Noting that \(\alpha >\frac{N}{p}>1\), one has \(W^{1,p}(\Omega )\) is embedded in \(L^{\alpha '}(\Omega )\), where \(\alpha '=\frac{\alpha}{\alpha -1}\) is the conjugate of α, thus \(W^{1,p}_{0}(\Omega )\) is embedded in \(L^{\alpha '}(\Omega )\). Similarly, \(W^{1,p}_{0}(\Omega )\) is embedded in \(L^{\alpha 's}(\Omega ), 1< s\leq p\). Let \(c_{\alpha '}\) be the embedding constant of the compact embedding \(W^{1,p}(\Omega )\hookrightarrow L^{\alpha '}(\Omega )\), and \(c_{\alpha 's}\) be the embedding constant of the compact embedding \(W^{1,p}(\Omega )\hookrightarrow L^{\alpha 's}(\Omega )\), \(1< s\leq p\).
Noting that \(q\in (1, p)\), \(\beta >\frac{(N-1)}{p-1}\), one has \(1<\beta ' q<\frac{(N-1)p}{N-p}\), where \(\beta '=\frac{\beta}{\beta -1}\) is the conjugate exponent of β, so \(W^{1,p}_{0}(\Omega )\) is embedded in \(L^{\beta 'q}( \partial \Omega )\) (see [4], Theorem 2.79). Let \(c_{\beta ' q}\) be the continuous embedding constant of \(W^{1,p}(\Omega )\hookrightarrow L^{\beta 'q}(\partial \Omega )\).
Putting
$$ l(x)=\sup \bigl\{ l>0: B(x,l)\subseteq \Omega \bigr\} $$
for all \(x \in \Omega \), we can show that there exists \(x_{0} \in \Omega \) such that \(B(x_{0},d) \subseteq \Omega \), where
$$ d = \sup_{x\in \Omega} l(x).$$
Suppose there exist two positive constants δ, r, with \(r= \frac{a_{1}}{p}(\frac{2}{d})^{p-N}\delta ^{p}(2^{N}-1)|B(0,1)|\), such that
$$ \begin{aligned} &\frac{pd^{p}C}{2^{p-N}\delta ^{p}(a_{0}+a_{2}C)d^{N}(2^{N}-1)} \operatorname*{ess\, inf}_{ B(x_{0},\frac{d}{2})} F(x, \delta ) \\ &\quad := \frac{1}{\check{\lambda}}> \frac{M_{1}\gamma}{r} c_{\alpha '} \Vert b \Vert _{\alpha}+ \frac{M_{2}\gamma ^{s}}{rs}c_{\alpha 's}^{s} \Vert b \Vert _{\alpha} := \frac{1}{\hat{\lambda}}, \end{aligned} $$
(3.1)
where \(\gamma =(\frac{pr}{a_{1}})^{\frac{1}{p}}\), \(C=(\frac{N-p}{p})^{p}\).
With the above notations we present the following results.
Theorem 3.1
Assume conditions \((\mathbf{f})\), \((\mathbf{g})\), and
-
(H1)
\(F(x,\xi )\geq 0\), \(\forall (x,\xi )\in B(x_{0},d) \times [0,\delta ]\);
-
(H2)
\(\limsup_{|\xi |\to +\infty} \frac{\sup_{x \in \Omega}F(x,\xi )}{\xi ^{p}} \le \frac{a_{1}}{pc} ( \frac{M_{1}\gamma}{r}c_{\alpha '}\|b\|_{\alpha}+ \frac{M_{2}\gamma ^{s}}{rs}c_{\alpha 's}\|b\|_{\alpha})\), where \(c=|\Omega |^{\frac{p^{*}-p}{p^{*}}}T^{p}\), \(r=\frac{a_{1}}{p}( \frac{2}{d})^{p-N}\delta ^{p}(2^{N}-1)|B(0,1)|\)
hold, then for every \(\lambda \in (\check{\lambda},\hat{\lambda}) \), when \(\mu \in [0, \frac{q(rs-\lambda s \gamma M_{1}c_{\alpha '}\|b\|_{\alpha}-\lambda \gamma ^{s} M_{2}c_{\alpha 's}\|b\|_{\alpha})}{\gamma ^{q}c_{q}h_{0}s})\), where \(h_{0}= \operatorname{ess}\sup_{\Gamma _{2}}h(x) \), the problem (1.1) has at least three weak solutions.
