In this section we prove the existence of solutions to problem (1.1). Our method is based on the properties of the operator \(\mathcal{L}_{K}\) and again on the nonlinear alternative of Leray-Schauder.
Let \(W_{0}^{*}\) denote the dual space of \(W_{0}\). We first give some properties to operator \(\mathcal{L}_{K}\).
Lemma 3.1
-
(1)
\(\mathcal{L}_{K}: W_{0}\rightarrow W_{0}^{*}\)
is a continuous, bounded and strictly monotone operator;
-
(2)
\(\mathcal{L}_{K}\)
is a mapping of type
\((S_{+})\), i.e. if
\(u_{n}\rightharpoonup u\)
in
\(W_{0}\)
and
$$\limsup_{n\rightarrow\infty}\langle\mathcal{L}_{K} u_{n}-\mathcal {L}_{K}u, u_{n}-u\rangle\leq0, $$
then
\(u_{n}\rightarrow u\)
in
\(W_{0}\);
-
(3)
\(\mathcal{L}: W_{0}\rightarrow W_{0}^{*}\)
is a homomorphism.
Proof
(1) Note that, by the Hölder inequality, one has
$$\begin{aligned} \bigl\vert \langle\mathcal{L}_{K}u,v\rangle\bigr\vert &= \biggl\vert \iint_{Q} \bigl\vert u(x)-u(y)\bigr\vert ^{p-2} \bigl(u(x)-u(y)\bigr) \bigl(v(x)-v(y)\bigr)K(x-y)\,dx\,dy\biggr\vert \\ &\leq\|u\|_{W_{0}}^{p-1}\|v\|_{W_{0}} \end{aligned}$$
for all \(u,v\in W_{0}\). Following this inequality, we easily see that \(\mathcal{L}_{K}\) is continuous and bounded.
Let us now recall the well-known Simon inequality (see [26] and [28]): for all \(\xi,\eta\in\mathbb{R}\), there exists \(C_{p}>0\) such that
$$ C_{p} \bigl(|\xi|^{p-2}\xi-|\eta|^{p-2} \eta \bigr)\cdot(\xi-\eta )\geq \textstyle\begin{cases} |\xi-\eta|^{p} &\mbox{if } p\geq2, \\ |\xi-\eta|^{2}(|\xi|^{p}+|\eta|^{p})^{(p-2)/p} &\mbox{if } 1< p< 2. \end{cases} $$
(3.1)
It follows from (3.1) that the operator \(\mathcal{L}_{K}\) is strictly monotone.
(2) By (1), if \(u_{n}\rightharpoonup u\) and \(\limsup_{n\rightarrow \infty}\langle\mathcal{L}_{K} u_{n}-\mathcal{L}_{K}u, u_{n}-u\rangle\leq 0\), then \(\lim_{n\rightarrow\infty}\langle\mathcal{L}_{K} u_{n}-\mathcal{L}_{K}u, u_{n}-u\rangle=0 \). Using the well-known vector inequalities (3.1), we obtain, for \(p>2\),
$$ \|u_{n}-u\|_{W_{0}}^{p}\leq C_{p} \langle\mathcal{L}_{K}u_{n}-\mathcal {L}_{K}u,u_{n}-u\rangle=o(1), $$
(3.2)
and, for \(1< p<2\),
$$\begin{aligned} \|u_{n}-u\|_{W_{0}}^{p}& \leq C_{p}^{p/2} \bigl[\langle\mathcal {L}_{K}u_{n}- \mathcal{L}_{K}u,u_{n}-u\rangle\bigr]^{p/2} \bigl( \|u_{n}\|_{W_{0}}^{p}+\|u\| _{W_{0}}^{p} \bigr)^{(2-p)/2} \\ &\leq C_{p}^{p/2} \bigl[\langle\mathcal{L}_{K}u_{n}- \mathcal{L}_{K}u,u_{n}-u\rangle \bigr]^{p/2} \bigl( \|u_{n}\|_{W_{0}}^{p(2-p)/2}+\|u\|_{W_{0}}^{p(2-p)/2} \bigr) \\ &\leq C \bigl[\langle\mathcal{L}_{K}u_{n}- \mathcal{L}_{K}u,u_{n}-u\rangle\bigr]^{p/2}=o(1), \end{aligned}$$
(3.3)
where \(C>0\) is a constant. Combining (3.2) with (3.3), we obtain \(u_{n}\rightarrow u\) in \(W_{0}\) as \(n\rightarrow\infty\), i.e.
\(\mathcal{L}_{K}\) is of type \((S_{+})\).
