A fractional Kirchhoff problem involving a singular term and a critical nonlinearity

TL;DR

This paper studies a complex fractional Kirchhoff problem with a singular term and critical nonlinearity, establishing the existence of two solutions using variational methods and truncation techniques.

Contribution

It introduces a novel approach to handle the degenerate Kirchhoff case with critical nonlinearity and singular terms, proving existence of solutions.

Findings

Existence of two solutions for the problem.

Effective handling of the degenerate Kirchhoff case.

Application of variational methods with truncation.

Abstract

In this paper we consider the following critical nonlocal problem $$ \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-\Delta)^s u = \displaystyle\frac{\lambda}{u^\gamma}+u^{2^*_s-1}&\quad\mbox{in } \Omega,\\ u>0&\quad\mbox{in } \Omega,\\ u=0&\quad\mbox{in } \mathbb{R}^N\setminus\Omega, \end{array}\right. $$ where $\Omega$ is an open bounded subset of $\mathbb R^N$ with continuous boundary, dimension $N>2s$ with parameter $s\in (0,1)$, $2^*_s=2N/(N-2s)$ is the fractional critical Sobolev exponent, $\lambda>0$ is a real parameter, exponent $\gamma\in(0,1)$, $M$ models a Kirchhoff type coefficient, while $(-\Delta)^s$ is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is when the Kirchhoff function $M$ is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.

Equations

$$\left\{\begin{array}[]{ll}M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy\right)(-\Delta)^{s}u=\displaystyle\frac{\lambda}{u^{\gamma}}+u^{2^{*}_{s}-1}&\quad\mbox{in }\Omega,\\ u>0&\quad\mbox{in }\Omega,\\ u=0&\quad\mbox{in }\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$$

$$\left\{\begin{array}[]{ll}\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy\right)^{\theta-1}(-\Delta)^{s}u=\displaystyle\frac{\lambda}{u^{\gamma}}+u^{2^{*}_{s}-1}&\quad\mbox{in }\Omega,\\ u>0&\quad\mbox{in }\Omega,\\ u=0&\quad\mbox{in }\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$$

$$(-\Delta {)}^s\phi (x)={\int }_{{\mathbb{R}}^N}\frac{2\phi (x)-\phi (x+y)-\phi (x-y)}{|y{|}^{N+2s}}dy,x\in {\mathbb{R}}^N,$$

$$\left\{\begin{array}[]{ll}-M\left(\int_{\Omega}|\nabla u(x)|^{2}dx\right)\Delta u=\displaystyle\frac{\lambda}{u^{\gamma}}+u^{2^{*}-1}&\quad\mbox{in }\Omega,\\ u>0&\quad\mbox{in }\Omega,\\ u=0&\quad\mbox{in }\partial\Omega,\end{array}\right.$$

$$M(t)=a+b{t}^{\theta -1},a,b\ge 0,a+b>0,\theta \ge 1.$$

$$\left\|u\right\|_{H^{s}(\mathbb{R}^{N})}=\left\|u\right\|_{L^{2}(\mathbb{R}^{N})}+\Big{(}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{2}}{\left|x-y\right|^{N+2s}}dxdy\Big{)}^{1/2}.$$

$$X_{0}=\big{\{}u\in H^{s}(\mathbb{R}^{N}):\,\,u=0\mbox{ a.e. in }\mathbb{R}^{N}\setminus\Omega\big{\}}.$$

$$\left\|u\right\|_{X_{0}}=\Big{(}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{2}}{\left|x-y\right|^{N+2s}}dxdy\Big{)}^{1/2},$$

$${⟨u,v⟩}_{X_0}={\iint }_{{\mathbb{R}}^{2N}}\frac{(u(x)-u(y))(v(x)-v(y))}{{\left|x-y\right|}^{N+2s}}dxdy.$$

$$\left\{\begin{array}[]{ll}\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy\right)^{\theta-1}(-\Delta)^{s}u=\displaystyle\frac{\lambda}{(u^{+})^{\gamma}}+(u^{+})^{2^{*}_{s}-1}&\quad\mbox{in }\Omega,\\ u=0&\quad\mbox{in }\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$$

$$\Vert u{\Vert }^{2(\theta -1)}⟨u,\phi ⟩=\lambda {\int }_{\Omega }\frac{\phi }{(u^{+}{)}^{\gamma }}dx+{\int }_{\Omega }(u^{+}{)}^{2_s^{\ast }-1}\phi ,$$

$$J_{\lambda }(u)=\frac{1}{2\theta }\Vert u{\Vert }^{2\theta }-\frac{\lambda }{1-\gamma }{\int }_{\Omega }(u^{+}{)}^{1-\gamma }dx-\frac{1}{2_s^{\ast }}\Vert u^{+}{\Vert }_{2_s^{\ast }}^{2_s^{\ast }},$$

$$\left\{\begin{array}[]{ll}\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy\right)^{\theta-1}(-\Delta)^{s}u=\displaystyle\frac{\lambda}{(u^{+}+1/n)^{\gamma}}+(u^{+})^{2^{*}_{s}-1}\quad\mbox{in }\Omega,\\ u=0\quad\mbox{in }\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$$

$$\Vert u{\Vert }^{2(\theta -1)}⟨u,\phi ⟩=\lambda {\int }_{\Omega }\frac{\phi }{(u^{+}+1/n{)}^{\gamma }}dx+{\int }_{\Omega }(u^{+}{)}^{2_s^{\ast }-1}\phi ,$$

$$J_{n,\lambda }(u)=\frac{1}{2\theta }\Vert u{\Vert }^{2\theta }-\frac{\lambda }{1-\gamma }{\int }_{\Omega }[(u^{+}+1/n{)}^{1-\gamma }-(1/n{)}^{1-\gamma }]dx-\frac{1}{2_s^{\ast }}\Vert u^{+}{\Vert }_{2_s^{\ast }}^{2_s^{\ast }}.$$

