0}^{s})$ there is $v_{λ}^{s} \in X_{0}^{s, p} (Ω) ∖ {0}$ such that $I^{λ} (v_{λ}^{s}) = c_{λ}^{s}$ and ${(I^{λ})}^{'} (v_{λ}^{s}) = 0$ , where $\begin{matrix} c_{λ}^{s} : = inf_{h \in Γ_{λ}^{s}} max_{ζ \in [0, 1]} I^{λ} (h (ζ)) \end{matrix}$ and $\begin{matrix} Γ_{λ}^{s} : = {h \in C ([0, 1], X_{0}^{s, p} (Ω)) : h (0) = 0, h (1) = w_{λ}^{s}} . \end{matrix}$

2. Abstract framework and preliminary results

We consider the potential operator $A_{p}$ associated to the functional $u \mapsto ‖ u ‖^{p} / p$ on $X_{0}^{s, p} (Ω)$ , i.e. the operator $A_{p} : X_{0}^{s, p} (Ω) \to {(X_{0}^{s, p} (Ω))}^{*}$ such that $\begin{matrix} ⟨ A_{p} (u), v ⟩ = \int_{R^{N} \times R^{N}} \frac{| u (x) - u (y) |^{p - 2} (u (x) - u (y)) (v (x) - v (y))}{| x - y |^{N + s p}} d x d y \end{matrix}$ for every u, $v \in X_{0}^{s, p} (Ω)$ . Trivially, $\begin{matrix} ⟨ A_{p} (u), u ⟩ = ‖ u ‖^{p}, | ⟨ A_{p} (u), v ⟩ | ⩽ ‖ u ‖^{p - 1} ‖ v ‖ . \end{matrix}$

Lemma 2.1.
If a sequence ${(u_{n})}_{n}$ converges weakly to u in $X_{0}^{s, p} (Ω)$ and $\begin{matrix} ⟨ A_{p} (u_{n}), u_{n} - u ⟩ \to 0, \end{matrix}$ then $‖ u_{n} - u ‖ \to 0$ .
Proof.
We refer to [3] for a proof. □

The following Theorem is a classical result on fractional Sobolev spaces. For a proof we refer to [16].
Theorem 2.2.
Let $s \in (0, 1)$ and $p \in [1, \infty,)$ be such that $N > p s$ . Let $q \in [1, p_{s}^{})$ , $Ω \subset R^{N}$ be a bounded Lipschitz domain and* $F$ a bounded subset of $L^{p} (Ω)$ such that $\begin{matrix} sup_{u \in F} \int_{R^{2 N}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + p s}} d x d y < \infty . \end{matrix}$ Then $F$ is relatively compact in $L^{q} (Ω)$ .
Lemma 2.3.
Let $q \in R^{N}$ , $ε \in (0, 1)$ and $u \in L^{p_{s}^{}} (R^{N})$ . Suppose that either* $U = B (q, ε)$ and $V = R^{N}$ , or $U = R^{N}$ and $V = B (q, ε)$ . Then $\begin{matrix} lim_{ε \to 0} \frac{1}{ε^{p}} \int_{U} \int_{V \cap {| x - y | ⩽ ε}} \frac{| u (y) |^{p}}{| x - y |^{N + p s - p}} d x d y = 0 \end{matrix}$ and $\begin{matrix} lim_{ε \to 0} \int_{U} \int_{V \cap {| x - y | > ε}} \frac{| u (y) |^{p}}{| x - y |^{N + p s}} d x d y = 0 . \end{matrix}$
Proof.
The verification of the two limits is similar to [18, Proposition 7]. We omit the details. □
Proposition 2.4.
Let ${(u_{n})}_{n} \subset X_{0}^{s, p} (Ω)$ a bounded sequence. Suppose that $ϑ \in C^{\infty} (R^{N})$ is such that $0 ⩽ ϑ ⩽ 1$ , $ϑ = 1$ in $B (0, 1)$ and $ϑ = 0$ in $R^{N} ∖ B (0, 2)$ . For $q \in R^{N}$ , let $ϑ_{ε} (x) = ϑ (\frac{x - q}{ε})$ . Then $\begin{matrix} lim_{ε \to 0} lim_{n \to \infty} \int_{R^{2 N}} {| u_{n} (y) |}^{p} \frac{| ϑ_{ε} (x) - ϑ_{ε} (y) |^{p}}{| x - y |^{N + p s}} d x d y = 0 . \end{matrix}$
Proof.
The verification of the limit is similar to [18, Theorem 2]. We omit the details. □

We conclude this section recalling the best Sobolev constant is defined as $\begin{matrix} (2.1) & S_{s, p} : = inf_{u \in W^{s, p} (R^{N}) ∖ {0}} \frac{‖ u ‖_{W_{s, p} (R^{N})}^{p}}{‖ u ‖_{p_{s}^{*}}^{p}} . \end{matrix}$ A natural conjecture is that all the minimizers for $S_{s, p}$ are of the form $V (| \cdot - x_{0} | / ε)$ , where $\begin{matrix} V (x) = \frac{1}{{(1 + | x |^{\frac{p}{p - 1}})}^{\frac{N - p s}{p}}} \end{matrix}$ in analogy to the case $p = 2$ and $s = 1$ (see for instance [22]). Unfortunately this problem is still open and we can only rely on some asymptotic estimates at infinity proved by Mosconi et al. in [28].
3. Weakly sequentially lower semicontinuity and validity of the Palais–Smale