(3) By the strictly monotonicity, \(\mathcal{L}_{K}\) is an injection. Since
$$\begin{aligned} \lim_{\|u\|_{W_{0}}\rightarrow\infty}\frac{\langle\mathcal {L}_{K}u,u\rangle}{\|u\|_{W_{0}}} &=\lim_{\|u\|_{W_{0}}\rightarrow\infty} \frac{ \iint _{Q}|u(x)-u(y)|^{p}K(x-y)\,dx\,dy}{\|u\|_{W_{0}}} \\ &=\lim_{\|u\|_{W_{0}}\rightarrow\infty}\|u\|_{W_{0}}^{p-1}=\infty, \end{aligned}$$
thanks to \(1< p<\frac{N}{s}\). Hence \(\mathcal{L}_{K}\) is coercive on \(W_{0}\). Furthermore, by the Minty-Browder theorem (see Theorem 26A of [29]), we know \(\mathcal {L}_{K}\) is a surjection. Thus \(\mathcal{L}_{K}\) has an inverse mapping \(\mathcal {L}_{K}^{-1}:W_{0}^{*}\rightarrow W_{0}\). Now we check the continuity of \(\mathcal{L}_{K}^{-1}\). Assume \(g_{n},g\in W_{0}^{*}\), with \(g_{n}\rightarrow g\) in \(W_{0}^{*}\). Let \(u_{n}=\mathcal {L}_{K}^{-1}g_{n}\) and \(u=\mathcal{L}_{K}^{-1}g \), then \(\mathcal{L}_{K}u_{n}=g_{n}\) and \(\mathcal{L}_{K}u=g\). Clearly, \(\{ u_{n}\}\) is bounded in \(W_{0}\). Thus there exist \(u_{0}\in W_{0}\) and a subsequence of \(\{u_{n}\}\) still denoted by \(\{u_{n}\}\) such that \(u_{n}\rightharpoonup u_{0}\). Since \(g_{n}\rightarrow g\), we have
$$ \lim_{n\rightarrow\infty}\langle\mathcal{L}_{K}u_{n}- \mathcal {L}_{k}u_{0},u_{n}-u_{0} \rangle=\lim_{n\rightarrow\infty} \langle g_{n}, u_{n}-u_{0}\rangle=0. $$
In view of \(\mathcal{L}_{K}\) is of type \((S_{+})\), we get \(u_{n}\rightarrow u_{0}\) in \(W_{0}\). Moreover, \(u=u_{0}\) a.e. in Ω. Hence \(u_{n}\rightarrow u\) in \(W_{0}\), so that \(\mathcal{L}_{K}^{-1}\) is continuous. Thus, we complete the proof. □
To prove the existence of solutions for problem (1.1), we need the following theorem.
Theorem 3.1
(Alternative of Leray-Schauder; see [30])
Let
\(B(0,R)\)
denote the closed ball in a Banach space
X, \(\{u\in X : u\leq R\}\), and let
\(I:B(0,R)\rightarrow X\)
be a compact operator. Then either:
-
(i)
the equation
\(\lambda Iu=u\)
has a solution in
\(B(0,R)\)
for
\(\lambda =1\)
or
-
(ii)
there exists
\(u\in X \)
with
\(\|u\|=R\)
satisfying
\(\lambda Iu=u\)
for some
\(\lambda\in(0,1)\).
Proof of Theorem 1.1
Following the idea of [31], for simplicity of notation, we set
$$ Y=L^{q}(\Omega),\qquad Y^{*}=L^{q^{\prime}}(\Omega),\qquad \|\cdot \|_{Y}=\|\cdot\| _{L^{q}(\Omega)} . $$
By Lemma 2.1 and \(1< q<p_{s}^{*}\), \(W_{0}\) is compactly embedded in Y. Denote by i the compact injection of \(W_{0}\) in Y and by \(i^{*}:Y^{*}\rightarrow W_{0}^{*}\), \(i^{*}v=v\circ i\) for all \(v\in Y^{*}\), its adjoint. It follows from assumption (f1) that the Nemytskii operator \(N_{f}\) generated by f, \((N_{f}u)(x)=f(x,u(x))\), is well defined from Y into \(Y^{*}\), continuous, and bounded (see for example [32]). In order to prove that problem (1.1) has a weak solution in \(W_{0}\) it is sufficient to prove that the equation
$$ \mathcal{L}_{K}u=\bigl(i^{*}N_{f}i\bigr)u $$
(3.4)
has a solution in \(W_{0}\). Indeed, if \(u\in W_{0}\) satisfies (3.4), then for all \(v\in W_{0}\), one has
$$ \langle\mathcal{L}_{K}u,v\rangle=\bigl\langle \bigl(i^{*}N_{f}i \bigr)u,v\bigr\rangle =\bigl\langle N_{f}(iu),iv\bigr\rangle , $$
which can be rewritten
$$ \iint_{Q}\bigl\vert u(x)-u(y)\bigr\vert ^{p-2} \bigl(u(x)-u(y)\bigr) \bigl(v(x)-v(y)\bigr)K(x-y)\,dx\,dy= \int_{\Omega}f(x,u)v\, dx, $$
this means that u is a weak solution in \(W_{0}\) to problem (1.1).
By Lemma 3.1, \(\mathcal{L}_{K}\) is a homeomorphism of \(W_{0}\) onto \(W_{0}^{*}\). Equation (3.4) can be equivalently rewritten
$$ u=\mathcal{L}_{K}^{-1}\bigl(i^{*}N_{f}i\bigr)u. $$
(3.5)
Therefore, proving problem (1.1) has a weak solution in \(W_{0}\) reduces to proving that the compact operator
$$ \mathscr{L}=\mathcal{L}_{K}^{-1}\bigl(i^{*}N_{f}i \bigr):W_{0}\rightarrow W_{0}^{*} $$
has a fixed point.
