We consider the energy functional \(J: W_{0}^{1,p}(\Omega)\to\mathbb{R}\) defined by
$$ J(u) = \frac{1}{p}G\bigl(\|u\|^{p}\bigr)-\lambda \int_{\Omega}H(x,u)\, dx-\frac{1}{p^{\ast}} \int_{\Omega}|u|^{p^{\ast}}\, dx, $$
where \(W_{0}^{1,p}(\Omega)\) is the Sobolev space endowed with the norm \(\|u\|^{p}=\int_{\Omega}|\nabla u|^{p}\, dx\). Standard arguments [24] show that a critical point of J is a weak solution of problem (1.1). We try to use a new version of the symmetric mountain-pass lemma due to Kajikiya [22]. But since the functional \(J(u)\) is not bounded from below, we could not use the theory directly. So we follow [23] to consider a truncated functional of \(J(u)\). Denote by \(J' : E \rightarrow E^{\ast}\) the derivative operator of J in the weak sense. Then
$$\begin{aligned} \bigl\langle J'(u), v\bigr\rangle =& g \bigl(\|u\|^{p} \bigr) \int_{\Omega} \bigl(|\nabla u|^{p-2}\nabla u \cdot\nabla v \bigr)\, dx - \int_{\Omega}|u|^{p^{\ast}-2}uv \, dx \\ &{}- \lambda \int_{\Omega}h(x, u)v\, dx,\quad \forall u, v \in W_{0}^{1,p}(\Omega). \end{aligned}$$
To use variational methods, we give some results related to the Palais-Smale compactness condition. Recall that a sequence \((u_{n})\) is a Palais-Smale sequence of J at the level c, if \(J(u_{n})\to c\) and \(J'(u_{n})\to0\).
We recall the second concentration-compactness principle of Lions [25, 26].
Lemma 2.1
[25, 26]
Let
\(\{u_{n}\}\)
be a weakly convergent sequence to
u
in
\(W_{0}^{1,p}(\Omega)\)
such that
\(|u_{n}|^{p^{\ast}}\rightharpoonup\nu\)
and
\(|\nabla u_{n}| \rightharpoonup\mu\)
in the sense of measures. Then, for some at most countable index set
I,
-
(i)
\(\nu= |u|^{p^{\ast}} + \sum_{j \in I} \delta_{x_{j}}\nu_{j}\), \(\nu_{j} > 0\),
-
(ii)
\(\mu\geq |\nabla u|^{p} + \sum_{j \in I} \delta_{x_{j}}\mu_{j}\), \(\mu_{j} > 0\),
-
(iii)
\(\mu_{j} \geq S \nu_{j}^{p/p^{\ast}}\),
where
S
is the best Sobolev constant, i.e. \(S = \inf \{ \int_{\mathbb{R}^{N}}|\nabla u|^{p}\, dx: \int_{\mathbb{R}^{N}}|u|^{p^{\ast}}\, dx = 1 \}\), \(x_{j} \in\mathbb{R}^{N}\), \(\delta_{x_{j}}\)
are Dirac measures at
\(x_{j}\)
and
\(\mu_{j}\), \(\nu_{j}\)
are constants.
Under assumptions (H1) and (H2), we have
$$ h(x, s)s = o \bigl(|s|^{p^{\ast}} \bigr), \qquad H(x, s) = o \bigl(|s|^{p^{\ast}} \bigr), $$
which means that, for all \(\varepsilon> 0\), there exist \(a(\varepsilon), b(\varepsilon) > 0\) such that
$$\begin{aligned}& \bigl\vert h(x,s)s\bigr\vert \leq a(\varepsilon) + \varepsilon|s|^{p^{\ast}}, \end{aligned}$$
(2.1)
$$\begin{aligned}& \bigl\vert H(x,s)\bigr\vert \leq b(\varepsilon) + \varepsilon|s|^{p^{\ast}}. \end{aligned}$$
(2.2)
Hence,
$$ H(x,s) - \frac{\sigma}{p}h(x, s)s \leq c(\varepsilon) + \varepsilon |s|^{p^{\ast}} $$
(2.3)
for some \(c(\varepsilon) > 0\).