Proof of Theorem 1.2.
We start assuming $\begin{matrix} a^{\frac{N - 2 p s}{p s}} b ⩾ L (N, p, s) . \end{matrix}$ Take a sequence ${(u_{n})}_{n} \subset X_{0}^{s, p} (Ω)$ such that $u_{n} ⇀ u$ in $X_{0}^{s, p} (Ω)$ . Recalling that the embedding $X_{0}^{s, p} (Ω) ↪ L^{q} (Ω)$ is compact for every $q \in [1, p_{s}^{})$ by Theorem 2.2, we deduce $u_{n} \to u$ in $L^{q} (Ω)$ for all $q \in [1, p_{s}^{})$ and in particular $u_{n} (x) \to u (x)$ a.e. in $R^{N}$ as $n \to \infty$ . At this point we use [30, Lemma 3.2] getting $\begin{matrix} (3.1) & ‖ u_{n} - u ‖^{p} = ‖ u_{n} ‖^{p} - ‖ u ‖^{p} + o (1) as n \to \infty . \end{matrix}$ Furthermore, we observe $\begin{array}{l} ‖ u_{n} ‖^{2 p} - ‖ u ‖^{2 p} & = (‖ u_{n} ‖^{p} - ‖ u ‖^{p}) (‖ u_{n} ‖^{p} + ‖ u ‖^{p}) \\ (3.2) & = (‖ u_{n} - u ‖^{p} + o (1)) (‖ u_{n} - u ‖^{p} + 2 ‖ u ‖^{p} + o (1)), \end{array}$ where we used (3.1). We also apply the classical Brezis–Lieb Lemma (see [11, Theorem 1]) to get $\begin{matrix} (3.3) & ‖ u_{n} - u ‖_{p_{s}^{}}^{p_{s}^{}} = ‖ u_{n} ‖_{p_{s}^{}}^{p_{s}^{}} - ‖ u ‖_{p_{s}^{}}^{p_{s}^{}} + o (1) . \end{matrix}$ Now, we assemble (3.1), (3.2), (3.3) and we compute $\begin{array}{l} I (u_{n}) - I (u) & = \frac{a}{p} (‖ u_{n} ‖^{p} - ‖ u ‖^{p}) + \frac{b}{2 p} (‖ u_{n} ‖^{2 p} - ‖ u ‖^{2 p}) - \frac{1}{p_{s}^{}} (‖ u_{n} ‖_{p_{s}^{}}^{p_{s}^{}} - ‖ u ‖_{p_{s}^{}}^{p_{s}^{}}) \\ = \frac{a}{p} ‖ u_{n} - u ‖^{p} + \frac{b}{2 p} (‖ u_{n} - u ‖^{2 p} + 2 ‖ u ‖^{p} ‖ u_{n} - u ‖^{p}) \\ - \frac{1}{p_{s}^{}} ‖ u_{n} - u ‖_{p_{s}^{}}^{p_{s}^{}} + o (1) \\ ⩾ \frac{a}{p} ‖ u_{n} - u ‖^{p} + \frac{b}{2 p} ‖ u_{n} - u ‖^{2 p} - \frac{S_{s, p}^{- \frac{p_{s}^{}}{p}}}{2_{s}^{}} ‖ u_{n} - u ‖^{p_{s}^{}} + o (1) \\ (3.4) & = ‖ u_{n} - u ‖^{p} [\frac{a}{p} + \frac{b}{2 p} ‖ u_{n} - u ‖^{p} - \frac{S_{s, p}^{- \frac{p_{s}^{}}{p}}}{p_{s}^{}} ‖ u_{n} - u ‖^{p_{s}^{} - p}] + o (1) \end{array}$ as $n \to \infty$ , where we also used the Sobolev inequality given in (2.1). We introduce the auxiliary function $\begin{matrix} f_{s, p} (ζ) = \frac{a}{p} + \frac{b}{2 p} ζ^{p} - \frac{S_{s, p}^{- \frac{p_{s}^{}}{p}}}{p_{s}^{}} ζ^{p_{s}^{} - p}, ζ ⩾ 0, \end{matrix}$ and we notice that $f_{s, p}$ attains its global minimum at the point $\begin{matrix} m_{s, p} = {(\frac{b}{2} \frac{p_{s}^{}}{p_{s}^{} - p} S_{s, p}^{\frac{p_{s}^{}}{p}})}^{\frac{1}{p_{s}^{} - 2 p}} . \end{matrix}$ Besides, one easily verifies that $\begin{matrix} (3.5) & a^{\frac{N - 2 p s}{p s}} b ⩾ L (N, p, s) \Leftrightarrow f_{s, p} (m_{s, p}) = \frac{1}{p} (a - b^{- \frac{p s}{N - 2 p s}} L {(N, p, s)}^{\frac{2 s}{N - 2 p s}}) ⩾ 0 . \end{matrix}$ From (3.4) and (3.5) it follows that $\begin{matrix} \underset{n \to \infty}{lim inf} (I (u_{n}) - I (u)) ⩾ \underset{n \to \infty}{lim inf} ‖ u_{n} - u ‖^{p} f_{s, p} (‖ u_{n} - u ‖) ⩾ 0, \end{matrix}$ proving the sufficiency implication. In order to prove the other part of the theorem. we argue by contradiction. Under the assumption that $I$ is sequentially weakly lower semicontinuous we suppose that $\begin{matrix} (3.6) & a^{\frac{N - 2 p s}{p s}} b < L (N, p, s) . \end{matrix}$ Consider a minimizing sequence ${(u_{n})}_{n} \subset X_{0}^{s, p} (Ω)$ for (2.1). Since problem (2.1) is homogeneous, we can assume the sequence ${(u_{n})}_{n}$ is bounded on $X_{0}^{s, p} (Ω)$ . As a consequence, up to a subsequence, we have $u_{n} ⇀ u$ in $X_{0}^{s, p} (Ω)$ for some $u \in X_{0}^{s, p} (Ω) ∖ {0}$ . Set $L : = {lim inf}_{n \to \infty} ‖ u_{n} ‖$ and observe that exploiting the sequentially weakly semicontinuity of the norm we get $0 < ‖ u ‖ ⩽ L$ . Now, there exists a subsequence ${(u_{n_{k}})}_{k}$ such that ${lim}_{k \to \infty} ‖ u_{n_{k}} ‖ = L$ . We have already seen that the function $f_{s, p}$ has a minimum in $m_{s, p}$ which is global since $p > p_{s}^{} - p > 0$ implies ${lim}_{ζ \to + \infty} f_{s, p} (ζ) = + \infty$ . At this point we set $c = m_{s, p} / L$ . On one hand, also ${(c u_{n_{j}})}_{j}$ is a minimizing sequence for $S_{s, p}$ , so $\begin{array}{l} \underset{n \to \infty}{lim inf} I (c u_{n}) & ⩽ \underset{k \to \infty}{lim inf} I (c u_{n_{k}}) \\ = \underset{k \to \infty}{lim inf} ‖ c u_{n_{k}} ‖^{p} f_{s, p} (‖ c u_{n_{k}} ‖) = {(c L)}^{p} f_{s, p} (c L) \\ (3.7) & = {(c L)}^{p} f_{s, p} (m_{s, p}) ⩽ ‖ c u ‖^{p} f_{s, p} (m_{s, p}) ⩽ ‖ c u ‖^{p} f_{s, p} (‖ c u ‖), \end{array}$ where in the second to last inequality we used the inequality $f_{s, p} (m_{s, p}) < 0$ , since $a^{\frac{N - 2 p s}{p s}} b < L (N, p, s)$ . On the other hand, from the Sobolev inequality it follows $\begin{array}{l} ‖ c u ‖^{p} f_{s, p} (‖ c u ‖) & = \frac{a}{p} ‖ c u ‖^{p} + \frac{b}{2 p} ‖ c u ‖^{2 p} - \frac{S_{s, p}^{- \frac{p_{s}^{}}{p}}}{p_{s}^{}} ‖ c u ‖ \\ (3.8) & ⩽ \frac{a}{p} ‖ c u ‖^{p} + \frac{b}{2 p} ‖ c u ‖^{2 p} - \frac{1}{p_{s}^{}} \int_{Ω} | c u |^{p_{s}^{}} d x = I (c u) . \end{array}$ Coupling (3.7) and (3.8) we get $\begin{matrix} (3.9) & \underset{n \to \infty}{lim inf} I (c u_{n}) ⩽ I (c u) . \end{matrix}$ The strict inequality in (3.9) would contradict the weakly sequentially lower sequentially of the functional $I$ , so in (3.9) the equality must hold. However, this means that $c u$ would be a minimazer for (2.1), but recalling that $u = 0$ in $R^{N} ∖ Ω$ , we have a contradiction with [10, Theorem 1.1] since $0 < ‖ u ‖ ⩽ L$ . □
Proof of Theorem 1.3.
Let ${(u_{n})}_{n} \subset X_{0}^{s, p} (Ω)$ be a ${(PS)}_{c}$ sequence, i.e. $I_{a, b} (u_{n}) \to c$ , and $I_{a, b}^{'} (u_{n}) \to 0$ as $n \to \infty$ . From (2.1) it follows $\begin{matrix} I (u) = \frac{a}{p} ‖ u ‖^{p} + \frac{b}{2 p} ‖ u ‖^{2 p} - \frac{1}{p_{s}^{}} \int_{Ω} | u |^{p_{s}^{}} d x ⩾ \frac{a}{p} ‖ u ‖^{p} + \frac{b}{2 p} ‖ u ‖^{2 p} - \frac{S_{s, p}^{- \frac{p_{s}^{}}{p}}}{p_{s}^{}} ‖ u ‖^{p_{s}^{}} . \end{matrix}$ Recalling $2 p > p_{s}^{}$ , we can deduce that the functional $I$ is bounded from below. As a consequence of that, we have that the sequence ${(u_{n})}_{n}$ is bounded since $I (u_{n}) \to c$ as $n \to \infty$ . Thus, we are allowed to suppose $\begin{matrix} \{\begin{matrix} u_{n} ⇀ u & in X_{0}^{s, p} (Ω), \\ u_{n} \to u & in L^{q} (Ω) for all q \in [1, p_{s}^{}), \\ u_{n} \to u & a.e in R^{N} . \end{matrix} \end{matrix}$ Exploiting the Hölder inequality, we can deduce the boundedness of the sequence ${(u_{n})}_{n}$ also in the space of measures $M (Ω)$ . At this point, invoking [29, Theorem 2.5] there exist two Borel regular measures μ and ν such that $\begin{matrix} \int_{R^{N}} \frac{| u_{n} (x) - u_{n} (y) |^{p}}{| x - y |^{N + s p}} d y ⇀^{} μ and | u_{n} |^{2_{s}^{}} ⇀^{} ν in M (Ω), \end{matrix}$ where $\begin{matrix} (3.10) & ν = | u |^{p_{s}^{}} + \sum_{j \in J} ν_{j} δ_{x_{j}} \end{matrix}$ and $\begin{matrix} (3.11) & μ ⩾ \int_{R^{N}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + s p}} d y + \sum_{j \in J} μ_{j} δ_{x_{j}} \end{matrix}$ with $\begin{matrix} ν_{j} = ν ({x_{j}}) μ_{j} = μ ({x_{j}}) \end{matrix}$ and the set J at most countable. We also have $\begin{matrix} (3.12) & μ_{j} ⩾ S_{s, p} ν_{j}^{\frac{p}{p_{s}^{}}} . \end{matrix}$ We claim that the set J is empty. If the claim were false, there would exist at least an index $j_{0} \in J$ and a point $x_{j_{0}}$ with $ν_{j_{0}} \neq 0$ associated to it. Pick $ε > 0$ and consider a cut-off function such that $\begin{matrix} \{\begin{matrix} 0 ⩽ ϑ_{ε} ⩽ 1 & in Ω, \\ ϑ_{ε} = 1 & in B (x_{j_{0}}, ε), \\ ϑ_{ε} = 0 & in Ω ∖ B (x_{j_{0}}, 2 ε) . \end{matrix} \end{matrix}$ We notice that also the sequence ${(u_{n} ϑ_{ε})}_{n}$ is bounded, hence $\begin{matrix} lim_{n \to \infty} I^{'} (u_{n}) [u_{n} ϑ_{ε}] = 0 . \end{matrix}$ As a consequence of that $\begin{array}{l} o (1) & = I^{'} (u_{n}) [u_{n} ϑ_{ε}] \\ = (a + b ‖ u_{n} ‖^{p}) \\ \times \int_{Q} \frac{| u_{n} (x) - u_{n} (y) |^{p - 2} (u_{n} (x) - u_{n} (y)) (u_{n} (x) ϑ_{ε} (x) - u_{n} (y) ϑ_{ε} (y))}{| x - y |^{N + p s}} d x d y \\ - \int_{Ω} | u_{n} |^{p_{s}^{}} ϑ_{ε} d x \\ = (a + b ‖ u_{n} ‖^{p}) [\int_{Q} u_{n} (y) \frac{| u_{n} (x) - u_{n} (y) |^{p - 2} (u_{n} (x) - u_{n} (y)) (ϑ_{ε} (x) - ϑ_{ε} (y))}{| x - y |^{N + p s}} d x d y \\ (3.13) & + \int_{Q} ϑ_{ε} (x) \frac{| u_{n} (x) - u_{n} (y) |^{p}}{| x - y |^{N + p s}} d x d y] - \int_{Ω} | u_{n} |^{p_{s}^{}} ϑ_{ε} d x . \end{array}$ We estimate the first term of $I^{'} (u_{n}) [u_{n} ϑ_{ε}]$ with the Hölder inequality, obtaining $\begin{array}{l} (a + b ‖ u_{n} ‖^{p}) \int_{Q} u_{n} (y) \frac{| u_{n} (x) - u_{n} (y) |^{p - 2} (u_{n} (x) - u_{n} (y)) (ϑ_{ε} (x) - ϑ_{ε} (y))}{| x - y |^{N + p s}} d x d y \\ ⩽ C {(\int_{Q} \frac{| u_{n} (x) - u_{n} (y) |^{p}}{| x - y |^{N + p s}} d x d y)}^{\frac{p - 1}{p}} {(\int_{Q} {| u_{n} (y) |}^{p} \frac{| ϑ_{ε} (x) - ϑ_{ε} (y) |^{p}}{| x - y |^{N + p s}} d x d y)}^{\frac{1}{p}} \\ ⩽ \tilde{C} {(\int_{Q} {| u_{n} (y) |}^{p} \frac{| ϑ_{ε} (x) - ϑ_{ε} (y) |^{p}}{| x - y |^{N + p s}} d x d y)}^{\frac{1}{p}} \end{array}$ for some constants $C > 0$ , $\tilde{C} > 0$ . Now, Proposition 2.4 yields $\begin{matrix} lim_{ε \to 0} lim_{n \to \infty} \int_{R^{2 N}} {| u_{n} (y) |}^{p} \frac{| ϑ_{ε} (x) - ϑ_{ε} (y) |^{p}}{| x - y |^{N + p s}} d x d y = 0, \end{matrix}$ so that $\begin{array}{l} lim_{ε \to 0} lim_{n \to \infty} (a + b ‖ u_{n} ‖^{p}) \\ (3.14) & \times \int_{Q} u_{n} (y) \frac{| u_{n} (x) - u_{n} (y) |^{p - 2} (u_{n} (x) - u_{n} (y)) (ϑ_{ε} (x) - ϑ_{ε} (y))}{| x - y |^{N + p s}} d x d y = 0 . \end{array}$ Now, exploiting (3.11), we have $\begin{array}{l} lim_{n \to \infty} (a + b ‖ u_{n} ‖^{p}) \int_{Q} ϑ_{ε} (x) \frac{| u_{n} (x) - u_{n} (y) |^{p}}{| x - y |^{N + p s}} d x d y \\ ⩾ lim_{n \to \infty} [a \int_{R^{2 N} ∖ B {(x_{j_{0}}, 2 ε)}^{c} \times Ω^{c}} ϑ_{ε} (x) \frac{| u_{n} (x) - u_{n} (y) |^{p}}{| x - y |^{N + p s}} d x d y \\ + b {(\int_{Q} ϑ_{ε} (x) \frac{| u_{n} (x) - u_{n} (y) |^{p}}{| x - y |^{N + p s}} d x d y)}^{2}] \\ ⩾ a \int_{R^{2 N} ∖ B {(x_{j_{0}}, 2 ε)}^{c} \times Ω^{c}} ϑ_{ε} (x) \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + p s}} d x d y + a μ_{j_{0}} \\ + b {(\int_{Q} ϑ_{ε} (x) \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + p s}} d x d y)}^{2} + b μ_{j_{0}}^{2} . \end{array}$ Thus $\begin{matrix} (3.15) & lim_{ε \to 0} lim_{n \to \infty} (a + b ‖ u_{n} ‖^{p}) \int_{Q} ϑ_{ε} (x) \frac{| u_{n} (x) - u_{n} (y) |^{p}}{| x - y |^{N + p s}} d x d y ⩾ a μ_{j_{0}} + b μ_{j_{0}}^{2} . \end{matrix}$ Furthermore, it follows from (3.10) that $\begin{matrix} (3.16) & lim_{ε \to 0} lim_{n \to \infty} \int_{Ω} | u_{n} |^{p_{s}^{}} ϑ_{ε} d x = lim_{ε \to 0} \int_{Ω} | u |^{p_{s}^{}} ϑ_{ε} d x + ν_{j_{0}} = ν_{j_{0}} . \end{matrix}$ At this point, from (3.13), taking into account (3.14), (3.15), (3.16) and using (3.12) we deduce $\begin{matrix} (3.17) & 0 ⩾ a μ_{j_{0}} + b μ_{j_{0}}^{2} - ν_{j_{0}} ⩾ a μ_{j_{0}} + b μ_{j_{0}}^{2} - S_{s, p}^{- \frac{p_{s}^{}}{p}} μ_{j_{0}}^{\frac{p_{s}^{}}{p}} = μ_{j_{0}} (a + b μ_{j_{0}} - S_{s, p}^{- \frac{p_{s}^{}}{p}} μ_{j_{0}}^{\frac{p_{s}^{}}{p} - 1}) . \end{matrix}$ We define $\begin{matrix} {\tilde{f}}_{s, p} (ζ) = a + b ζ - S_{s, p}^{- \frac{p_{s}^{}}{p}} ζ^{\frac{p_{s}^{}}{p} - 1} for ζ ⩾ 0 . \end{matrix}$ We observe that the function ${\tilde{f}}_{s, p}$ has a global minimum in $\begin{matrix} {\tilde{m}}_{s, p} : = {(b S_{s, p}^{\frac{p_{s}^{}}{p}} \frac{p}{p_{s}^{} - p})}^{\frac{p}{p_{s}^{} - 2 p}} \end{matrix}$ and that $\begin{matrix} a^{\frac{N - 2 p s}{p s}} b > PS (N, p, s) \Leftrightarrow {\tilde{f}}_{s, p} ({\tilde{m}}_{s, p}) = a - b^{- \frac{p s}{N - 2 p s}} PS {(N, p, s)}^{\frac{p s}{N - 2 p s}} > 0 . \end{matrix}$ Hence $\begin{matrix} a + b μ_{j_{0}} - S_{N, s}^{- \frac{p_{s}^{}}{2}} μ_{j_{0}}^{\frac{p_{s}^{}}{2} - 1} > 0, \end{matrix}$ and the only admissible case in (3.17) is $μ_{j_{0}} = 0$ . From this, recalling (3.12), we also have $ν_{j_{0}} = 0$ that is absurd. Hence $J = \emptyset$ , which means $\begin{matrix} lim_{n \to \infty} \int_{Ω} | u_{n} |^{p_{s}^{}} d x = \int_{Ω} | u |^{p_{s}^{}} d x . \end{matrix}$ This coupled with (3.3) implies $\begin{matrix} u_{n} \to u in L^{p_{s}^{}} (Ω) . \end{matrix}$ From this and the Hölder inequality it follows that $\begin{matrix} (3.18) & lim_{n \to \infty} \int_{Ω} | u_{n} |^{p_{s}^{} - 2} u_{n} (u - u_{n}) d x = 0 . \end{matrix}$ Computing the derivative of $I (u_{n})$ along $u_{n} - u$ , we get $\begin{array}{l} 0 & = lim_{n \to \infty} I^{'} (u_{n}) [u_{n} - u] \\ = lim_{n \to \infty} [(a + b ‖ u_{n} ‖^{2}) \\ \times \int_{Q} \frac{| u_{n} (x) - u_{n} (y) |^{p - 2} (u_{n} (x) - u_{n} (y)) ((u_{n} - u) (x) - (u_{n} - u) (y))}{| x - y |^{N + p s}} d x d y \\ - \int_{Ω} | u_{n} |^{2_{s}^{} - 2} u_{n} (u_{n} - u) d x] \\ = lim_{n \to \infty} (a + b ‖ u_{n} ‖^{2}) \\ \times \int_{Q} \frac{| u_{n} (x) - u_{n} (y) |^{p - 2} (u_{n} (x) - u_{n} (y)) ((u_{n} - u) (x) - (u_{n} - u) (y))}{| x - y |^{N + p s}} d x d y \end{array}$ which implies $\begin{matrix} (3.19) & \int_{Q} \frac{| u_{n} (x) - u_{n} (y) |^{p - 2} (u_{n} (x) - u_{n} (y)) ((u_{n} - u) (x) - (u_{n} - u) (y))}{| x - y |^{N + p s}} d x d y \to 0 \end{matrix}$ since ${(u_{n})}_{n}$ is bounded in $X_{0}^{s, p} (Ω)$ . Along a subsequence, $u_{n}$ converges weakly to u in $X_{0}^{s, p} (Ω)$ , and Lemma 2.1 implies that $u_{n} \to u$ in $X_{0}^{s, p} (Ω)$ as $n \to \infty$ . □
4. The perturbed problem