Proof
Let \(u_{0}(x)=0\), δ be a constant, and
$$ \bar{u}(x)= \textstyle\begin{cases} 0, & x\in \Omega \setminus \bar{B}(x_{0},d), \\ \frac{2\delta}{d}(d- \vert x-x_{0} \vert ), & x\in B(x_{0},d)\setminus \bar{B}(x_{0}, \frac{d}{2}), \\ \delta, & x\in B(x_{0}, \frac{d}{2}), \end{cases} $$
thus \(u_{0}, \bar{u}\in W^{1,p}_{0}(\Omega ) \), \(\Phi (u_{0})=\Psi (u_{0})=0\), and
$$ \begin{aligned} \Phi (\bar{u})&= \int _{\Omega}\mathcal{A}\bigl(x, \nabla \bar{u}(x)\bigr) \,dx + \frac{1}{p} \int _{\Omega}\frac{a(x) \vert \bar{u}(x) \vert ^{p}}{ \vert x \vert ^{p}} \,dx \\ &\geq \frac{a_{1}}{p}\frac{(2\delta )^{p}}{d^{p}} \biggl\vert B(x_{0},d) \bigm\backslash \bar{B}\biggl(x_{0}, \frac{d}{2}\biggr) \biggr\vert \\ &= \frac{a_{1}}{p}\frac{(2\delta )^{p}}{d^{p}} \bigl\vert B(0,1) \bigr\vert \biggl(d^{N}-\biggl( \frac{d}{2}\biggr)^{N} \biggr) \\ & =\frac{a_{1}}{p}\biggl(\frac{2}{d}\biggr)^{p-N} \delta ^{p}\bigl(2^{N}-1\bigr) \bigl\vert B(0,1) \bigr\vert =r. \end{aligned} $$
(3.2)
Taking account of the following Hardy inequality (see [8, Lemma 2.1] for more details),
$$ \int _{\Omega}\frac{ \vert \bar{u}(x) \vert ^{p}}{ \vert x \vert ^{p}} \,dx \leq \frac{1}{C} \int _{ \Omega} \bigl\vert \nabla \bar{u}(x) \bigr\vert ^{p} \,dx ,\quad \forall u\in W^{1,p}_{0}(\Omega ), $$
where \(C=(\frac{N-p}{p})^{p}\) is the optimal constant, one has
$$ \begin{aligned} \Phi (\bar{u})&= \int _{\Omega}\mathcal{A}\bigl(x, \nabla \bar{u}(x)\bigr) \,dx + \frac{1}{p} \int _{\Omega}\frac{a(x) \vert \bar{u}(x) \vert ^{p}}{ \vert x \vert ^{p}} \,dx \\ &\leq \frac{a_{2}}{p}\frac{(2\delta )^{p}}{d^{p}} \biggl\vert B(x_{0},d) \bigm\backslash \bar{B}\biggl(x_{0}, \frac{d}{2}\biggr) \biggr\vert +\frac{a_{0}}{p} \int _{\Omega} \frac{ \vert \bar{u}(x) \vert ^{p}}{ \vert x \vert ^{p}} \,dx \\ & \leq \frac{a_{2}}{p}\frac{(2\delta )^{p}}{d^{p}} \biggl\vert B(x_{0},d) \bigm\backslash \bar{B}\biggl(x_{0}, \frac{d}{2}\biggr) \biggr\vert +\frac{a_{0}}{pC} \int _{\Omega} \bigl\vert \nabla \bar{u}(x) \bigr\vert ^{p} \,dx \\ & \leq \frac{2^{p-N}\delta ^{p}(a_{0}+a_{2}C)}{pd^{p}C}d^{N}\bigl(2^{N}-1\bigr) \bigl\vert B(0,1) \bigr\vert . \end{aligned} $$