Lemma 2.2
Suppose that (G1)-(G2), (H1)-(H3) hold. Then, for any
\(\lambda> 0\), the functional
J
satisfies the local
\((\mathit{PS})_{c}\)
condition in
$$c \in \biggl(-\infty, \frac{p^{\ast}\sigma-p}{2pp^{\ast}}(\alpha_{0} S)^{N/p} - \lambda c \biggl(\frac{p^{\ast}\sigma-p}{2pp^{\ast}\lambda} \biggr)|\Omega| \biggr) $$
in the following sense: if
$$J(u_{n}) \rightarrow c < \frac{p^{\ast}\sigma-p}{2pp^{\ast}}(\alpha_{0} S)^{N/p} - \lambda c \biggl(\frac{p^{\ast}\sigma-p}{2pp^{\ast}\lambda} \biggr)|\Omega| $$
and
\(J'(u_{n}) \rightarrow0\)
for some sequence in
\(W_{0}^{1,p}(\Omega)\), then
\(\{u_{n}\}\)
contains a subsequence converging strongly in
\(W_{0}^{1,p}(\Omega)\).
Proof
Let \(\{u_{n}\}\) be a sequence in \(W_{0}^{1,p}(\Omega)\) such that
$$\begin{aligned}& J(u_{n}) = \frac{1}{p}G\bigl(\|u_{n} \|^{p}\bigr)-\lambda \int_{\Omega}H(x,u_{n})\, dx-\frac{1}{p^{\ast}} \int_{\Omega}|u_{n}|^{p^{\ast}}\, dx = c + o(1), \end{aligned}$$
(2.4)
$$\begin{aligned}& \bigl\langle J'(u_{n}), v\bigr\rangle = g \bigl(\|u_{n}\|^{p} \bigr) \int_{\Omega} \bigl(|\nabla u_{n}|^{p-2}\nabla u_{n} \cdot\nabla v \bigr)\, dx - \int_{\Omega}|u_{n}|^{p^{\ast}-2}u_{n}v \, dx \\& \hphantom{\bigl\langle J'(u_{n}), v\bigr\rangle ={}}{} - \lambda \int_{\Omega}h(x, u_{n})v\, dx = o(1)\|u_{n}\|. \end{aligned}$$
(2.5)
By (2.4) and (2.5), we have
$$\begin{aligned} c+ o(1)\|u_{n}\| =& J(u_{n}) - \frac{\sigma}{p}\bigl\langle J'(u_{n}), u_{n}\bigr\rangle \\ =&\frac{1}{p}G\bigl(\|u_{n}\|^{p}\bigr)- \frac{\sigma}{p}g \bigl(\|u_{n}\|^{p} \bigr) \|u_{n}\|^{p} + \biggl(\frac{\sigma }{p}-\frac{1}{p^{\ast}} \biggr) \int_{\Omega}|u_{n}|^{p^{\ast}}\,dx \\ &{}- \lambda \int_{\Omega}H(x, u_{n})\,dx + \frac{\sigma}{p}\lambda \int _{\Omega}h(x, u_{n})u_{n} \,dx \\ \geq&\frac{p^{\ast}\sigma-p}{pp^{\ast}} \int_{\Omega}|u_{n}|^{p^{\ast}}\,dx- \lambda \int_{\Omega}H(x, u_{n})\,dx \\ &{}+ \frac{\sigma}{p}\lambda \int_{\Omega}h(x, u_{n})u_{n} \,dx, \end{aligned}$$
i.e.