In this section, applying the result obtained in Theorem 1.2, we investigate the existence of solutions of different kind of the perturbed problem $\begin{matrix} (P_{a, b}^{λ}) & \{\begin{matrix} (a + b \int_{Q} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + p s}} d x d y) {(- Δ_{p})}^{s} u = | u |^{p_{s}^{*} - 2} u + λ g (x, u) & in Ω, \\ u = 0 & in R^{N} ∖ Ω, \end{matrix} \end{matrix}$ where as before a, b are real positive parameter, Ω is a bounded domain and $λ > 0$ . As for g, we make the same assumptions present in [8] but with the general Sobolev critical exponent. Namely, we make the following assumptions:

$g : Ω \times R \to R$ is a Carathéodory function such that $g (x, 0) = 0$ a.e. in Ω;

$g (x, t) > 0$ for every $t > 0$ and $g (x, t) < 0$ for every $t < 0$ a.e. in Ω. In addition, we require that there is a $μ > 0$ such that $g (x, t) ⩾ μ > 0$ a.e in Ω and for every $t \in I$ , where I is some open interval of $(0, \infty)$ ;

there is a constant $c > 0$ and $q \in (p, p_{s}^{*})$ such that $g (x, t) ⩽ c (1 + | t |^{q - 1})$ a.e. in Ω;

${lim}_{t \to 0} g (x, t) / | t |^{p - 1} = 0$ uniformly with respect to $x \in Ω$ .

Using a variational approach, we investigate the existence of critical points of the functional defined on the space

X_{0}^{s} (Ω)

\begin{matrix} I^{λ} (u) : = \frac{a}{p} ‖ u ‖^{p} + \frac{b}{2 p} ‖ u ‖^{2 p} - \frac{1}{p_{s}^{*}} ‖ u ‖_{p_{s}^{*}}^{p_{s}^{*}} - λ \int_{Ω} G (x, u) d x, \end{matrix}

where we denote with

G (x, t) = \int_{0}^{t} g (x, τ) d τ

. Before starting the analysis of our problem we need to prove some technical results that will be useful up to the end of the section.

Remark 4.1.

For the reader convenience, we remember the definitions of the functions $\begin{matrix} f_{s, p} (ζ) : = \frac{a}{p} + \frac{b}{2 p} ζ^{p} - \frac{S_{s, p}^{- \frac{p_{s}^{*}}{p}}}{p_{s}^{*}} ζ^{p_{s}^{*} - p} \end{matrix}$ and $\begin{matrix} {\tilde{f}}_{s, p} (ζ) = a + b ζ - S_{s, p}^{- \frac{p_{s}^{*}}{p}} ζ^{\frac{p_{s}^{*}}{p} - 1} \end{matrix}$ defined in the previous section. We also recall that these functions have a unique local minimum attained respectively at $\begin{matrix} m_{s, p} = {[\frac{b}{2} \frac{p_{s}^{*}}{p_{s}^{*} - p} S_{s, p}^{\frac{p_{s}^{*}}{p}}]}^{\frac{1}{p_{s}^{*} - 2 p}}, \end{matrix}$ and $\begin{matrix} {\tilde{m}}_{s, p} = {[b S_{s, p}^{\frac{p_{s}^{*}}{p}} \frac{p}{p_{s}^{*} - p} S_{s, p}^{\frac{p_{s}^{*}}{p}}]}^{\frac{1}{p_{s}^{*} - 2 p}} . \end{matrix}$ Besides, $f_{s, p} (m_{s, p}) > 0$ if and only if $a^{\frac{N - 2 p s}{p s}} b > L (N, p, s)$ and $f_{s, p} (m_{s, p}) = 0$ when $a^{\frac{N - 2 p s}{p s}} b = L (N, p, s)$ . Similarly ${\tilde{f}}_{s, p} ({\tilde{m}}_{s, p}) > 0$ if and only if $a^{(N - 2 p s) / p s} b > PS (N, p, s)$ and ${\tilde{f}}_{s, p} ({\tilde{m}}_{s, p}) = 0$ when $a^{(N - 2 p s) / p s} b = PS (N, p, s)$ .

Proposition 4.2.

Let $u \in X_{0}^{s, p} (Ω) ∖ {0}$ . We have that:

for every $ζ > 0$ it holds $\begin{matrix} \frac{a}{p} ‖ u ‖^{p} + \frac{b}{2 p} ζ^{p} ‖ u ‖^{2 p} - \frac{1}{p_{s}^{*}} ζ^{p_{s}^{*} - p} > f_{s, p} (ζ ‖ u ‖) ‖ u ‖^{p}; \end{matrix}$

for every $ζ > 0$ it holds $\begin{matrix} a ‖ u ‖^{p} + b ζ^{p} ‖ u ‖^{2 p} - ‖ u ‖_{p_{s}^{*}}^{p_{s}^{*}} ζ^{p_{s}^{*} - p} > {\tilde{f}}_{s, p} (ζ ‖ u ‖) ‖ u ‖^{2} . \end{matrix}$

Proof.

We only prove (i), since (ii) follows in a similar way. Considering $u = 0$ in $R^{N} ∖ Ω$ , taking into account [10, Theorem 1.1] and the Sobolev inequality, we have $\begin{array}{l} ζ^{p} [\frac{a}{p} ‖ u ‖^{p} + \frac{b}{2 p} ζ^{p} ‖ u ‖^{2 p} - \frac{1}{p_{s}^{*}} ζ^{p_{s}^{*} - p} ‖ u ‖_{p_{s}^{*}}^{p_{s}^{*}}] \\ = \frac{a}{p} {(ζ ‖ u ‖)}^{p} + \frac{b}{2 p} {(ζ ‖ u ‖)}^{2 p} - \frac{‖ u ‖_{p_{s}^{*}}^{p_{s}^{*}}}{‖ u ‖^{p_{s}^{*}}} \frac{{(ζ ‖ u ‖)}^{p_{s}^{*}}}{p_{s}^{*}} \\ (4.1) & > \frac{a}{p} {(ζ ‖ u ‖)}^{p} + \frac{b}{2 p} {(ζ ‖ u ‖)}^{2 p} - S_{s, p}^{- \frac{p_{s}^{*}}{p}} \frac{{(ζ ‖ u ‖)}^{p_{s}^{*}}}{p_{s}^{*}} . \end{array}$ □

We are now going to prove that the functional is sequentially lower semi continuous and satisfies the Palais–Smale condition for a and b large enough.

Lemma 4.3.

Let $a, b \in R^{+}$ , ${(u_{k})}_{k} \subset X_{0}^{s, p} (Ω)$ and $λ_{k} \to λ ⩾ 0$ as $k \to \infty$ :

if $a^{(N - 2 p s) / p s} b ⩾ L (N, p, s)$ and $u_{k} ⇀ u$ in $X_{0}^{s, p} (Ω)$ then $\begin{matrix} I^{λ} (u) ⩽ \underset{k \to \infty}{lim inf} I^{λ_{k}} (u_{k}); \end{matrix}$

if $a^{(N - 2 p s) / 2 s} b > PS (N, p, s)$ , $I^{λ} (u_{k}) \to c$ and ${(I^{λ})}^{'} (u_{k}) \to 0$ then ${(u_{k})}_{k}$ is convergent to some u in $X_{0}^{s, p} (Ω)$ up to subsequence.

Proof.

Since the proof is essentially the same of Theorems 1.2 and 1.3 we omit it. □

At this point fix $λ ⩾ 0$ and $u \in X_{0}^{s} (Ω)$ . For all $ζ > 0$ we define the fiber map $\begin{matrix} J^{λ, u} (ζ) : = I^{λ} (ζ u) = \frac{a}{p} ζ^{p} ‖ u ‖^{p} + \frac{b}{2 p} ζ^{2 p} ‖ u ‖^{2 p} - \frac{ζ^{p_{s}^{*}}}{p_{s}^{*}} ‖ u ‖_{p_{s}^{*}}^{p_{s}^{*}} - \int_{Ω} G (x, ζ u) d x . \end{matrix}$

Proposition 4.4.

Let $λ \in R$ be nonnegative and $u \in X_{0}^{s, p} (Ω) {0}$ . Then it is possible to find a neighbourhood $V_{λ}$ of 0 such that $J^{λ, u} (ζ) > 0$ for every $ζ \in V_{λ} \cap (0, \infty)$ . Furthermore $J^{λ, u} (ζ) \to \infty$ as $ζ \to \infty$ . In particular, the map $ζ \mapsto J^{λ, u} (ζ)$ is bounded from below.

Proof.