(3.3)
In view of the conditions (H1) and \((\mathbf{g})\), one has
$$ \begin{aligned} \Psi (\bar{u})&= \int _{\Omega}F(x, \bar{u})\,dx +\frac{\mu}{\lambda} \int _{\partial \Gamma _{2}}G\bigl(x,\gamma (\bar{u})\bigr)\,d\sigma \\ &\geq \int _{ B(x_{0},\frac{d}{2})} F(x, \delta )\,dx \\ &\geq \biggl\vert B\biggl(x_{0},\frac{d}{2}\biggr) \biggr\vert \operatorname*{ess\, inf}_{ B(x_{0},\frac{d}{2})} F(x, \delta ).\end{aligned} $$
(3.4)
Noting that \(0< b\in L^{\alpha}(\Omega )\), \(\alpha >N/p\), since \(p< N\), \(W^{1,p}(\Omega )\) is embedded in \(L^{\alpha '}(\Omega )\), where \(\alpha '=\frac{\alpha}{\alpha -1}\) is the conjugate exponent of α, so \(W^{1,p}_{0}(\Omega )\) is embedded in \(L^{\alpha '}(\Omega )\). Similarly, \(W^{1,p}_{0}(\Omega )\) is embedded in \(L^{\alpha 's}(\Omega ), 1< s\leq p\).
By the Hölder inequality, we can obtain
$$ \begin{aligned} &\int _{\Omega}b(x)u\, dx\leq \Vert b \Vert _{\alpha} \Vert u \Vert _{\alpha '},\qquad \int _{ \Omega}b(x)u^{s}\,dx \leq \Vert b \Vert _{\alpha} \Vert u \Vert _{\alpha 's}^{s}, \\ &\int _{ \partial \Omega} h \vert u \vert ^{q}\,d\sigma \leq \Vert h \Vert _{\beta ,\partial \Omega} \Vert u \Vert _{\beta 'q,\partial \Omega}^{q}. \end{aligned} $$
For every \(u \in \Phi ^{-1}(- \infty ,r]\), one has \(\Phi (u)\leq r\), and \(\|u\|\leq (\frac{pr}{a_{1}})^{\frac{1}{p}}=\gamma \). Thus,
$$ \begin{aligned} &\sup_{u \in \Phi ^{-1}(- \infty ,r]}\Psi (u) \\ &\quad = \sup _{u \in \Phi ^{-1}(- \infty ,r]}\biggl( \int _{\Omega} F(x, u) \,dx +\frac{\mu}{\lambda } \int _{ \Gamma _{2}}G\bigl(x,\gamma (u)\bigr)\,d\sigma \biggr) \\ &\quad \leq \int _{\Omega}\sup_{ \Vert u \Vert \leq (\frac{pr}{a_{1}})^{\frac{1}{p}}} \biggl(F(x, u) \,dx + \frac{\mu}{\lambda } \int _{\Gamma _{2}}G\bigl(x,\gamma (u)\bigr)\,d\sigma \biggr) \\ &\quad \leq \int _{\Omega}\sup_{ \Vert u \Vert \leq (\frac{pr}{a_{1}})^{\frac{1}{p}}}\biggl(b(x) \biggl(M_{1} \vert u \vert + \frac{M_{2}}{s} \vert u \vert ^{s}\biggr) \,dx +\frac{\mu}{\lambda q} \int _{\partial \Omega} h \vert u \vert ^{q}\,d\sigma \biggr) \\ &\quad \leq \int _{\Omega}\sup_{ \Vert u \Vert \leq (\frac{pr}{a_{1}})^{\frac{1}{p}}}\biggl( M_{1} \Vert b \Vert _{\alpha} \Vert u \Vert _{\alpha '}+\frac{M_{2}}{s} \Vert b \Vert _{\alpha} \Vert u \Vert _{\alpha 's}^{s} +\frac{\mu }{\lambda q} \Vert h \Vert _{\beta ,\partial \Omega} \Vert u \Vert _{\beta 'q,\partial \Omega}^{q} \biggr) \\ &\quad \leq \int _{\Omega}\sup_{ \Vert u \Vert \leq (\frac{pr}{a_{1}})^{\frac{1}{p}}}\biggl( M_{1}c_{\alpha '} \Vert b \Vert _{\alpha} \Vert u \Vert +\frac{M_{2}}{s}c_{\alpha 's}^{s} \Vert b \Vert _{\alpha} \Vert u \Vert ^{s} + \frac{c_{\beta 'q}^{q}\mu }{\lambda q} \Vert h \Vert _{ \beta ,\partial \Omega} \Vert u \Vert ^{q}\biggr) \\ &\quad \leq M_{1}\gamma c_{\alpha '} \Vert b \Vert _{\alpha}+ \frac{M_{2}\gamma ^{s}}{s}c_{\alpha 's}^{s} \Vert b \Vert _{\alpha}+ \frac{\mu \gamma ^{q}c_{\beta 'q}^{q}}{\lambda q} \Vert h \Vert _{\beta , \partial \Omega}. \end{aligned} $$
Thus, taking account of \(\mu \in [0, \frac{q(rs-\lambda s \gamma M_{1}c_{\alpha '}\|b\|_{\alpha}-\lambda \gamma ^{s} M_{2}c_{\alpha 's}^{s}\|b\|_{\alpha})}{s\gamma ^{q}c_{\beta 'q}^{q}\|h\|_{\beta ,\partial \Omega}})\), one has
$$ \begin{aligned} \frac{\sup_{u \in \Phi ^{-1}(- \infty ,r]}\Psi (u)}{r}&\leq \frac{M_{1}\gamma}{r} c_{\alpha '} \Vert b \Vert _{\alpha}+ \frac{M_{2}\gamma ^{s}}{rs}c_{\alpha 's}^{s} \Vert b \Vert _{\alpha}+ \frac{\mu \gamma ^{q}c_{\beta 'q}^{q}}{\lambda qr} \Vert h \Vert _{\beta , \partial \Omega} \\ &< \frac{1}{\lambda}. \end{aligned} $$
(3.5)
Combining (3.1), (3.3) with (3.4), one has
$$ \begin{aligned} \frac{\Psi (\bar{u})}{\Phi (\bar{u})}&\geq \frac{ \vert B(x_{0},\frac{d}{2}) \vert \operatorname*{ess\, inf}_{ B(x_{0},\frac{d}{2})} F(x, \delta )}{\frac{2^{p-N}\delta ^{p}(a_{0}+a_{2}C)}{pd^{p}C}d^{N}(2^{N}-1) \vert B(0,1) \vert } \\ &= \frac{pd^{p}C}{2^{p-N}\delta ^{p}(a_{0}+a_{2}C)d^{N}(2^{N}-1)} \operatorname*{ess\, inf}_{ B(x_{0},\frac{d}{2})} F(x, \delta )= \frac{1}{\check{\lambda}}>\frac{1}{\lambda}.\end{aligned} $$
(3.6)
Therefore, thanks to (3.5) and (3.6), one has that assumption (i) of Theorem 2.1 is satisfied.
Now, we prove the coercivity of the functional \(\mathcal{I}_{\lambda ,\mu}(u)\).