$$S=\underset{\begin{array}{c}v\in H^s({\mathbb{R}}^N)\\ v\not\equiv 0\end{array}}{\inf }\frac{{\iint }_{{\mathbb{R}}^{2N}}\frac{|v(x)-v(y){|}^2}{|x-y{|}^{N+2s}}dxdy}{{\left({\int }_{{\mathbb{R}}^N}|v(x){|}^{2_s^{\ast }}dx\right)}^{2/2_s^{\ast }}},$$

$$m_{\lambda }=\inf \left\{J_{\lambda }(u):u\in {\overline{B}}_{\rho }\right\},$$

$${\int }_{\Omega }(u^{+}{)}^{1-\gamma }dx\le |\Omega {|}^{\frac{2_s^{\ast }-1+\gamma }{2_s^{\ast }}}\Vert u{\Vert }_{2_s^{\ast }}^{1-\gamma }\le |\Omega {|}^{\frac{2_s^{\ast }-1+\gamma }{2_s^{\ast }}}{S}^{-\frac{1-\gamma }{2}}\Vert u{\Vert }^{1-\gamma }.$$

$$J_{\lambda }(u)\ge \frac{1}{2\theta }\Vert u{\Vert }^{2\theta }-\frac{S^{-\frac{2_s^{\ast }}{2}}}{2_s^{\ast }}\Vert u{\Vert }^{2_s^{\ast }}-\frac{\lambda }{1-\gamma }|\Omega {|}^{\frac{2_s^{\ast }-1+\gamma }{2_s^{\ast }}}{S}^{-\frac{1-\gamma }{2}}\Vert u{\Vert }^{1-\gamma }.$$

$$\eta (t)=\frac{1}{2\theta }{t}^{2\theta -1+\gamma }-\frac{S^{-\frac{2_s^{\ast }}{2}}}{2_s^{\ast }}{t}^{2_s^{\ast }-1+\gamma },t\in [0,1]$$

$${\lambda }_0=\frac{(1-\gamma )S^{\frac{1-\gamma }{2}}}{2|\Omega {|}^{\frac{2_s^{\ast }-1+\gamma }{2_s^{\ast }}}}\eta (\rho ),$$

$$J_{\lambda }(tv)=\frac{t^{2\theta }}{2\theta }\Vert v{\Vert }^{2\theta }-t^{1-\gamma }\frac{\lambda }{1-\gamma }{\int }_{\Omega }(v^{+}{)}^{1-\gamma }dx-\frac{t^{2_s^{\ast }}}{2_s^{\ast }}\Vert v^{+}{\Vert }_{2_s^{\ast }}^{2_s^{\ast }}<0,$$

$$\underset{k\to \infty }{\lim }{J}_{\lambda }(u_k)=m_{\lambda }.$$

$$\begin{array}[]{ll}u_{k}\rightharpoonup u_{0}\text{ in }X_{0},&u_{k}\rightharpoonup u_{0}\text{ in }L^{2^{*}_{s}}(\Omega),\\ u_{k}\to u_{0}\text{ in }L^{p}(\Omega)\mbox{ for any }p\in[1,2^{*}_{s}),&u_{k}\to u_{0}\text{ a.e. in }\Omega.\end{array}$$

$$\left|{\int }_{\Omega }(u_k^{+}{)}^{1-\gamma }dx-{\int }_{\Omega }(u_0^{+}{)}^{1-\gamma }dx\right|\le {\int }_{\Omega }{\left|u_k^{+}-u_0^{+}\right|}^{1-\gamma }dx\le \Vert u_k^{+}-u_0^{+}{\Vert }_2^{1-\gamma }|\Omega {|}^{\frac{1+\gamma }{2}},$$

$$\underset{k\to \infty }{\lim }{\int }_{\Omega }(u_k^{+}{)}^{1-\gamma }dx={\int }_{\Omega }(u_0^{+}{)}^{1-\gamma }dx.$$

$$\Vert u_k{\Vert }^2=\Vert w_k{\Vert }^2+\Vert u_0{\Vert }^2+o(1),\Vert u_k{\Vert }_{2_s^{\ast }}^{2_s^{\ast }}=\Vert w_k{\Vert }_{2_s^{\ast }}^{2_s^{\ast }}+\Vert u_0{\Vert }_{2_s^{\ast }}^{2_s^{\ast }}+o(1)$$

$$\frac{1}{2\theta }\Vert u{\Vert }^{2\theta }-\frac{1}{2_s^{\ast }}\Vert u^{+}{\Vert }_{2_s^{\ast }}^{2_s^{\ast }}\ge \alpha >0,$$

$$\frac{1}{2\theta }\Vert w_k{\Vert }^{2\theta }-\frac{1}{2_s^{\ast }}\Vert w_k^{+}{\Vert }_{2_s^{\ast }}^{2_s^{\ast }}>0.$$

$$m_{\lambda }$$

Full text

[TABLE]

where $\Omega$ is an open bounded subset of $\mathbb{R}^{N}$ with continuous boundary, dimension $N>2s$ with parameter $s\in(0,1)$, $2^{*}_{s}=2N/(N-2s)$ is the fractional critical Sobolev exponent, $\lambda>0$ is a real parameter, exponent $\gamma\in(0,1)$, $M$ models a Kirchhoff type coefficient, while $(-\Delta)^{s}$ is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is when the Kirchhoff function $M$ is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.

aa 2010 AMS Subject Classification. Primary: 35J75, 35R11, 49J35; Secondary: 35A15, 45G05, 35S15.