Choose $ε > 0$ . From $(H_{4})$ , for ζ small enough, it follows $\begin{array}{l} J^{λ, u} (ζ) & = ζ^{p} (\frac{a}{p} ‖ u ‖^{p} + \frac{b}{2 p} ζ^{p} ‖ u ‖^{2 p} - \frac{ζ^{p_{s}^{*} - p}}{p_{s}^{*}} ‖ u ‖_{p_{s}^{*}}^{p_{s}^{*}} - λ \int_{Ω} \frac{G (x, ζ u)}{ζ^{p}} d x) \\ ⩾ ζ^{p} (\frac{a}{p} ‖ u ‖^{p} + \frac{b}{2 p} ζ^{p} ‖ u ‖^{2 p} - \frac{ζ^{p_{s}^{*} - p}}{p_{s}^{*}} ‖ u ‖_{p_{s}^{*}}^{p_{s}^{*}} - λ \frac{ε}{p} ‖ u ‖_{p}^{p}) . \end{array}$ Applying the Sobolev inequality, selecting ε adequately and taking ζ even smaller if needed we get the first part of the assertion. To conclude, it is sufficient to notice that G has subcritical growth and that $p < q < p_{s}^{*} < 2 p$ . □

Now we fix $u \in X_{0}^{s, p} (Ω)$ and we consider the system $\begin{matrix} (4.2) & \{\begin{matrix} J^{λ, u} (ζ) = 0, \\ {(J^{λ, u})}^{'} (ζ) = 0, \\ J^{λ, u} (ζ) = {inf}_{ϱ > 0} J^{λ, u} (ϱ) \end{matrix} \end{matrix}$ in the unknowns λ and ζ.

Proposition 4.5.

Let $a, b \in R^{+}$ such that $a^{(N - 2 p s) / p s} b ⩾ L (N, p, s)$ . For any $u \in X_{0}^{s, p} (Ω) ∖ {0}$ there is a unique $λ = λ_{0}^{s} (u)$ that solves ( 4.2 ).

Proof.

The statement follows as in [8, Proposition 4]. □

Corollary 4.6.

Let $u \in X_{0}^{s, p} (Ω) ∖ {0}$ . The number $λ_{0}^{s} (u)$ is the only parameter such that $\begin{matrix} inf_{ζ \in (0, \infty)} J^{λ_{0}^{s} (u), u} (ζ) = 0 . \end{matrix}$ In addition, $\begin{matrix} inf_{ζ \in (0, \infty)} J^{λ, u} (ζ) \{\begin{matrix} < 0 & if λ > λ_{0}^{s} (u), \\ = 0 & if 0 ⩽ λ ⩽ λ_{0}^{s} (u) . \end{matrix} \end{matrix}$

Proof.

The assertion comes as an immediate consequence of the proof of Proposition 4.5. □

Now we define $\begin{matrix} {\overline{λ}}_{0}^{s} : = inf_{u \in X_{0}^{s, p} (Ω) ∖ {0}} λ_{0}^{s} (u) . \end{matrix}$ We emphasize that ${\overline{λ}}_{0}^{s}$ is independent from u. In addition, as we are going to see, ${\overline{λ}}_{0}^{s}$ has a key importance in determining at what level of energy the minimum is attained. The next Proposition exhibits the relation between ${\overline{λ}}_{0}^{s}$ and the parameters a and b.

Proposition 4.7.

The following statements hold:

if $a^{(N - 2 p s) / p s} b > L (N, p, s)$ then ${\overline{λ}}_{0}^{s} > 0$ ;

if $a^{(N - 2 p s) / p s} b = L (N, p, s)$ then ${\overline{λ}}_{0}^{s} = 0$ . Furthermore, if ${(u_{k})}_{k} \subset X_{0}^{s, p} (Ω) ∖ {0}$ is a sequence such that $λ_{0}^{s} (u_{k}) \to 0$ as $k \to \infty$ , we have that $u_{k} ⇀ 0$ and $\begin{matrix} \frac{‖ u_{k} ‖_{p}^{p}}{‖ u_{k} ‖_{p_{s}^{*}}^{p}} \to S_{s, p} . \end{matrix}$

Before giving the proof we need some estimates on the minimizers of (2.1). Consider the function $u_{ε, δ} (r)$ defined on [28, Lemma 2.7]. In particular, the support of $u_{ε, δ} (r)$ is compact and there exists $R > 0$ such that $\begin{matrix} u_{ε, δ} (r) = U_{ε} (r), \end{matrix}$ for $r ⩽ R$ , where $\begin{matrix} U_{ε} (r) = U_{ε} (x) = \frac{1}{ε^{\frac{N - p s}{p}}} U (\frac{| x |}{ε}) \end{matrix}$ and U is a minimizer for (2.1) whose existence is guaranteed by [28, Proposition 2.1].

Now, take the rescaled function $\begin{matrix} w_{ε} (x) : = {(ε^{\frac{1}{p}})}^{- \frac{N - p s}{p (p - 1)}} u_{\sqrt[p]{ε}, δ} (x) . \end{matrix}$ We point out that we omitted the dependence of δ since is not relevant for our purposes. Taking under consideration this rescaling, from [28, Lemma 2.7] it follows $\begin{matrix} ‖ w_{ε} ‖^{p} ⩽ S_{s, p} ε^{- \frac{N - p s}{p (p - 1)}} + O (1), ‖ w_{ε} ‖_{p_{s}^{*}}^{p_{s}^{*}} ⩾ S_{s, p}^{\frac{N}{p s}} ε^{- \frac{N}{p (p - 1)}} - O (1) . \end{matrix}$ From this, denoting with $v_{ε} : = w_{ε} / ‖ w_{ε} ‖$ the normalized function we get $\begin{matrix} (4.3) & ‖ v_{ε} ‖ = 1, ‖ v_{ε} ‖_{p_{s}^{*}}^{p_{s}^{*}} ⩾ S_{s, p}^{- \frac{p_{s}^{*}}{p}} + O (ε^{\frac{N - p s}{p (p - 1)}}), ‖ w_{ε} ‖ ⩽ S_{s, p}^{\frac{1}{p}} ε^{- \frac{N - p s}{p^{2} (p - 1)}} + O (ε^{\frac{N - p s}{p^{2}}}) \end{matrix}$ as $ε \to 0$ .

Proof of Proposition 4.7.

(i) We start noticing that the function $u \to λ_{0}^{s} (u)$ is well defined and homogeneous of degree zero. Indeed, considering a solution $(ζ, λ_{0}^{s} (u))$ of (4.2) and $α > 0$ , observing that $J^{λ, α u} (ζ) = J^{λ, u} (α ζ)$ and ${(J^{λ, α u})}^{'} (ζ) = {(J^{λ, u})}^{'} (α ζ)$ we get that also $(\frac{ζ}{α}, λ_{0}^{s} (u))$ solves (4.2) with $α u$ . From the uniqueness of the parameter $λ_{0}^{s} (α u)$ it follows that $λ_{0}^{s} (α u) = λ_{0}^{s} (u)$ . To see the positivity of $λ_{0}^{s}$ , we argue by contradiction supposing ${\overline{λ}}_{0}^{s} = 0$ . If so, there would be a sequence ${(u_{k})}_{k} \subset X_{0}^{s, p} (Ω) ∖ {0}$ such that $λ_{k} : = λ_{0}^{s} (u_{k}) \to 0$ . Exploiting the homogeneity of the map $u \to λ_{0}^{s} (u)$ , it is not restrictive to assume $‖ u_{k} ‖ = 1$ . Now, Proposition 4.5 implies the existence of $ζ_{k} > 0$ such that $J^{λ_{k}, u_{k}} (ζ_{k}) = 0$ , that is $\begin{matrix} \frac{a}{p} + \frac{b}{2 p} ζ_{k}^{p} - \frac{1}{p_{s}^{*}} ‖ u_{k} ‖_{p_{s}^{*}}^{p_{s}^{*}} ζ_{k}^{p_{s}^{*} - p} - λ_{k} \int_{Ω} \frac{G (x, ζ_{k} u)}{ζ_{k}^{p}} d x = 0 . \end{matrix}$ Applying Proposition 4.4, we obtain $\begin{matrix} (4.4) & f_{s, p} (ζ_{k}) < \frac{a}{p} + \frac{b}{2 p} ζ_{k}^{p} - \frac{1}{p_{s}^{*}} ‖ u_{k} ‖_{p_{s}^{*}}^{p_{s}^{*}} ζ_{k}^{p_{s}^{*} - p} = λ_{k} \int_{Ω} \frac{G (x, ζ_{k} u)}{ζ_{k}^{p}} d x . \end{matrix}$ From hypotheses $(H_{3})$ and $(H_{4})$ it follows that for any $ε > 0$ there is a positive constant $c > 0$ such that $| G (x, t) | < \frac{ε}{p} t^{p} + \frac{c}{q} | t |^{q}$ for a.e. $x \in Ω$ and all $t \in R$ . As a consequence, the sequence ${(ζ_{k})}_{k}$ must be bounded, and up to subsequence it converges to some $\overline{ζ} > 0$ . Finally, letting $k \to \infty$ and taking into account Remark 4.1, from (4.4) we obtain $\begin{matrix} 0 < f_{s, p} (\overline{ζ}) = lim_{k \to \infty} λ_{k} \int_{Ω} \frac{G (x, ζ_{k} u_{k})}{ζ_{k}^{p}} d x = 0 \end{matrix}$ which is impossible.