$$\begin{aligned} \frac{p^{\ast}\sigma-p}{pp^{\ast}} \int_{\Omega}|u_{n}|^{p^{\ast}}\,dx \leq \lambda \int_{\Omega} \biggl(H(x, u_{n}) - \frac{\sigma}{p}h(x, u_{n})u_{n} \biggr)\,dx + c+ o(1)\|u_{n}\|. \end{aligned}$$
Then by (2.3), we have
$$ \biggl(\frac{p^{\ast}\sigma-p}{pp^{\ast}}- \lambda\varepsilon \biggr) \int_{\Omega}|u_{n}|^{p^{\ast}}\,dx \leq\lambda c( \varepsilon)|\Omega| + c + o(1)\|u_{n}\|. $$
Setting \(\varepsilon= \frac{p^{\ast}\sigma-p}{2pp^{\ast}\lambda}\), we get
$$ \int_{\Omega}|u_{n}|^{p^{\ast}}\,dx \leq M + o(1) \|v_{n}\|, $$
(2.6)
where \(o(1) \rightarrow0\) and M is a some positive number. On the other hand, by (2.2) and (2.6), we have
$$\begin{aligned} c + o(1)\|u_{n}\| =& J(u_{n}) \\ =& \frac{1}{p}G\bigl(\|u_{n}\|^{p}\bigr)-\lambda \int _{\Omega}H(x,u_{n})\,dx-\frac{1}{p^{\ast}} \int_{\Omega}|u_{n}|^{p^{\ast}}\,dx \\ \geq& \frac{\alpha_{0}\sigma}{p}\|u_{n}\|^{p} -\lambda b( \varepsilon)|\Omega| - \biggl[\frac{1}{p^{\ast}} + \lambda\varepsilon \biggr] \int_{\Omega}|u_{n}|^{p^{\ast}}\,dx. \end{aligned}$$
(2.7)
Therefore, the inequalities (2.6) and (2.7) imply that \(\{u_{n}\}\) is bounded in \(W_{0}^{1,p}(\Omega)\). Hence, up to a subsequence, we may assume that
$$\begin{aligned}& u_{n}\rightharpoonup u \quad \text{weakly in } W_{0}^{1,p}(\Omega), \\& u_{n}\to u\quad \text{a.e. in }\Omega, \\& u_{n}\to u \quad \text{in }L^{s}(\Omega), 1\leq s< p^{\ast}, \\& |\nabla u_{n}|^{p}\rightharpoonup\mu \quad \bigl( \text{weak}^{*}\mbox{-sense of measures}\bigr), \\& |u_{n}|^{p^{\ast}}\rightharpoonup\nu \quad \bigl(\text{weak}^{*} \mbox{-sense of measures}\bigr), \end{aligned}$$
(2.8)
where μ and ν are a nonnegative bounded measures on Ω̅. Then, by the concentration-compactness principle due to Lions [25, 26], there exists some at most countable index set I such that
$$\begin{aligned}& \nu=|u|^{p^{\star}}+\sum_{j\in I}\nu_{j} \delta_{x_{j}} , \quad \nu_{j}>0, \\& \mu\geq|\nabla u|^{p}+\sum_{j\in I} \mu_{j}\delta_{x_{j}} ,\quad \mu_{j}>0, \\& S\nu_{j}^{p/p^{\star}}\leq\mu_{j}, \end{aligned}$$
where \(\delta_{x_{j}}\) is the Dirac measure mass at \(x_{j}\in\overline{\Omega}\). Let \(\psi(x)\in C_{0}^{\infty}\) such that \(0\leq\psi\leq1\),
$$ \psi(x)= \textstyle\begin{cases} 1&\text{if }|x|< 1, \\ 0&\text{if }|x|\geq2, \end{cases} $$
(2.9)
and \(|\nabla\psi|_{\infty}\leq2\).