1. Introduction

This paper is devoted to the study of a class of Kirchhoff type problems driven by a nonlocal fractional operator and involving a singular term and a critical nonlinearity. More precisely, we consider

[TABLE]

where $\Omega$ is an open bounded subset of $\mathbb{R}^{N}$ with continuous boundary, dimension $N>2s$ with parameter $s\in(0,1)$, $2^{*}_{s}=2N/(N-2s)$ is the fractional critical Sobolev exponent, $\lambda>0$ is a real parameter, exponent $\theta\in(1,2^{*}_{s}/2)$ while $\gamma\in(0,1)$. Here $(-\Delta)^{s}$ is the fractional Laplace operator defined, up to normalization factors, by the Riesz potential as

[TABLE]

along any $\varphi\in C^{\infty}_{0}(\Omega)$; we refer to [11] and the recent monograph [22] for further details on the fractional Laplacian and the fractional Sobolev spaces $H^{s}(\mathbb{R}^{N})$ and $H^{s}_{0}(\Omega)$.

As well explained in [11], problem (1.1) is the fractional version of the following nonlinear problem

[TABLE]

where $\Delta$ denotes the classical Laplace operator while, just for a general discussion, $M(t)=t^{\theta-1}$ for any $t\in\mathbb{R}^{+}_{0}$. In literature, problems like (1.1) and (1.2) are called of Kirchhoff type whenever the function $M:\mathbb{R}^{+}_{0}\to\mathbb{R}^{+}_{0}$ models the Kirchhoff prototype, given by

[TABLE]

In particular, when $M(t)\geq\mbox{{ constant}}>0$ for any $t\in\mathbb{R}^{+}_{0}$, Kirchhoff problems are said to be non–degenerate and this happens for example if $a>0$ in the model case (1.3). While, if $M(0)=0$ but $M(t)>0$ for any $t\in\mathbb{R}^{+}$, Kirchhoff problems are called degenerate. Of course, for (1.3) this occurs when $a=0$.

This kind of nonlocal problems has been widely studied in recent years. We refer to [17, 18, 19, 20, 21] for different Kirchhoff problems with $M$ like in (1.3), driven by the Laplace operator and involving a singular term of type $u^{-\gamma}$. In [20], the authors study a Kirchhoff problem with a singular term and a Hardy potential, by using the Nehari method. The same approach is used in [19] for a singular Kirchhoff problem with also a subcritical term. In [17], strongly assuming $a>0$ in (1.3), the authors prove the existence of two solutions for a Kirchhoff problem like (1.2), by combining perturbation and variational methods. While in [18], they provide a uniqueness result for a singular Kirchhoff problem involving a negative critical nonlinearity, by a minimization argument. By arguing similarly to [17], the authors of [21] give the existence of two solutions for a critical Kirchhoff problem with a singular term of type $|x|^{-\beta}u^{-\gamma}$.

Problem (1.1) has been studied in [3] when $\theta=1$, namely without a Kirchhoff coefficient. In [3], they prove the existence of two solutions by applying the sub/supersolutions and Sattinger methods. In [8], the authors generalize the results of Section 3 of [3] to the delicate case of the $p$–fractional Laplace operator $(-\Delta)^{s}_{p}$. While in the last section of [1], the authors provide the existence of a solution for nonlinear fractional problems with a singularity like $u^{-\gamma}$ and a fractional Hardy term, by perturbation methods. Concerning fractional Kirchhoff problems involving critical nonlinearities, we refer to [2, 9, 13, 14, 16, 23] for existence results and to [5, 12, 24, 25, 29] for multiplicity results. In particular, in [9, 13, 14, 23] different singular terms appear, but given by the fractional Hardy potential.

Inspired by the above works, we study a multiplicity result for problem (1.1). As far as we know, a fractional Kirchhoff problem involving a singular term of type $u^{-\gamma}$ has not been studied yet. Thus, we can state our result as follows.

Theorem 1.1**.**

Let $s\in(0,1)$, $N>2s$, $\theta\in(1,2^{*}_{s}/2)$, $\gamma\in(0,1)$ and let $\Omega$ be an open bounded subset of $\mathbb{R}^{N}$ with $\partial\Omega$ continuous. Then, there exists $\overline{\lambda}>0$ such that for any $\lambda\in(0,\overline{\lambda})$ problem (1.1) has at least two different solutions.

The first solution of problem (1.1) is obtained by a suitable minimization argument, where we must pay attention to the nonlocal nature of the fractional Laplacian. Concerning the second solution, because of the presence of $u^{-\gamma}$, we can not apply the usual critical point theory to problem (1.1). For this, we first study a perturbed problem obtained truncating the singular term $u^{-\gamma}$. Then, by approximation we get our second solution of (1.1).

Finally, we observe that Theorem 1.1 generalizes in several directions the first part of [3, Theorem 4.1] and [17, Theorem 1.1].

The paper is organized as follows. In Section 2, we discuss the variational formulation of problem (1.1) and we introduce the perturbed problem. In Section 3, we prove the existence of the first solution of (1.1) and we give a possible generalization of this existence result, at the end of the section. In Section 4, we prove the existence of a mountain pass solution for the perturbed problem. In Section 5, we prove Theorem 1.1.

2. Variational setting

Throughout the paper we assume that $s\in(0,1)$, $N>2s$, $\theta\in(1,2^{*}_{s}/2)$, $\gamma\in(0,1)$ and $\Omega$ is an open bounded subset of $\mathbb{R}^{N}$ with $\partial\Omega$ continuous, without further mentioning. As a matter of notations, we denote with $\varphi^{+}=\max\{\varphi,0\}$ and $\varphi^{-}=\max\{-\varphi,0\}$ respectively the positive and negative part of a function $\varphi$.