(ii) Up to a translation, we can suppose that $0 \in Ω$ . In virtue of the estimates in (4.3), we have $\begin{array}{l} J^{λ, v_{ε}} (ζ) & = \frac{a}{p} ζ^{p} + \frac{b}{2 p} ζ^{2 p} - \frac{ζ^{p_{s}^{*}}}{p_{s}^{*}} ‖ v_{ε} ‖_{2_{s}^{*}}^{2_{s}^{*}} - λ \int_{Ω} G (x, ζ v_{ε} (| x |)) d x \\ ⩽ ζ^{p} f_{s, p} (ζ) - \frac{ζ^{p_{s}^{*}}}{p_{s}^{*}} O (ε^{\frac{N - p s}{p (p - 1)}}) - λ \int_{Ω} G (x, ζ v_{ε} (| x |)) d x . \end{array}$ Selecting as $ζ = m_{s, p}$ we get $\begin{matrix} (4.5) & J^{λ, v_{ε}} (m_{s, p}) ⩽ - \frac{m_{s, p}^{p_{s}^{*}}}{p_{s}^{*}} O (ε^{\frac{N - p s}{p (p - 1)}}) - λ \int_{Ω} G (x, m_{s, p} v_{ε} (| x |)) d x . \end{matrix}$

Claim.

There exists a constant $C_{1} > 0$ such that $\int_{Ω} G (x, m_{N, s} u_{ε}) d x ⩾ C_{1} ε^{\frac{N}{p^{2}}}$ as $ε \to 0$ .

Indeed, hypothesis $(H_{2})$ asserts the existence of $μ > 0$ such that $g (x, t) ⩾ χ_{I}$ where I is an open interval of $(0, \infty)$ and $χ_{I}$ is its characteristic function. Hence we can find a $β > 0$ such that $G (x, t) ⩾ \tilde{G} (t) : = μ \int_{0}^{t} χ_{I} (τ) d τ ⩾ β$ for any $t ⩾ α$ where $α : = inf I > 0$ . At this point, we have $\begin{array}{l} \int_{Ω} G (x, m_{s, p} v_{ε} (| x |)) d x & ⩾ \int_{| x | ⩽ R} G (x, m_{s, p} v_{ε} (| x |)) d x = \int_{| x | ⩽ R} G (x, \frac{m_{s, p}}{‖ w_{ε} ‖} w_{ε} (| x |)) d x \\ ⩾ \int_{| x | ⩽ R} \tilde{G} (\frac{m_{s, p}}{‖ w_{ε} ‖} w_{ε} (| x |)) d x \\ ⩾ \int_{| x | ⩽ R} \tilde{G} (\frac{m_{s, p}}{‖ w_{ε} ‖} ε^{- \frac{N - p s}{p (p - 1)}} U (| \frac{x}{\sqrt[p]{ε}} |)) d x \\ = ω_{N} \int_{0}^{R} \tilde{G} (m_{s, p} \frac{w_{ε} (w)}{‖ w_{ε} ‖}) w^{N - 1} d w \\ (4.6) & ⩾ ω_{N} \int_{0}^{\sqrt[p]{ε} R} \tilde{G} (m_{s, p} \frac{w_{ε} (w}{‖ w_{ε} ‖}) w^{N - 1} d w . \end{array}$ With the change of variable $x = \sqrt[p]{ε} y$ (4.6) becomes $\begin{array}{l} \int_{Ω} G (x, m_{s, p} v_{ε} (| x |)) d x & ⩾ ε^{\frac{N}{p}} \int_{| x | ⩽ R ε^{- \frac{1}{p}}} \tilde{G} (\frac{m_{s, p}}{‖ w_{ε} ‖} ε^{- \frac{N - p s}{p (p - 1)}} U (| y |)) d y \\ = ω_{N} ε^{\frac{N}{p}} \int_{0}^{R ε^{- \frac{1}{p}}} \tilde{G} (\frac{m_{s, p}}{‖ w_{ε} ‖} ε^{- \frac{N - p s}{p (p - 1)}} U (w)) w^{N - 1} d w \\ ⩾ ω_{N} ε^{\frac{N}{p}} \int_{0}^{R ε^{- \frac{p - 1}{p^{2}}}} \tilde{G} (\frac{m_{s, p}}{‖ w_{ε} ‖} ε^{- \frac{N - p s}{p (p - 1)}} U (w)) w^{N - 1} d w . \end{array}$ We point out that if $\begin{matrix} \frac{m_{s, p}}{‖ w_{ε} ‖} ε^{- \frac{N - p s}{p (p - 1)}} U (w) ⩾ α for w \in [0, ε^{- \frac{p - 1}{p^{2}}} R] \end{matrix}$ then $\begin{matrix} \int_{0}^{ε^{- \frac{p - 1}{p^{2}}} R} \tilde{G} (m_{s, p} \frac{w_{ε} (w)}{‖ w_{ε} ‖}) w^{N - 1} d w ⩾ β \int_{0}^{ε^{- \frac{p - 1}{p^{2}}} R} w^{N - 1} d w = \frac{C_{1}}{ω_{N}} ε^{- N \frac{p - 1}{p^{2}}} . \end{matrix}$ Since $w \in [0, ε^{- \frac{p - 1}{p^{2}}} R]$ , and recalling that $w_{ε}$ is monotone decreasing by [28, Proposition 2.1] and [10, Theorem 1.1], we have $\begin{matrix} (4.7) & \frac{m_{s, p}}{‖ w_{ε} ‖} ε^{- \frac{N - p s}{p (p - 1)}} U (w) ⩾ \frac{m_{s, p}}{‖ w_{ε} ‖} ε^{- \frac{N - p s}{p (p - 1)}} U (ε^{- \frac{p - 1}{p^{2}}} R) . \end{matrix}$ Now, applying again [10, Theorem 1.1], for ε small enough we have $\begin{matrix} \frac{1}{2} U_{\infty} {(ε^{- \frac{p - 1}{p^{2}}} R)}^{- \frac{N - p s}{p - 1}} ⩽ U (ε^{- \frac{p - 1}{p^{2}}} R) ⩽ \frac{3}{2} U_{\infty} {(ε^{- \frac{p - 1}{p^{2}}} R)}^{- \frac{N - p s}{p - 1}}, \end{matrix}$ where $U_{\infty}$ is a constant that can be supposed positive without restrictions. From this, (4.3) and (4.7) we get $\begin{array}{l} \frac{m_{s, p}}{‖ w_{ε} ‖} ε^{- \frac{N - p s}{p (p - 1)}} U (w) & ⩾ \frac{U_{\infty}}{2} \frac{m_{s, p}}{‖ w_{ε} ‖} ε^{- \frac{N - p s}{p^{2} (p - 1)}} R^{- \frac{N - p s}{p - 1}} \\ ⩾ m_{s, p} \frac{U_{\infty}}{2} \frac{R^{- \frac{N - p s}{p - 1}}}{S_{s, p}^{\frac{1}{p}} + O (ε^{\frac{N - p s}{p (p - 1)}})} ⩾ α \end{array}$ restricting eventually R, and the claim is proved.

At this point, exploiting the claim, from (4.5) it follows $\begin{matrix} J^{λ, v_{ε}} (m_{s, p}) ⩽ ε^{\frac{N}{p^{2}}} (- \frac{m_{s, p}^{p_{s}^{*}}}{p_{s}^{*}} O (ε^{\frac{N - p^{2} s}{p^{2} (p - 1)}}) - λ C_{1}) < 0 \end{matrix}$ for ε sufficiently small. As a consequence, $λ_{0}^{s} (u_{ε}) < λ$ . Since all arguments above are independent of the choice of λ, we may let $λ \to 0$ and obtain ${\overline{λ}}_{0}^{s} = 0$ . To prove the remaining part of the Proposition, take a sequence ${(u_{k})}_{k} \subset X_{0}^{s, p} (Ω) ∖ {0}$ such that $λ_{k} : = λ_{0}^{s} (u_{k}) \to {\overline{λ}}_{0}^{s} = 0$ . Analogously to part (i), it is not restrictive to assume $‖ u_{k} ‖ = 1$ , $u_{k} ⇀ u$ and that there exists $ζ_{k} > 0$ such that $\begin{matrix} (4.8) & \frac{a}{p} + \frac{b}{2 p} ζ_{k}^{p} + \frac{ζ^{p_{s}^{*} - p}}{p_{s}^{*}} ‖ u_{k} ‖_{p_{s}^{*}}^{p_{s}^{*}} - λ_{k} \int_{Ω} \frac{G (x, ζ_{k} u_{k})}{ζ_{k}^{p}} d x = 0 . \end{matrix}$ Putting together assumptions $(H_{3})$ , $(H_{4})$ and (4.8), we can see that, up to subsequence, $ζ_{k} \to \overline{ζ}$ and $‖ u_{k} ‖_{p_{s}^{*}}^{p_{s}^{*}} \to γ$ as $k \to \infty$ . Letting $k \to \infty$ in (4.8), we get $\begin{matrix} \frac{a}{p} + \frac{b}{2 p} {\overline{ζ}}^{p} - \frac{{\overline{ζ}}^{p_{s}^{*} - p}}{p_{s}^{*}} γ = 0 . \end{matrix}$ Since $a^{(N - 2 p s) / p s} b = L (N, p, s)$ it is easy to see that $γ = S_{s, p}^{- \frac{p_{s}^{*}}{p}}$ , implying that ${(u_{k})}_{k}$ is a minimizing sequence for $S_{s, p}$ . Finally, suppose by contradiction $u \neq 0$ . By the lower semicontinuity of the norm we have $‖ u ‖ ⩽ 1$ . From this, taking under consideration Remark 4.1, we obtain $\begin{array}{l} 0 & ⩽ \frac{a}{p} + \frac{b}{2 p} {\overline{ζ}}^{p} - \frac{S_{s, p}^{- \frac{p_{s}^{*}}{p}}}{p_{s}^{*}} {\overline{ζ}}^{p_{s}^{*} - p} ‖ u ‖^{p_{s}^{*}} ⩽ \frac{a}{p} + \frac{b}{2 p} {\overline{ζ}}^{p} - \frac{{\overline{ζ}}^{p_{s}^{*} - 2}}{p_{s}^{*}} ‖ u ‖_{p_{s}^{*}}^{p_{s}^{*}} \\ ⩽ \underset{k \to \infty}{lim sup} (\frac{a}{p} + \frac{b}{2 p} ζ_{k}^{p} - \frac{ζ_{k}^{p_{s}^{*}}}{p_{s}^{*}} ‖ u_{k} ‖_{p_{s}^{*}}^{p_{s}^{*}} - λ_{k} \int_{Ω} \frac{G (x, ζ_{k} u_{k})}{ζ_{k}^{p}} d x) = 0, \end{array}$ which would imply that u is a minimizer for (2.1). However, this is not possible if we compare [10, Theorem 1.1] and the fact that $u = 0$ in $R^{N} ∖ Ω$ . □