In view of condition (H2), we can choose a constant θ satisfying
$$ \limsup_{ \vert \xi \vert \to +\infty} \frac{\sup_{x \in \Omega}F(x,\xi )}{\xi ^{p}} < \theta < \frac{a_{1}}{pc} \biggl(\frac{M_{1}\gamma}{r}c_{\alpha '} \Vert b \Vert _{\alpha}+ \frac{M_{2}\gamma ^{s}}{rs}c_{\alpha 's} \Vert b \Vert _{\alpha}\biggr), $$
(3.7)
then, there exists a function \(k_{\theta}(x)\in L^{1}(\Omega )\) such that
$$ F(x,\xi )\leq \theta \vert \xi \vert ^{p}+k_{\theta}(x), \quad \forall x \in \Omega , \xi \in \mathbb{R}.$$
(3.8)
Combining (3.7), (3.8) with the Hölder inequality, we have
$$\begin{aligned} \mathcal{I}_{\lambda ,\mu}(u)&=\frac{1}{\lambda}\Phi (u)-\Psi (u) \\ & =\frac{1}{\lambda} \int _{\Omega}\mathcal{A}(x, \nabla u) \,dx + \frac{1}{p} \int _{\Omega} a(x) \vert u \vert ^{p} \,dx - \int _{\Omega} F(x, u) \,dx - \frac{\mu}{\lambda} \int _{\Gamma _{2}}G(x,u)\,d\sigma \\ &\geq \frac{a_{1}}{\lambda p} \Vert u \Vert ^{p}-\theta \Vert u \Vert _{p}^{p}- \Vert k_{ \theta} \Vert _{1}-\frac{\mu}{\lambda q} \int _{\partial \Omega} h \vert u \vert ^{q}\,d\sigma \\ &\geq \frac{a_{1}}{p} \biggl(\frac{M_{1}}{r}c_{\alpha '} \Vert b \Vert _{\alpha}\biggl( \frac{pr}{a_{1}} \biggr)^{\frac{1}{p}}+\frac{M_{2}}{rs}c_{\alpha 's} \Vert b \Vert _{ \alpha}\biggl(\frac{pr}{a_{1}}\biggr)^{\frac{s}{p}}\biggr) \Vert u \Vert ^{p}-\theta T^{p} \vert \Omega \vert ^{\frac{p^{*}-p}{p^{*}}} \Vert u \Vert ^{p} \\ &\quad{}- \Vert k_{\theta} \Vert _{1}- \frac{\mu }{\lambda q} \Vert h \Vert _{\beta , \partial \Omega} \Vert u \Vert _{\beta 'q,\partial \Omega}^{q} \\ & \geq \frac{a_{1}}{p} \biggl(\frac{M_{1}}{r}c_{\alpha '} \Vert b \Vert _{\alpha}\biggl( \frac{pr}{a_{1}} \biggr)^{\frac{1}{p}}+\frac{M_{2}}{rs}c_{\alpha 's} \Vert b \Vert _{ \alpha}\biggl(\frac{pr}{a_{1}}\biggr)^{\frac{s}{p}}\biggr) \Vert u \Vert ^{p} \\ &\quad {}-\theta c \Vert u \Vert ^{p}- \Vert k_{\theta} \Vert _{1}-\frac{c_{\beta 'q}^{q}\mu }{\lambda q} \Vert h \Vert _{ \beta ,\partial \Omega} \Vert u \Vert ^{q} \\ & =\biggl(\frac{a_{1}}{pc} \biggl(\frac{M_{1}\gamma}{r}c_{\alpha '} \Vert b \Vert _{\alpha}+ \frac{M_{2}\gamma ^{s}}{rs}c_{\alpha 's} \Vert b \Vert _{\alpha}\biggr) -\theta \biggr) c \Vert u \Vert ^{p} \\ &\quad {}- \Vert k_{\theta} \Vert _{1}- \frac{c_{\beta 'q}^{q}\mu }{\lambda q} \Vert h \Vert _{ \beta ,\partial \Omega} \Vert u \Vert ^{q}. \end{aligned}$$
Thus, the coercivity of \(\mathcal{I}_{\lambda ,\mu}(u)\) is obtained according to (3.7) and \(q< p\). Hence, combining Lemma 2.2 with Lemma 2.3, Theorem 2.1 ensures the conclusion. □
As special cases of Theorem 3.1, we can obtain the following results.