Problem (1.1) has a variational structure and the natural space where finding solutions is the homogeneous fractional Sobolev space $H^{s}_{0}(\Omega)$. In order to study (1.1), it is important to encode the “boundary condition” $u=0$ in $\mathbb{R}^{N}\setminus\Omega$ in the weak formulation, by considering also that the interaction between $\Omega$ and its complementary in $\mathbb{R}^{N}$ gives a positive contribution in the so called Gagliardo norm, given as

[TABLE]

The functional space that takes into account this boundary condition will be denoted by $X_{0}$ and it is defined as

[TABLE]

We refer to [26] for a general definition of $X_{0}$ and its properties. We also would like to point out that, when $\partial\Omega$ is continuous, by [15, Theorem 6] the space $C^{\infty}_{0}(\Omega)$ is dense in $X_{0}$, with respect to the norm (2.1). This last point will be used to overcome the singularity in problem (1.1).

In $X_{0}$ we can consider the following norm

[TABLE]

which is equivalent to the usual one defined in (2.1) (see [26, Lemma 6]). We also recall that $(X_{0},\left\|\,\cdot\,\right\|_{X_{0}})$ is a Hilbert space, with the scalar product defined as

[TABLE]

From now on, in order to simplify the notation, we will denote $\|\cdot\|_{X_{0}}$ and $\left\langle\cdot,\cdot\right\rangle_{X_{0}}$ by $\|\cdot\|$ and $\left\langle\cdot,\cdot\right\rangle$ respectively, and $\|\cdot\|_{L^{q}(\Omega)}$ by $\|\cdot\|_{q}$ for any $q\in[1,\infty]$.

In order to present the weak formulation of (1.1) and taking into account that we are looking for positive solutions, we will consider the following Kirchhoff problem

[TABLE]

We say that $u\in X_{0}$ is a (weak) solution of problem (2.2), if $u$ satisfies

[TABLE]

for any $\varphi\in X_{0}$. Problem (2.2) has a variational structure and $J_{\lambda}:X_{0}\to\mathbb{R}$, defined by

[TABLE]

is the underlying functional associated to (2.2). Because of the presence of a singular term in (2.2), functional $J_{\lambda}$ is not differentiable on $X_{0}$. Therefore, we can not apply directly the usual critical point theory to $J_{\lambda}$, in order to solve problem (2.2). However, it is possible to find a first solution of (2.2) by using a local minimization argument. In order to get the second solution of (2.2), we have to study an associated approximating problem. That is, for any $n\in\mathbb{N}$, we consider the following perturbed problem

[TABLE]

For this, we say that $u\in X_{0}$ is a (weak) solution of problem (2.4), if $u$ satisfies

[TABLE]

for any $\varphi\in X_{0}$. In this case, solutions of (2.4) correspond to the critical points of functional $J_{n,\lambda}:X_{0}\to\mathbb{R}$, set as

[TABLE]

It is immediate to see that $J_{n,\lambda}$ is of class $C^{1}(X_{0})$.

We conclude this section recalling the best constant of the fractional Sobolev embedding, which will be very useful to study the compactness property of functional $J_{n,\lambda}$. That is, we consider

[TABLE]

which is well defined and strictly positive, as shown in [10, Theorem 1.1].

3. A first solution for problem (1.1)

In this section we prove the existence of a solution for problem (1.1) by a local minimization argument. For this, we first study the geometry of functional $J_{\lambda}$.

Lemma 3.1**.**

There exist numbers $\rho\in(0,1]$, $\lambda_{0}=\lambda_{0}(\rho)>0$ and $\alpha=\alpha(\rho)>0$ such that $J_{\lambda}(u)\geq\alpha$ for any $u\in X_{0}$, with $\|u\|=\rho$, and for any $\lambda\in(0,\lambda_{0}]$.

Furthermore, set

[TABLE]

where $\overline{B}_{\rho}=\left\{u\in X_{0}:\,\,\|u\|\leq\rho\right\}$. Then, $m_{\lambda}<0$ for any $\lambda\in(0,\lambda_{0}]$.

Proof.

Let $\lambda>0$. From the Hölder inequality and (2.7), for any $u\in X_{0}$ we have

[TABLE]

Hence, by using again (2.7) and (3.1) we get

[TABLE]

Since $1-\gamma<1<2\theta<2^{*}_{s}$, the function

[TABLE]

admits a maximum at some $\rho\in(0,1]$ small enough, that is $\displaystyle\max_{t\in[0,1]}\eta(t)=\eta(\rho)>0$. Thus, let

[TABLE]

then for any $u\in X_{0}$ with $\|u\|=\rho\leq 1$ and for any $\lambda\leq\lambda_{0}$, we get $J_{\lambda}(u)\geq\rho^{1-\gamma}\eta(\rho)/2=\alpha>0$.

Furthermore, fixed $v\in X_{0}$ with $v^{+}\not\equiv 0$, for $t\in(0,1)$ sufficiently small

[TABLE]

being $1-\gamma<1<2\theta<2^{*}_{s}$. This concludes the proof. ∎

We are now ready to prove the existence of the first solution of (1.1).

Theorem 3.2**.**

Let $\lambda_{0}$ be given as in Lemma 3.1. Then, for any $\lambda\in(0,\lambda_{0}]$ problem (1.1) has a solution $u_{0}\in X_{0}$, with $J_{\lambda}(u_{0})<0$.

Proof.