For \(\varepsilon>0\) and \(j\in I\), denote \(\psi_{\varepsilon}^{j}(x)=\psi((x-x_{j})/\varepsilon)\). Since \(J'(u_{n})\to 0\) and \((\psi_{\varepsilon}^{j}u_{n})\) is bounded, \(\langle J'(u_{n}),\psi_{\varepsilon}^{j}u_{n}\rangle\to0\) as \(n\to\infty\); that is,
$$\begin{aligned}& g\bigl(\|u_{n}\|^{p}\bigr) \int_{\Omega}|\nabla u_{n}|^{p} \psi_{\varepsilon}^{j}\,dx \\& \quad =-g\bigl(\|u_{n}\|^{p}\bigr) \int_{\Omega}u_{n}|\nabla u_{n}|^{p-2} \nabla u_{n}\nabla\psi_{\varepsilon}^{j}\,dx \\& \qquad {}+\lambda \int_{\Omega}h(x,u_{n})u_{n} \psi_{\varepsilon}^{j}\,dx + \int_{\Omega}|u_{n}|^{p^{\ast}}\psi_{\varepsilon}^{j} \,dx+o_{n}(1). \end{aligned}$$
(2.10)
By (2.8) and Vitali’s theorem, we see that
$$ \lim_{n\to\infty} \int_{\Omega}\bigl\vert u_{n}\nabla \psi_{\varepsilon}^{j}\bigr\vert ^{p} \,dx = \int_{\Omega}\bigl\vert u\nabla\psi_{\varepsilon}^{j} \bigr\vert ^{p} \,dx. $$
Hence, by Hölder’s inequality we obtain
$$\begin{aligned}& \limsup_{n\to\infty} \biggl\vert \int_{\Omega}u_{n}|\nabla u_{n}|^{p-2} \nabla u_{n}\nabla\psi_{\varepsilon}^{j}\,dx\biggr\vert \\& \quad \leq\limsup_{n\to\infty} \biggl( \int_{\Omega}|\nabla u_{n}|^{p}\,dx \biggr)^{(p-1)/p} \biggl( \int_{\Omega}\bigl\vert u_{n}\nabla \psi_{\varepsilon}^{j}\bigr\vert ^{p} \,dx \biggr)^{1/p} \\& \quad \leq C_{1} \biggl( \int_{B(x_{j},2\varepsilon)}|u|^{p}\bigl\vert \nabla \psi_{\varepsilon}^{j}\bigr\vert ^{p} \,dx \biggr)^{1/p} \\& \quad \leq C_{1} \biggl( \int_{B(x_{j},2\varepsilon)}\bigl\vert \nabla \psi_{\varepsilon}^{j} \bigr\vert ^{N}\,dx \biggr)^{1/N} \biggl( \int_{B(x_{j},2\varepsilon)}|u|^{p^{\ast}} \,dx \biggr)^{1/p^{\ast}} \\& \quad \leq C_{2} \biggl( \int_{B(x_{j},2\varepsilon)}|u|^{p^{\ast}} \,dx \biggr)^{1/p^{\ast}} \to0 \quad \text{as } \varepsilon\to0 . \end{aligned}$$
(2.11)
Since \(\psi_{\varepsilon}^{j}\) has compact support, letting \(n\to\infty\) in (2.10) we deduce from (2.10) and (2.11) that
$$\alpha_{0} \int_{\Omega}\psi_{\varepsilon}^{j}\, d\mu\leq C_{2} \biggl( \int_{B(x_{j},2\varepsilon)}|u|^{p^{\ast}} \,dx \biggr)^{1/ p^{\star}} + \lambda \int_{B(x_{j},2\varepsilon)} f(x,u)u\,dx+ \int_{\Omega}\psi_{\varepsilon}^{j}\, d\nu. $$
Letting \(\varepsilon\to0\), we obtain \(\alpha_{0}\mu_{j}\leq\nu_{j}\). Therefore,
$$ (\alpha_{0} S)^{N/p}\leq\nu_{j}. $$
(2.12)
We will prove that this inequality is not possible. Let us assume that \((\alpha_{0} S)^{N/p}\leq\nu_{j_{0}}\) for some \(j_{0}\in I\). From (G2) we see that