Proposition 4.8.

If $λ ⩽ {\overline{λ}}_{0}^{s}$ then ${inf}_{ζ > 0} J^{λ, u} (ζ) = 0$ for any $u \in X_{0}^{s, p} (Ω) ∖ {0}$ . On the other hand, if $λ > {\overline{λ}}_{0}^{s}$ there exists $u \in X_{0}^{s, p} (Ω) ∖ {0}$ such that ${inf}_{ζ > 0} J^{λ, u} (ζ) < 0$ .

Proof.

The proof follows closely the line of [8, Proposition 6]. □

We are now ready to prove Theorem 1.5 and Theorem 1.6.

Proof of Theorem 1.5.

The thesis comes as in [8, Theorem 2]. □

Proof of Theorem 1.6.

(i) Consider a sequence ${(λ_{k})}_{k} \subset R^{+}$ such that $λ_{k} ↘ {\overline{λ}}_{0}^{s}$ . In virtue of Theorem 1.5 we can find a sequence ${(u_{k})}_{k} \subset X_{0}^{s, p} (Ω) ∖ {0}$ such that $ι_{λ_{k}}^{s} = I^{λ_{k}} (u_{k}) < 0$ . Similarly to what we have done in Proposition 4.8, after choosing $ε > 0$ we have $\begin{matrix} (4.9) & | G (x, t) | ⩽ \frac{ε}{p} | t |^{p} + \frac{c}{q} | t |^{q} \end{matrix}$ for all $t \in R$ and a.e. in Ω. As a consequence of that $\begin{array}{l} \frac{a}{p} ‖ u_{k} ‖^{p} + \frac{b}{2 p} ‖ u_{k} ‖^{2 p} - \frac{1}{p_{s}^{*}} ‖ u_{k} ‖_{p_{s}^{*}}^{p_{s}^{*}} \\ (4.10) & < λ_{k} \int_{Ω} G (x, u_{k}) d x ⩽ λ_{k} (\frac{ε}{p} ‖ u_{k} ‖_{p}^{p} + \frac{c}{q} ‖ u_{k} ‖_{q}^{q}) ⩽ \tilde{C} (‖ u_{k} ‖^{p} + ‖ u_{k} ‖^{q}) \end{array}$ for some $\tilde{C} > 0$ since $X_{0}^{s, p} (Ω) ↪ L^{υ} (Ω)$ continuously for any $υ \in [1, p_{s}^{*}]$ . Since $2 p > p_{s}^{*}$ the sequence ${(‖ u_{k} ‖)}_{k}$ need to be bounded and we are allowed to suppose $u_{k} ⇀ u$ in $X_{0}^{s, p} (Ω)$ . Now, on one hand we use Lemma 4.3[ $(1)$ ] and we get $\begin{matrix} I^{{\overline{λ}}_{0}^{s}} (u) ⩽ \underset{k \to \infty}{lim inf} I^{{\overline{λ}}_{0}^{s}} (u_{k}) ⩽ 0 . \end{matrix}$ On the other hand, Proposition 4.8 implies that $I^{{\overline{λ}}_{0}^{s}} (v) ⩾ 0$ for any $v \in X_{0}^{s, p} (Ω)$ . Hence, the only admissible scenario is $\begin{matrix} (4.11) & ι_{{\overline{λ}}_{0}^{s}}^{s} = I_{a, b}^{{\overline{λ}}_{0}^{s}} (u) = 0 . \end{matrix}$ In order to show that u is a non trivial minimizer, we start noticing that $\begin{array}{l} \frac{a}{p} ‖ u_{k} ‖^{p} + \frac{b}{2 p} ‖ u_{k} ‖^{2 p} - \frac{S_{s, p}^{- \frac{p_{s}^{*}}{p}}}{p_{s}^{*}} ‖ u_{k} ‖^{p_{s}^{*}} \\ ⩽ \frac{a}{p} ‖ u_{k} ‖^{p} + \frac{b}{2 p} ‖ u_{k} ‖^{2 p} - \frac{1}{p_{s}^{*}} ‖ u_{k} ‖_{p_{s}^{*}}^{p_{s}^{*}} < λ_{k} \int_{Ω} G (x, u_{k}) d x, \end{array}$ where we used the fractional Sobolev inequality. Dividing by $‖ u_{k} ‖^{p}$ and exploiting (3.18), we obtain $\begin{matrix} f_{s, p} (‖ u_{k} ‖) ⩽ λ_{k} (\frac{ε}{p} + \frac{c}{q} ‖ u_{k} ‖_{q}^{q}) . \end{matrix}$ If $u = 0$ , recalling that $X_{0}^{s, p} (Ω) ↪ L^{p} (Ω)$ is compact, we would obtain $f_{s, p} (‖ u_{k} ‖) \to 0$ as $k \to \infty$ since $ε > 0$ is arbitrary. However, Remark 4.1 yields $\begin{matrix} f_{s, p} (‖ u_{k} ‖) ⩾ f_{s, p} (m_{s, p}) > 0 \end{matrix}$ since $a^{(N - 2 p s) / p s} b > L (N, p, s)$ . This contradiction shows that $u \neq 0$ .

(ii) Proposition 4.7(ii) implies ${\overline{λ}}_{0}^{s} = 0$ , so $\begin{matrix} I^{{\overline{λ}}_{0}^{s}} (u) = \frac{a}{p} ‖ u ‖^{p} + \frac{b}{2 p} ‖ u ‖^{2 p} - \frac{1}{p_{s}^{*}} ‖ u ‖_{p_{s}^{*}}^{p_{s}^{*}} . \end{matrix}$ At this point, keeping in mind Remark 4.1, we have $\begin{matrix} I^{{\overline{λ}}_{0}^{s}} (u) = ‖ u ‖^{p} f_{s, p} (‖ u ‖) > 0 \end{matrix}$ for all $u \in X_{0}^{s, p} (Ω) ∖ {0}$ . In virtue of the inequality above, recalling that (4.11) still holds, it is evident that the infimum can be achieved only if $u = 0$ . □

Corollary 4.9.

If $a^{(N - 2 p s) / p s} b > L (N, p, s)$ and $u \in X_{0}^{s, p} (Ω) ∖ {0}$ is such that $ι_{{\overline{λ}}_{0}^{s}} = I^{{\overline{λ}}_{0}^{s}} (u)$ then ${\overline{λ}}_{0}^{s} = λ_{0}^{s} (u)$ .

Proof.

Observe that $({\overline{λ}}_{0}^{s}, u)$ solves (4.2) and conclude recalling the uniqueness. □

We are now in position to give the proof of Theorem 1.7. We point out that in the next proof we will highlight the dependence of $I^{λ}$ and $J^{λ, u}$ from a and b writing respectively $I_{a, b}^{λ}$ and $J_{a, b}^{λ, u}$ .

Proof of Theorem 1.7.