Fix $\lambda\in(0,\lambda_{0}]$ and let $\rho$ be as given in Lemma 3.1. We first prove there exists $u_{0}\in\overline{B}_{\rho}$ such that $J_{\lambda}(u_{0})=m_{\lambda}<0$. Let $\{u_{k}\}_{k}\subset\overline{B}_{\rho}$ be a minimizing sequence for $m_{\lambda}$, that is such that

[TABLE]

Since $\{u_{k}\}_{k}$ is bounded in $X_{0}$, by applying [26, Lemma 8] and [6, Theorem 4.9], there exist a subsequence, still denoted by $\{u_{k}\}_{k}$, and a function $u_{0}\in\overline{B}_{\rho}$ such that, as $k\to\infty$ we have

[TABLE]

Since $\gamma\in(0,1)$, by the Hölder inequality, for any $k\in\mathbb{N}$ we have

[TABLE]

which yields, by (3.3)

[TABLE]

Let $w_{k}=u_{k}-u_{0}$, by [7, Theorem 2] it holds true that

[TABLE]

as $k\to\infty$. Since $\{u_{k}\}_{k}\subset\overline{B}_{\rho}$, by (3.5) for $k$ sufficiently large, we have $w_{k}\in\overline{B}_{\rho}$. Lemma 3.1 implies that for any $u\in X_{0}$, with $\|u\|=\rho$, we get

[TABLE]

and from this, being $\rho\leq 1$, for $k$ sufficiently large we have

[TABLE]

Thus, by (3.2), (3.4)–(3.6) and considering $\theta\geq 1$, it follows that as $k\to\infty$

[TABLE]

being $u_{0}\in\overline{B}_{\rho}$. Hence, $u_{0}$ is a local minimizer for $J_{\lambda}$, with $J_{\lambda}(u_{0})=m_{\lambda}<0$ which implies that $u_{0}$ is nontrivial.

Now, we prove that $u_{0}$ is a positive solution of (2.2). For any $\psi\in X_{0}$, with $\psi\geq 0$ a.e. in $\mathbb{R}^{N}$, let us consider a $t>0$ sufficiently small so that $u_{0}+t\psi\in\overline{B}_{\rho}$. Since $u_{0}$ is a local minimizer for $J_{\lambda}$, we have

[TABLE]

From this, dividing by $t>0$ and passing to the limit as $t\to 0^{+}$, it follows that

[TABLE]

We observe that

[TABLE]

with $\xi\in(0,1)$ and $((u_{0}+\xi t\psi)^{+})^{-\gamma}\to(u^{+}_{0})^{-\gamma}\psi$ a.e. in $\Omega$, as $t\to 0^{+}$. Thus, by the Fatou lemma, we obtain

[TABLE]

Therefore, combining (3.7) and (3.8) we get

[TABLE]

for any $\psi\in X_{0}$, with $\psi\geq 0$ a.e. in $\mathbb{R}^{N}$.

Since $J_{\lambda}(u_{0})<0$ and by Lemma 3.1, we have $u_{0}\in B_{\rho}$. Hence, there exists $\delta\in(0,1)$ such that $(1+t)u_{0}\in\overline{B}_{\rho}$ for any $t\in[-\delta,\delta]$. Let us define $I_{\lambda}(t)=J_{\lambda}((1+t)u_{0})$. Since $u_{0}$ is a local minimizer for $J_{\lambda}$ in $\overline{B}_{\rho}$, functional $I_{\lambda}$ has a minimum at $t=0$, that is

[TABLE]

For any $\varphi\in X_{0}$ and any $\varepsilon>0$, let us define $\psi_{\varepsilon}=u^{+}_{0}+\varepsilon\varphi$. Then, by (3.9) we have

[TABLE]

We observe that, for a.e. $x$, $y\in\mathbb{R}^{N}$, we obtain

[TABLE]

from which we immediately get

[TABLE]

From the last inequality, it follows that

[TABLE]

Hence, denoting with $\Omega_{\varepsilon}=\left\{x\in\mathbb{R}^{N}:\,\,u^{+}_{0}(x)+\varepsilon\varphi(x)\leq 0\right\}$ and by combining (3.11) with (3.13), we get

[TABLE]

where last equality follows from (3.10). Arguing similarly to (3.12), for a.e. $x$, $y\in\mathbb{R}^{N}$ we have

[TABLE]

Thus, denoting with

[TABLE]

by the symmetry of the fractional kernel and (3.15), we get

[TABLE]

where we set

[TABLE]

Clearly $\mathcal{U}\in L^{1}(\mathbb{R}^{2N})$, so that for any $\sigma>0$ there exists $R_{\sigma}$ sufficiently large such that

[TABLE]

Also, by definition of $\Omega_{\varepsilon}$, we have $\Omega_{\varepsilon}\subset\mbox{\small supp }\varphi$ and $|\Omega_{\varepsilon}\times B_{R_{\sigma}}|\to 0$ as $\varepsilon\to 0^{+}$. Thus, since $\mathcal{U}\in L^{1}(\mathbb{R}^{2N})$, there exists $\delta_{\sigma}>0$ and $\varepsilon_{\sigma}>0$ such that for any $\varepsilon\in(0,\varepsilon_{\sigma}]$

[TABLE]

Therefore, for any $\varepsilon\in(0,\varepsilon_{\sigma}]$

[TABLE]

from which we get

[TABLE]

Combining (3.14) with (3.16), dividing by $\varepsilon$, letting $\varepsilon\to 0^{+}$ and using (3.17), we obtain

[TABLE]

for any $\varphi\in X_{0}$. By the arbitrariness of $\varphi$, we prove that $u_{0}$ verifies (2.3), that is $u_{0}$ is a nontrivial solution of (2.2).