By Theorem 3.1, a sufficient condition for \(\mathscr{L}\) to have a fixed point is that there exists a constant \(R>0\) such that
$$\mathcal{S}=\bigl\{ u\in W_{0}: u=\lambda\mathscr{L}u \mbox{ for some } t\in [0,1]\bigr\} \subset B(0,R). $$
Since for \(\lambda=0\) the only solution of equation \(u=\lambda \mathscr{L}u\) is \(u =0\), it is enough to show that there exists a constant such that any \(u\in W_{0}\) which satisfies
$$ u=\lambda\mathcal{L}_{K}^{-1}\bigl[ \bigl(i^{*}N_{f}i\bigr)u\bigr] $$
(3.6)
for some \(\lambda\in(0,1]\) belongs to \(B(0,R)\).
Indeed, if \(u\in W_{0}\) satisfies (3.6) for some \(\lambda\in(0,1]\), then we have
$$ \langle\mathcal{L}_{K} u,u\rangle= \lambda\bigl\langle \bigl(i^{*}N_{f}i\bigr)u,u\bigr\rangle . $$
(3.7)
It follows from (3.7) that
$$\begin{aligned} \Vert u\Vert _{W_{0}}^{p}&= \lambda\bigl\langle \bigl(i^{*}N_{f}i\bigr)u,u\bigr\rangle \leq\bigl\langle \bigl(i^{*}N_{f}i\bigr)u,u\bigr\rangle \\ &\leq\bigl\Vert i^{*}\bigr\Vert \bigl\Vert N_{f}(iu)\bigr\Vert _{Y^{*}}\Vert u\Vert _{W_{0}}. \end{aligned}$$
(3.8)
In order to estimate \(\|N_{f}(iu)\|_{Y^{*}}\), we deduce first from assumption (f1) that
$$ \bigl\vert (N_{f}v) (x)\bigr\vert =\bigl\vert f(x,v)\bigr\vert \leq c_{1}\bigl\vert v(x)\bigr\vert ^{q/q^{\prime}}+a(x). $$
Hence,
$$ \Vert N_{f}v\Vert _{Y^{*}}\leq\bigl\Vert c_{1}\bigl\vert v(x)\bigr\vert ^{q/q^{\prime}}+a(x)\bigr\Vert _{Y^{*}} \leq c_{1}\Vert v\Vert _{Y}^{q-1}+ \bigl\Vert a(x)\bigr\Vert _{Y^{*}}. $$
(3.9)
By taking (3.9) for \(v=iu\), \(u\in W_{0}\), we have
$$ \bigl\Vert N_{f}(iu)\bigr\Vert _{Y^{*}}\leq c_{1}\Vert i\Vert ^{q-1}\Vert u\Vert _{W_{0}}^{q-1}+\Vert a\Vert _{Y^{*}}. $$
(3.10)
In particular, if \(u\in W_{0}\) and satisfies (3.6) for some \(\lambda\in(0,1]\), we derive from (3.8) and (3.10) that
$$ \Vert u\Vert _{W_{0}}^{p}\leq\bigl\Vert i^{*} \bigr\Vert \bigl(c_{1}\Vert i\Vert ^{q-1}\Vert u\Vert _{W_{0}}^{q-1}+\Vert a\Vert _{Y^{*}}\bigr)\Vert u \Vert _{W_{0}} =C_{1}\Vert u\Vert _{W_{0}}^{q}+C_{2} \Vert u\Vert _{W_{0}}, $$
(3.11)
where \(C_{1}=c_{1}\|i^{*}\|\|i\|^{q}\), \(C_{2}=\|i^{*}\|\|a\|_{Y^{*}}\). Applying the Young inequality, we get
$$ \Vert u\Vert _{W_{0}}^{q}C_{1}\leq \frac{1}{4}\Vert u\Vert _{W_{0}}^{p}+4^{\frac {q}{p-q}}C_{1}^{\frac{p}{p-q}} $$
and
$$ \Vert u\Vert _{W_{0}}C_{2}\leq\frac{1}{4}\Vert u \Vert _{W_{0}}^{p}+4^{\frac {1}{p-1}}C_{2}^{\frac{p}{p-1}}. $$
Inserting these two inequalities into (3.11), one has
$$ \frac{1}{2}\|u\|_{W_{0}}^{p}\leq4^{\frac{q}{p-q}}C_{1}^{\frac {p}{p-q}}+4^{\frac{1}{p-1}}C_{2}^{\frac{p}{p-1}}. $$
Set
$$R= \bigl(2^{\frac{p+q}{p-q}}C_{1}^{\frac{p}{p-q}}+2^{\frac {p+1}{p-1}}C_{2}^{\frac{p}{p-1}} \bigr)^{1/p}. $$
Then \(\|u\|_{W_{0}}\leq R\), this implies the set \(\mathcal{S}\) is bounded. □