Recall the function $v_{ε}$ considered after the statement of Proposition 4.7 and select $ζ > 0$ . We have $\begin{array}{l} J_{a_{k}, b_{k}}^{λ, v_{ε}} (ζ) & = \frac{a_{k}}{p} ζ^{p} + \frac{b_{k}}{2 p} ζ^{2 p} - \frac{ζ^{p_{s}^{*}}}{p_{s}^{*}} ‖ v_{ε} ‖_{p_{s}^{*}}^{p_{s}^{*}} - λ \int_{Ω} G (x, ζ v_{ε} (| x |)) d x \\ ⩽ ζ^{p} f_{s, p}^{k} (ζ) - \frac{ζ^{p_{s}^{*}}}{p_{s}^{*}} O (ε^{\frac{N - p s}{p (p - 1)}}) - λ \int_{Ω} G (x, ζ v_{ε} (| x |)) d x, \end{array}$ where we denoted with $f_{s, p}^{k}$ the map $f_{s, p}$ emphasizing the dependence on the parameters $a_{k}$ , $b_{k}$ . We choose $ζ = m_{s, p}^{k}$ where we called $m_{s, p}^{k}$ the point in which $f_{s, p}^{k}$ attains its minimum, and since $m_{s, p}^{k} \to m_{s, p}$ as $k \to \infty$ , we get $\begin{matrix} (4.12) & lim_{k \to \infty} J_{a_{k}, b_{k}}^{λ, v_{ε}} (m_{s, p}^{k}) ⩽ - \frac{m_{s, p}^{p_{s}^{*}}}{p_{s}^{*}} O (ε^{\frac{N - p s}{p (p - 1)}}) - λ \int_{Ω} G (x, m_{s, p} v_{ε} (| x |)) d x . \end{matrix}$ At this point, as we did in Proposition 4.7, we estimate $\begin{matrix} \int_{Ω} G (x, m_{s, p} v_{ε} (| x |)) d x ⩾ C_{1} ε^{\frac{N}{p^{2}}}, \end{matrix}$ and from (4.12) we get $\begin{matrix} lim_{k \to \infty} J_{a_{k}, b_{k}}^{λ, v_{ε}} (m_{s, p}^{k}) ⩽ J^{λ, v_{ε}} (m_{s, p}) ⩽ ε^{\frac{N}{p^{2}}} (- \frac{m_{s, p}^{p_{s}^{*}}}{p_{s}^{*}} O (ε^{\frac{N - p^{2} s}{p^{2} (p - 1)}}) - λ C_{1}) < 0 . \end{matrix}$ Thus, choosing k big enough and a small ε $\begin{matrix} J_{a_{k}, b_{k}}^{λ, v_{ε}} (m_{s, p}^{k}) < 0 . \end{matrix}$ Hence, from Corollary 4.6 $λ_{k} ⩽ λ_{0}^{s} (v_{ε}) (a_{k}, b_{k}) < λ$ . Now, we point out that no restrictions were made on λ so we are free to let $λ \to 0$ and deduce that $λ_{k} \to 0$ as $k \to \infty$ . In order to prove the remaining part of the statement, we recall that in Proposition 4.7 we proved that the map $u \to λ_{0}^{s} (u)$ is homogeneous degree zero. As a consequence of that, it is not restrictive to suppose $‖ u_{k} ‖ = 1$ and $u_{k} ⇀ u$ . Arguing as for (4.8), it is possible to find $ζ_{k} > 0$ such that $\begin{matrix} (4.13) & \frac{a_{k}}{p} + \frac{b_{k}}{2 p} ζ_{k}^{p} - \frac{ζ_{k}^{p_{s}^{*} - p}}{p_{s}^{*}} ‖ u_{k} ‖_{p_{s}^{*}}^{p_{s}^{*}} - λ_{k} \int_{Ω} \frac{G (x, ζ_{k} u_{k})}{ζ_{k}^{p}} d x = 0 . \end{matrix}$ Furthermore, combining and (4.9) and (4.13),we can deduce the boundedness of ${(ζ_{k})}_{k}$ and suppose up to a subsequence that $ζ_{k} \to \overline{ζ} > 0$ and that $‖ u_{k} ‖_{p_{s}^{*}}^{p^{*} s} \to γ$ as $k \to \infty$ . Hence, passing to the limit in (4.13) we obtain $\begin{matrix} \frac{a}{p} + \frac{b}{2 p} {\overline{ζ}}^{p} - \frac{1}{p_{s}^{*}} γ {\overline{ζ}}^{p_{s}^{*} - p} = 0 . \end{matrix}$ From $a^{(N - 2 p s) / p s} b = L (N, p, s)$ it follows $γ = S_{p, s}$ implying that ${(u_{k})}_{k}$ is a minimizing sequence for the optimal Sobolev constant. Finally we can see $u = 0$ . In fact, if we assume $u \neq 0$ we have that $‖ u ‖ ⩽ 1$ exploiting the sequentially lower semicontinuity of the norm. From this, Lemma 2.3[ $(1)$ ] and Remark 4.1, we obtain $\begin{array}{l} 0 & ⩽ \frac{a}{p} + \frac{b}{2 p} {\overline{ζ}}^{p} - \frac{S_{p, s}^{- \frac{p_{s}^{*}}{p}}}{p_{s}^{*}} {\overline{ζ}}^{p_{s}^{*} - p} ‖ u ‖^{p_{s}^{*}} ⩽ \frac{a}{p} + \frac{b}{2 p} {\overline{ζ}}^{p} - \frac{{\overline{ζ}}^{p_{s}^{*} - p}}{p_{s}^{*}} ‖ u ‖_{p_{s}^{*}}^{p_{s}^{*}} \\ ⩽ \underset{k \to \infty}{lim inf} (\frac{a_{k}}{p} + \frac{b_{k}}{2 p} ζ_{k}^{p} - \frac{ζ_{k}^{p_{s}^{*} - p}}{p_{s}^{*}} ‖ u_{k} ‖_{p_{s}^{*}}^{p_{s}^{*}} - λ_{k} \int_{Ω} \frac{G (x, ζ_{k} u_{k})}{ζ_{k}^{p}} d x) = 0 . \end{array}$ So, u is a minimizer for $S_{s, p}$ but this is in not admissible since $u = 0$ in $R^{N} ∖ Ω$ as shown in [10, Theorem 1.1]. □

At this point, we start giving the proofs regarding the existence of mountain pass solutions. Namely we are going to prove Theorem 1.8.

Proof of Theorem 1.8.

Fix $ε > 0$ . Recalling (3.18) and that $X_{0}^{s, p} (Ω) ↪ L^{υ} (Ω)$ continuously for any $υ \in [1, p_{s}^{*}]$ we obtain $\begin{matrix} (4.14) & I^{λ} (u) ⩾ (\frac{a}{p} - λ C ε) ‖ u ‖^{p} + \frac{b}{2 p} ‖ u ‖^{2 p} - C ‖ u ‖^{p_{s}^{*}} - λ C ‖ u ‖^{q} \end{matrix}$ selecting $C > 0$ appropriately. At this point, to conclude the proof it suffices to argue as in [8, Theorem 5]. □

After analyzing the situation to the case $λ ⩾ {\overline{λ}}_{0}^{s}$ we focus to the case $λ ⩽ {\overline{λ}}_{0}^{s}$ . In particular we are interested in looking for local minimizer or mountain pass critical point of $I^{λ}$ .

Proposition 4.10.

If $λ ⩽ {\overline{λ}}_{0}^{s}$ then it is possible to find $r = r (s)$ , $M = M (s) > 0$ such that $\begin{matrix} (4.15) & inf {I^{λ} (u) : u \in X_{0}^{s, p} (Ω), ‖ u ‖ = r} ⩾ M . \end{matrix}$

Proof.

Fix $ε > 0$ . From (4.14) and $λ ⩽ {\overline{λ}}_{0}^{s}$ it follows $\begin{matrix} I^{λ} (u) ⩾ (\frac{a}{p} - {\overline{λ}}_{0}^{s} C ε) ‖ u ‖^{p} + \frac{b}{2 p} ‖ u ‖^{2 p} - C ‖ u ‖^{p_{s}^{*}} - {\overline{λ}}_{0}^{s} C ‖ u ‖^{q} \end{matrix}$ for any $u \in X_{0}^{s, p} (Ω)$ . Choosing ε such that $a / p - {\overline{λ}}_{0}^{s} C ε > 0$ we obtain the assertion. □

After showed the validity of the previous proposition we can finally prove the remaining two theorems.

Proof of Theorem 1.9.

Consider the number r given by Proposition 4.10, and argue as in [8, Theorem 6]. □

Remark 4.11.

It is immediate to see that ${\hat{ι}}_{0}^{s} \to 0$ as $λ \to {\overline{λ}}_{0}^{s}$ . In fact, take a function $u \in X_{0}^{s, p} (Ω)$ such that ${\overline{λ}}_{0}^{s} = λ_{0}^{s} (u)$ whose existence was shown in Theorem 1.6) and notice that $\begin{matrix} 0 ⩽ {\hat{ι}}_{λ}^{s} ⩽ I^{λ} (u) \to 0 as λ \to {\overline{λ}}_{0}^{s} . \end{matrix}$

Remark 4.12.

The function $w_{λ}^{s}$ obtained in the previous theorem is a critical point for the functional $I^{λ}$ , and more precisely it is a local minimizer.

Proof of Theorem 1.10.

Observe that $min {I^{λ} (0), I^{λ} (w_{λ}^{s})} < M$ , recall $‖ w_{λ}^{s} ‖ > M$ and (4.15). Hence, we have a mountain pass geometry. Furthermore, the Palais–Smale condition holds as showed in Lemma 4.3. At this point, in order to conclude, it suffices to apply the Mountain Pass Theorem (see [2]). □

Footnotes

Acknowledgements

The first and third author are supported by GNAMPA project “Equazioni alle derivate parziali: problemi e modelli”. A. Fiscella realized the manuscript within the auspices of the INdAM-GNAMPA project titled “Equazioni differenziali alle derivate parziali in fenomeni non lineari” (CUP_E55F22000270001) and of the FAPESP Thematic Project titled “Systems and partial differential equations” (2019/02512-5).

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