Finally, by considering $\varphi=u^{-}_{0}$ in (2.3) and using (3.12), we see that $\|u^{-}_{0}\|=0$, which implies that $u_{0}$ is nonnegative. Moreover, by the maximum principle in [28, Proposition 2.2.8], we can deduce that $u_{0}$ is a positive solution of (2.2) and so also solves problem (1.1). This concludes the proof. ∎

We end this section observing that the result in Theorem 3.2 can be extended to more general Kirchhoff problems. That is, we can consider the following problem

[TABLE]

where $p^{*}_{s}=pN/(N-ps)$, with here $N>ps$ and $p>1$, while the Kirchhoff coefficient $M$ satisfies condition

1. $(\mathcal{M})$

$M:\mathbb{R}^{+}_{0}\rightarrow\mathbb{R}^{+}_{0}$* is continuous and nondecreasing. There exist numbers $a>0$ and $\vartheta$ such that for any $t\in\mathbb{R}^{+}_{0}$*

[TABLE]

The main operator $(-\Delta)^{s}_{p}$ is the fractional $p$–Laplacian which may be defined, for any function $\varphi\in C^{\infty}_{0}(\Omega)$, as

[TABLE]

where $B_{\varepsilon}(x)=\{y\in\mathbb{R}^{N}:\,\,|x-y|<\varepsilon\}$. Then, arguing as in the proof of Theorem 3.2 and observing that we have not used yet the assumption that $\partial\Omega$ is continuous, we can prove the following result.

Theorem 3.3**.**

Let $s\in(0,1)$, $p>1$, $N>ps$, $\gamma\in(0,1)$ and let $\Omega$ be an open bounded subset of $\mathbb{R}^{N}$. Let $M$ satisfy $(\mathcal{M})$. Then, there exists $\lambda_{0}>0$ such that for any $\lambda\in(0,\lambda_{0}]$ problem (3.18) admits a solution.

4. A mountain pass solution for problem (2.4)

In this section we prove the existence of a positive solution for perturbed problem (2.4), by the mountain pass theorem. For this, throughout this section we assume $n\in\mathbb{N}$, without further mentioning. Now, we first prove that the related functional $J_{n,\lambda}$ satisfies all the geometric features required by the mountain pass theorem.

Lemma 4.1**.**

Let $\rho\in(0,1]$, $\lambda_{0}=\lambda_{0}(\rho)>0$ and $\alpha=\alpha(\rho)>0$ be given as in Lemma 3.1. Then, for any $\lambda\in(0,\lambda_{0}]$ and any $u\in X_{0}$, with $\|u\|\leq\rho$, functional $J_{n,\lambda}(u)\geq\alpha$.

Furthermore, there exists $e\in X_{0}$, with $\|e\|>\rho$, such that $J_{n,\lambda}(e)<0$.

Proof.

Since $\gamma\in(0,1)$, by the subadditivity of $t\mapsto t^{1-\gamma}$, we have

[TABLE]

for any $u\in X_{0}$ and any $n\in\mathbb{N}$. Thus, we have $J_{n,\lambda}(u)\geq J_{\lambda}(u)$ for any $u\in X_{0}$ and the first part of lemma directly follows by Lemma 3.1.

For any $v\in X_{0}$, with $v^{+}\not\equiv 0$, and $t>0$ we have

[TABLE]

being $1-\gamma<1<2\theta<2^{*}_{s}$. Hence, we can find $e\in X_{0}$, with $\|e\|>\rho$ sufficiently large, such that $J_{n,\lambda}(e)<0$. This concludes the proof. ∎

We discuss now the compactness property for the functional $J_{n,\lambda}$, given by the Palais–Smale condition. We recall that $\{u_{k}\}_{k}\subset X_{0}$ is a Palais–Smale sequence for $J_{n,\lambda}$ at level $c\in\mathbb{R}$ if

[TABLE]

We say that $J_{n,\lambda}$ satisfies the Palais–Smale condition at level $c$ if any Palais–Smale sequence $\{u_{k}\}_{k}$ at level $c$ admits a convergent subsequence in $X_{0}$.

Before proving this compactness condition, we introduce the following positive constants which will help us for a better explanation

[TABLE]

Lemma 4.2**.**

Let $\lambda>0$. Then, the functional $J_{n,\lambda}$ satisfies the Palais–Smale condition at any level $c\in\mathbb{R}$ verifying

[TABLE]

with $D_{1}$, $D_{2}>0$ given as in (4.3).

Proof.

Let $\lambda>0$ and let $\{u_{k}\}_{k}$ be a Palais–Smale sequence in $X_{0}$ at level $c\in\mathbb{R}$, with $c$ satisfying (4.4). We first prove the boundedness of $\{u_{k}\}_{k}$. By (4.2) there exists $\sigma>0$ such that, as $k\to\infty$

[TABLE]

where the last two inequalities follow by (2.7), (4.1) and the Hölder inequality. Therefore, $\{u_{k}\}_{k}$ is bounded in $X_{0}$, being $1-\gamma<1<2\theta$. Also, $\{u^{-}_{k}\}_{k}$ is bounded in $X_{0}$ and by (4.2) we have

[TABLE]

Thus, by inequality (3.12) we deduce that $\|u^{-}_{k}\|\to 0$ as $k\to\infty$, which yields

[TABLE]

Hence, we can suppose that $\{u_{k}\}_{k}$ is a sequence of nonnegative functions.

By the boundedness of $\{u_{k}\}_{k}$ and by using [26, Lemma 8] and [6, Theorem 4.9], there exist a subsequence, still denoted by $\{u_{k}\}_{k}$, and a function $u\in X_{0}$ such that

[TABLE]

as $k\to\infty$, with $h\in L^{p}(\Omega)$ for a fixed $p\in[1,2^{*}_{s})$. If $\mu=0$, then immediately $u_{k}\to 0$ in $X_{0}$ as $k\to\infty$. Hence, let us assume that $\mu>0$.