Theorem 3.2
Assume conditions \((\mathbf{f})\), \((\mathbf{g})\), and
- \((H_{1})\):
-
\(F(x,\xi )\geq 0\), \(\forall (x,\xi )\in B(x_{0},d) \times [0,\delta ]\);
- \((H_{2})'\):
-
\(\limsup_{|\xi |\to +\infty} \frac{\sup_{x \in \Omega}F(x,\xi )}{\xi ^{p}}=0\)
hold, then for every \(\lambda \in (\check{\lambda},\hat{\lambda}) \), when \(\mu \in [0, \frac{q(rs-\lambda s \gamma M_{1}c_{\alpha '}\|b\|_{\alpha}-\lambda \gamma ^{s} M_{2}c_{\alpha 's}\|b\|_{\alpha})}{\gamma ^{q}c_{q}h_{0}s})\), where \(h_{0}= \operatorname{ess}\sup_{\Gamma _{2}}h(x) \), the problem (1.1) has at least three weak solutions.
Proof
We only need to prove the coercivity of the functional \(\mathcal{I}_{\lambda ,\mu}(u)\).
Fix \(0\le \varepsilon \le \frac{a_{1}}{\lambda pc}\). In view of condition \((H_{2})'\), there is a function \(k_{\varepsilon}(x)\in L^{1}(\Omega )\) such that
$$ F(x,\xi )\leq \varepsilon \vert \xi \vert ^{p}+k_{\varepsilon}(x), \quad \forall x \in \Omega , \xi \in \mathbb{R}.$$
(3.9)
Combining (3.9) with the Hölder inequality, we have
$$\begin{aligned} \mathcal{I}_{\lambda ,\mu}(u)&=\frac{1}{\lambda}\Phi (u)-\Psi (u) \\ & =\frac{1}{\lambda} \int _{\Omega}\mathcal{A}(x, \nabla u) \,dx + \frac{1}{p} \int _{\Omega} a(x) \vert u \vert ^{p} \,dx - \int _{\Omega} F(x, u) \,dx - \frac{\mu}{\lambda} \int _{\Gamma _{2}}G(x,u)\,d\sigma \\ &\geq \frac{a_{1}}{\lambda p} \Vert u \Vert ^{p}-\varepsilon \Vert u \Vert _{p}^{p}- \Vert k_{ \varepsilon} \Vert _{1}-\frac{c_{2}\mu h_{0}}{\lambda q} \Vert u \Vert ^{q} \\ &\geq \frac{a_{1}}{\lambda p} \Vert u \Vert ^{p}-\varepsilon c \Vert u \Vert ^{p}- \Vert k_{ \varepsilon} \Vert _{1}-\frac{c_{2}\mu h_{0}}{\lambda q} \Vert u \Vert ^{q}. \end{aligned}$$
Thus, the coercivity of \(\mathcal{I}_{\lambda ,\mu}(u)\) is obtained according to (3.7) and \(q< p\), \(0< \varepsilon < \frac{a_{1}}{\lambda pc}\). Hence, combining Lemma 2.2 with Lemma 2.3, Theorem 2.1 ensures the conclusion. □
Suppose there exist two positive constants δ, r, with \(r= \frac{a_{1}}{p}(\frac{2}{d})^{p-N}\delta ^{p}(2^{N}-1)|B(0,1)|\), such that
$$ \begin{aligned} &\frac{pd^{p}C}{2^{p-N}\delta ^{p}(a_{0}+a_{2}C)d^{N}(2^{N}-1)} \operatorname*{ess\, inf}_{ B(x_{0},\frac{d}{2})} F(x, \delta ) \\ &\quad := \frac{1}{\check{\alpha}} > \frac{M_{1}\gamma}{r}c_{1}+\frac{M_{2}\gamma ^{s}}{rs}c_{s} := \frac{1}{\hat{\alpha}}, \end{aligned} $$
(3.10)
where \(\gamma =(\frac{pr}{a_{1}})^{\frac{1}{p}}\), \(C=(\frac{N-p}{p})^{p}\).