$$ G\bigl(\|u_{n}\|^{p}\bigr)-\sigma g\bigl(\|u_{n} \|^{p}\bigr)\|u_{n}\|^{p}\geq0 \quad \text{for all }n. $$
Since
$$ c = J(u_{n})-\frac{\sigma}{p} \bigl\langle J'(u_{n}),u_{n} \bigr\rangle +o_{n}(1), $$
it follows that
$$\begin{aligned} c =& \lim_{n \rightarrow\infty} \biggl(J(u_{n}) - \frac{\sigma}{p}\bigl\langle J'(u_{n}), u_{n} \bigr\rangle \biggr) \\ \geq& \biggl(\frac{\sigma}{p}-\frac{1}{p^{\ast}} \biggr) \int_{\Omega }|u_{n}|^{p^{\ast}}\,dx - \lambda \int_{\Omega} \biggl[H(x, u_{n})-\frac{\sigma }{p}h(x,u_{n})u_{n} \biggr]\,dx \\ \geq& \biggl(\frac{p^{\ast}\sigma-p}{pp^{\ast}}-\lambda\varepsilon \biggr) \int_{\Omega}\psi_{\varepsilon}^{j_{0}}|u_{n}|^{p^{\ast}} \,dx- \lambda c(\varepsilon)|\Omega|. \end{aligned}$$
Letting \(\varepsilon= \frac{p^{\ast}\sigma-p}{2pp^{\ast}\lambda}\) and \(n\to\infty\), we obtain
$$\begin{aligned} \begin{aligned} c &\geq \frac{p^{\ast}\sigma-p}{2pp^{\ast}}\sum_{j\in J} \psi_{\varepsilon}^{j_{0}}(x_{j})\nu_{j} - \lambda c \biggl(\frac{p^{\ast}\sigma-p}{2pp^{\ast}\lambda} \biggr)|\Omega| \\ &\geq \frac{p^{\ast}\sigma-p}{2pp^{\ast}}(\alpha_{0} S)^{N/p} - \lambda c \biggl(\frac{p^{\ast}\sigma-p}{2pp^{\ast}\lambda} \biggr)|\Omega|. \end{aligned} \end{aligned}$$
This is impossible. Then \(I = \emptyset\), and hence \(u_{n}\to u\) in \(L^{p^{\ast}}(\Omega)\).
Then, using (2.8) and the fact that \(u_{n}\to u\) in \(L^{p^{\ast}}(\Omega)\), we have
$$ \lim_{n\to\infty} \int_{\Omega}h(x,u_{n}) (u_{n}-u)\,dx=0 $$
(2.13)
and
$$ \lim_{n\to\infty} \int_{\Omega}|u_{n}|^{p^{\ast}-2}u_{n}(u_{n}-u) \,dx=0. $$
(2.14)
From \(\langle J'(u_{n}),u_{n}-u\rangle=o_{n}(1)\), we deduce that
$$\begin{aligned} \bigl\langle J'(u_{n}),u_{n}-u\bigr\rangle =& g\bigl(\|u_{n}\|^{p}\bigr) \int_{\Omega}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla(u_{n}-u)\,dx \\ &{}-\lambda \int_{\Omega}h(x,u_{n}) (u_{n}-u)\,dx- \int_{\Omega}|u_{n}|^{p^{\ast}-2}u_{n}(u_{n}-u) \,dx =o_{n}(1). \end{aligned}$$
This, (2.13), and (2.14) imply
$$\lim_{n\to\infty} g\bigl(\|u_{n}\|^{p}\bigr) \int_{\Omega}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla(u_{n}-u)\,dx=0. $$
Since \(u_{n}\) is bounded and g is continuous, up to subsequence, there is \(t_{0}\geq0\) such that
$$g\bigl(\|u_{n}\|^{p}\bigr)\to g\bigl(t_{0}^{p} \bigr)\geq\alpha_{0}\quad \text{as } n\to\infty, $$
and so
$$\lim_{n\to\infty} \int_{\Omega}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla(u_{n}-u)\,dx=0. $$
Thus by the \((S_{+})\) property, \(u_{n}\to u\) strongly in \(W_{0}^{1,p}(\Omega)\). The proof is complete. □