Since $n\in\mathbb{N}$, by (4.5) it follows that

[TABLE]

so by the dominated convergence theorem and (4.5), we have

[TABLE]

By (4.5) and [7, Theorem 2] we have

[TABLE]

as $k\to\infty$. Consequently, we deduce from (4.2), (4.5), (4.6) and (4.7) that, as $k\to\infty$

[TABLE]

Therefore, we have proved the crucial formula

[TABLE]

If $\ell=0$, since $\mu>0$, by (4.5) and (4.8) we have $u_{k}\to u$ in $X_{0}$ as $k\to\infty$, concluding the proof.

Thus, let us assume by contradiction that $\ell>0$. By (2.7), the notation in (4.5) and (4.8), we get

[TABLE]

Noting that (4.8) implies in particular that

[TABLE]

using (4.9), it follows that

[TABLE]

From this, we obtain

[TABLE]

and considering $N<2s\theta/(\theta-1)=2s\theta^{\prime}$, we have

[TABLE]

Indeed, the restriction $N/(2\theta^{\prime})<s$ follows directly from the fact that $1<\theta<2^{*}_{s}/2=N/(N-2s)$. By (4.1) and considering that $n\in\mathbb{N}$, for any $k\in\mathbb{N}$ we have

[TABLE]

From this, as $k\to\infty$, being $\theta\geq 1$, by (4.2), (4.5), (4.7), (4.10), the Hölder inequality and the Young inequality, we obtain

[TABLE]

which contradicts (4.4), since (4.3). This concludes the proof. ∎

We now give a control from above for functional $J_{n,\lambda}$, under a suitable situation. For this, we assume, without loss of generality, that $0\in\Omega$. By [10] we know that the infimum in (2.7) is attained at the function

[TABLE]

that is, it holds true that

[TABLE]

Let us fix $r>0$ such that $B_{4r}\subset\Omega$, where $B_{4r}=\{x\in\mathbb{R}^{N}:\,|x|<4r\}$, and let us introduce a cut–off function $\phi\in C^{\infty}(\mathbb{R}^{N},[0,1])$ such that

[TABLE]

For any $\varepsilon>0$, we set

[TABLE]

Then, we can prove the following result.

Lemma 4.3**.**

There exist $\psi\in X_{0}$ and $\lambda_{1}>0$ such that for any $\lambda\in(0,\lambda_{1})$

[TABLE]

with $D_{1}$, $D_{2}>0$ given as in (4.3).

Proof.

Let $\lambda$, $\varepsilon>0$. Let $u_{\varepsilon}$ and $\psi_{\varepsilon}$ be as respectively in (4.11) and in (4.13). By (2.6), we have $J_{n,\lambda}(t\psi_{\varepsilon})\to-\infty$ as $t\to\infty$, so that there exists $t_{\varepsilon}>0$ such that $J_{n,\lambda}(t_{\varepsilon}\psi_{\varepsilon})=\max_{t\geq 0}J_{n,\lambda}(t\psi_{\varepsilon})$. By Lemma 4.1 we know that $J_{n,\lambda}(t_{\varepsilon}\psi_{\varepsilon})\geq\alpha>0$. Hence, by continuity of $J_{n,\lambda}$, there exist two numbers $t_{0}$, $t^{*}>0$ such that $t_{0}\leq t_{\varepsilon}\leq t^{*}$.

Now, since $\|u_{\varepsilon}\|_{L^{2^{*}_{s}}(\mathbb{R}^{N})}$ is independent of $\varepsilon$, by [27, Proposition 21] we have

[TABLE]

from which, by the elementary inequality

[TABLE]

with $p=2\theta$, it follows that, as $\varepsilon\to 0^{+}$

[TABLE]

Hence, by the last inequality, (2.6) and being $t_{0}\leq t_{\varepsilon}\leq t^{*}$, for any $\varepsilon>0$ sufficiently small, we have

[TABLE]

with $C_{1}$ a suitable positive constant. We observe that

[TABLE]

Thus, by (4.14) it follows that

[TABLE]

Now, let us consider a positive number $q$ satisfying

[TABLE]

that is, being $2<2\theta<2^{*}_{s}$, $N>2s$ and $\gamma\in(0,1)$, such that

[TABLE]

By the elementary inequality

[TABLE]

with $p=2^{*}_{s}/2$, and considering $\varepsilon<r^{1/q}$ sufficiently small, with $r$ given by (4.12), we have

[TABLE]

with $\widetilde{C}$ and $C_{2}$ two positive constant independent by $\varepsilon$. By combining (4.15) with (4.17), we get

[TABLE]

Thus, let us consider $\lambda^{*}>0$ such that

[TABLE]

and let us set

[TABLE]

where $r$ and $q$ are given respectively in (4.12) and (4.16), while we still consider $D_{1}$ and $D_{2}$ as defined in (4.3). Then, by considering $\varepsilon=\lambda^{\nu_{1}/q}$ in (4.18), since (4.16) implies that $\nu_{3}<0$, for any $\lambda\in(0,\lambda_{1})$ we have

[TABLE]

which concludes the proof. ∎

We can now prove the existence result for (2.4) by applying the mountain pass theorem.

Theorem 4.4**.**

There exists $\overline{\lambda}>0$ such that, for any $\lambda\in(0,\overline{\lambda})$, problem (2.4) has a positive solution $v_{n}\in X_{0}$, with

[TABLE]

where $\alpha$, $D_{1}$ and $D_{2}$ are given respectively in Lemma 3.1 and (4.3).

Proof.