Let \(c_{1}\) be the embedding constant of the compact embedding \(W^{1,p}(\Omega )\hookrightarrow L^{1}(\Omega )\); and \(c_{s}\) be the embedding constant of the compact embedding \(W^{1,p}(\Omega )\hookrightarrow L^{s}(\Omega )\), \(1< s\leq p\).
Similarly, we can obtain the following two theorems as special cases of Theorem 3.1.
Theorem 3.3
Assume the condition \((\mathbf{g})\), and
- \((H_{1})'\):
-
\(|f(u)|\leq M_{1}+M_{2}|u|^{s-1}\), \(\forall u\in \mathbb{R}\), \(1< s\le p\), \(F(\xi )\geq 0\), \(\forall \xi \in [0,\delta ]\);
- \((H_{2})''\):
-
$$ \limsup_{|\xi |\to +\infty} \frac{F(\xi )}{\xi ^{p}} \le \frac{a_{1}}{pc} \biggl(\frac{M_{1}\gamma}{r}c_{1}+ \frac{M_{2}\gamma ^{s}}{rs}c_{s}\biggr), $$
where \(c=|\Omega |^{\frac{p^{*}-p}{p^{*}}}T^{p}\), \(r=\frac{a_{1}}{p}( \frac{2}{d})^{p-N}\delta ^{p}(2^{N}-1)|B(0,1)|\)
hold, then for every \(\lambda \in (\check{\alpha},\hat{\alpha}) \), when \(\mu \in [0, \frac{q(rs-\lambda s \gamma M_{1}c_{1}-\lambda \gamma ^{s} M_{2}c_{s})}{\gamma ^{q}c_{q}h_{0}s})\), where \(h_{0}= \operatorname{ess}\sup_{\Gamma _{2}}h(x) \), the following elliptic problem
$$\begin{aligned} \textstyle\begin{cases} -\operatorname{div}\mathbf{A}(x,\nabla u) + \frac{ a(x)}{ \vert x \vert ^{p}} \vert u \vert ^{p-2}u= \lambda f(u) & \textit{in } \Omega , \\ u=0 & \textit{on } \Gamma _{1}, \\ \mathbf{A}(x,\nabla u)\cdot \nu =\mu g(x,\gamma (u))& \textit{on } \Gamma _{2}, \end{cases}\displaystyle \end{aligned}$$
(3.11)
has at least three weak solutions.
Theorem 3.4
Assume the condition \((\mathbf{g})\), and
- \((H_{1})'\):
-
\(|f(u)|\leq M_{1}+M_{2}|u|^{s-1}\), \(\forall u\in \mathbb{R}\), \(1< s\le p\), \(F(\xi )\geq 0\), \(\forall \xi \in [0,\delta ]\);
- \((H_{2})'''\):
-
$$ \limsup_{|\xi |\to +\infty} \frac{F(\xi )}{\xi ^{p}}=0 $$
hold, then for every \(\lambda \in (\check{\alpha},\hat{\alpha}) \), when \(\mu \in [0, \frac{q(rs-\lambda s \gamma M_{1}c_{1}-\lambda \gamma ^{s} M_{2}c_{s})}{\gamma ^{q}c_{q}h_{0}s})\), where \(h_{0}= \operatorname{ess}\sup_{\Gamma _{2}}h(x) \), the following elliptic problem
$$\begin{aligned} \textstyle\begin{cases} -\operatorname{div}\mathbf{A}(x,\nabla u) + \frac{ a(x)}{ \vert x \vert ^{p}} \vert u \vert ^{p-2}u= \lambda f(u) & \textit{in } \Omega , \\ u=0 & \textit{on } \Gamma _{1}, \\ \mathbf{A}(x,\nabla u)\cdot \nu =\mu g(x,\gamma (u)) & \textit{on } \Gamma _{2}, \end{cases}\displaystyle \end{aligned}$$
(3.12)
has at least three weak solutions.