Let $\overline{\lambda}=\min\{\lambda_{0},\lambda_{1}\}$, with $\lambda_{0}$ and $\lambda_{1}$ given respectively in Lemmas 3.1 and 4.3. Let us consider $\lambda\in(0,\overline{\lambda})$. By Lemma 4.1, functional $J_{n,\lambda}$ verifies the mountain pass geometry. For this, we can set the critical mountain pass level as

[TABLE]

where

[TABLE]

By Lemmas 4.1 and 4.3, we get

[TABLE]

Hence, by Lemma 4.2 functional $J_{n,\lambda}$ satisfies the Palais–Smale condition at level $c_{n,\lambda}$. Thus, the mountain pass theorem gives the existence of a critical point $v_{n}\in X_{0}$ for $J_{n,\lambda}$ at level $c_{n,\lambda}$. Since $J_{n,\lambda}(v_{n})=c_{n,\lambda}>\alpha>0=J_{n,\lambda}(0)$, $v_{n}$ is a nontrivial solution of (2.4). Furthermore, by (2.5) with test function $\varphi=v^{-}_{n}$ and inequality (3.12), we can see that $\|v^{-}_{n}\|=0$, which implies $v_{n}$ is nonnegative. By the maximum principle in [28, Proposition 2.2.8], we have that $v_{n}$ is a positive solution of (2.4), concluding the proof. ∎

5. A second solution for problem (1.1)

In this last section we prove the existence of a second solution for problem (1.1), as a limit of solutions of the perturbed problem (2.4). For this, here we need the assumption that $\partial\Omega$ is continuous, in order to apply a density argument for space $X_{0}$.

Proof of Theorem 1.1.

Let us consider $\overline{\lambda}$ as given in Theorem 4.4 and let $\lambda\in(0,\overline{\lambda})$. Since $\overline{\lambda}\leq\lambda_{0}$, by Theorem 3.2 we know that problem (1.1) admits a solution $u_{0}$ with $J_{\lambda}(u_{0})<0$.

In order to find a second solution for (1.1), let $\{v_{n}\}_{n}$ be a family of positive solutions of (2.4). By (2.7), (4.1), (4.19) and the Hölder inequality, we have

[TABLE]

which yields that $\{v_{n}\}_{n}$ is bounded in $X_{0}$, being $1-\gamma<1<2\theta$. Hence, by using [26, Lemma 8] and [6, Theorem 4.9], there exist a subsequence, still denoted by $\{v_{n}\}_{n}$, and a function $v_{0}\in X_{0}$ such that

[TABLE]

We want to prove that $v_{n}\to v_{0}$ in $X_{0}$ as $n\to\infty$. When $\mu=0$, by (5.1) we have $v_{n}\to 0$ in $X_{0}$ as $n\to\infty$. For this, we suppose $\mu>0$. We observe that

[TABLE]

so by the Vitali convergence theorem and (5.1), it follows that

[TABLE]

By using (2.5) for $v_{n}$ and test function $\varphi=v_{n}$, by (5.1) and (5.2), as $n\to\infty$ we have

[TABLE]

For any $n\in\mathbb{N}$, by an immediate calculation in (2.4) we see that

[TABLE]

Thus, since $\{v_{n}\}_{n}$ is bounded in $X_{0}$ and by using a standard comparison argument (see [4, Lemma 2.1]) and the maximum principle in [28, Proposition 2.2.8], for any $\widetilde{\Omega}\subset\subset\Omega$, there exists a constant $c_{\widetilde{\Omega}}>0$ such that

[TABLE]

Now, let $\varphi\in C^{\infty}_{0}(\Omega)$, with $\mbox{\small supp }\varphi=\widetilde{\Omega}\subset\subset\Omega$. By (5.4) we have

[TABLE]

so that by (5.1) and the dominated convergence theorem

[TABLE]

Thus, by considering (2.5) for $v_{n}$, sending $n\to\infty$ and using (5.1) and (5.5), for any $\varphi\in C^{\infty}_{0}(\Omega)$ it follows that

[TABLE]

However, since $\partial\Omega$ is continuous, by [15, Theorem 6] the space $C^{\infty}_{0}(\Omega)$ is dense in $X_{0}$. Thus, by a strandard density argument, (5.6) holds true for any $\varphi\in X_{0}$. By combining (5.3) and (5.6) with test function $\varphi=v_{0}$, as $n\to\infty$ we get

[TABLE]

and by (5.1) and [7, Theorem 2] we have

[TABLE]

If $\ell=0$, then $v_{n}\to v_{0}$ in $X_{0}$ as $n\to\infty$, since $\mu>0$.

Let us suppose $\ell>0$ by contradiction. Arguing as in Lemma 4.2, by (5.1) and (5.7) we get (4.10). Therefore, being $\theta\geq 1$, by (4.1), (4.10), (4.19), (5.1), the Hölder inequality and the Young inequality, we have

[TABLE]

which is the desired contradiction, thanks to (4.3).

Therefore, $v_{n}\to v_{0}$ in $X_{0}$ as $n\to\infty$ and by (2.3) and (2.5) we immediately see that $v_{0}$ is a solution of problem (2.2). Furthermore, by (4.19) we have $J_{\lambda}(v_{0})\geq\alpha>0$, which also implies that $v_{0}$ is nontrivial. Reasoning as at the end of the proof of Theorem 4.4, we conclude that $v_{0}$ is a positive solution of (2.2) and so $v_{0}$ also solves problem (1.1). Finally, $v_{0}$ is different from $u_{0}$, since $J_{\lambda}(v_{0})>0>J_{\lambda}(u_{0})$. ∎

Acknowledgments

The author is supported by Coordenação de Aperfeiçonamento de pessoal de nível superior through the fellowship PNPD–CAPES 33003017003P5. The author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica “G. Severi” (INdAM).