In this section, we formulate our main results.
First we give the following application of Theorem 2.1 as our first main result. Let
$$\begin{aligned} \lambda_{1}&=\inf \biggl\{ \frac{\int_{\Omega }\vert \Delta u(x) \vert ^{p} \,{\mathrm{d}}x+ \widehat{M} [ \int_{\Omega }\vert \nabla u(x) \vert ^{p} \,{\mathrm{d}}x ] +\varrho \int_{\Omega }\vert u(x) \vert ^{p}\,{\mathrm{d}}x}{p \int_{\Omega }F(x,u(x))\,\mathrm{d}x}\,\colon \\ & \quad {}u\in {\mathrm{E}}, \int_{\Omega }F \bigl(x,u(x) \bigr)\,\mathrm{d}x>0 \biggr\} \end{aligned}$$
and \(\lambda_{2}= \frac{1}{\max \{ 0,\lambda_{0},\lambda_{\infty } \} }\), where
$$\begin{aligned} \lambda_{0}=\limsup_{\vert u \vert \rightarrow 0} \frac{p\int_{\Omega }F(x,u(x)) \,\mathrm{d}x}{\int_{\Omega }\vert \Delta u(x) \vert ^{p}\,{\mathrm{d}}x+ \widehat{M} [ \int_{\Omega }\vert \nabla u(x) \vert ^{p} \,{\mathrm{d}}x ] + \varrho \int_{\Omega }\vert u(x) \vert ^{p}\,{\mathrm{d}}x} \end{aligned}$$
and
$$\begin{aligned} \lambda_{\infty }=\limsup_{\Vert u \Vert \rightarrow + \infty }\frac{p \int_{\Omega }F(x,u(x))\,\mathrm{d}x}{\int_{\Omega }\vert \Delta u(x) \vert ^{p}\,{\mathrm{d}}x+ \widehat{M} [ \int_{\Omega }\vert \nabla u(x) \vert ^{p} \,{\mathrm{d}}x ] +\varrho \int_{\Omega }\vert u(x) \vert ^{p}\,{\mathrm{d}}x}. \end{aligned}$$
Theorem 3.1
Assume that
-
\((\mathcal{A}_{1})\)
:
-
there exists a constant
\(\varepsilon >0\)
such that
$$ \max \biggl\{ \limsup_{u\rightarrow 0}\frac{\max_{x\in \overline{\Omega }}F(x,u(x))}{\vert u \vert ^{p}} ,\,\limsup _{\vert u \vert \to +\infty } \frac{ \max_{x\in \overline{\Omega }}F(x,u(x))}{\vert u \vert ^{p}} \biggr\} < \varepsilon ; $$
-
\((\mathcal{A}_{2})\)
:
-
there exists a function
\(w\in {\mathrm{E}}\)
such that
$$ K_{w}:= \int_{\Omega } \bigl\vert \Delta w(x) \bigr\vert ^{p}\,{\mathrm{d}}x+\widehat{M} \biggl[ \int_{\Omega } \bigl\vert \nabla w(x) \bigr\vert ^{p}\,{ \mathrm{d}}x \biggr] + \varrho \int_{\Omega } \bigl\vert w(x) \bigr\vert ^{p}\,{\mathrm{d}}x\neq 0 $$
and
$$ \varepsilon < \frac{M^{-}\int_{\Omega }F(x,w(x))\,\mathrm{d}x}{k^{p}K_{w}}. $$
Then, for each compact interval
\([c,d]\subset (\lambda_{1},\lambda _{2})\), there exists
\(R> 0\)
with the following property: for every
\(\lambda \in [c,d]\)
and every
\(\mathrm{L}^{1}\)-Carathéodory function
\(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists
\(\gamma >0\)
such that, for each
\(\mu \in [0,\gamma ]\), the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three weak solutions whose norms in E are less than
R.
Proof
Take \(X=\mathrm{E}\). Clearly, E is a separable and uniformly convex Banach space. Let the functionals Φ, J and Ψ be as given in (2.2), (2.3) and (2.4), respectively. The functional Φ is \(\mathrm{C}^{1}\), and by [38], Proposition 2.3, its derivative admits a continuous inverse on \(X^{*}\). Moreover, since \(m_{0}\leq K(s)\leq m_{1}\) for all \(s\in [0,+\infty[\), from (2.2) we have
$$\begin{aligned} \frac{M^{-}}{p}\Vert u \Vert ^{p}\leq \Phi (u) \leq \frac{M^{+}}{p}\Vert u \Vert ^{p} \end{aligned}$$
(3.1)
for all \(u \in X\), it follows \(\lim_{\Vert u \Vert \to +\infty }\Phi (u)=+ \infty \), namely, Φ is coercive. Furthermore, Φ is sequentially weakly lower semicontinuous. Moreover, let A be a bounded subset of X. That is, there exists a constant \(c>0\) such that \(\Vert u \Vert \leq c\) for each \(u\in A\). Then, by (3.1), we have
$$ \bigl\vert \Phi (u) \bigr\vert \leq \frac{M^{+}}{p}c^{p}. $$
Hence Φ is bounded on each bounded subset of X. Furthermore, by Remark 2.1, \(\Phi \in \mathcal{W}_{X}\). The functionals J and Ψ are two \(\mathrm{C}^{1}\)-functionals with compact derivatives. Moreover, Φ has a strict local minimum 0 with \(\Phi (0)=J(0)=0\). In view of \((\mathcal{A}_{1})\), there exist \(\tau_{1}\), \(\tau_{2}\) with \(0<\tau_{1}<\tau_{2}\) such that
$$ F(x,u)\leq \varepsilon \vert u \vert ^{p} $$
(3.2)
for every \(x\in \Omega \) and every u with \(\vert u \vert \in [0, \tau_{1}) \cup (\tau_{2}, +\infty )\). Since \(F(x,u)\) is continuous on \(\Omega \times \mathbb{R}\), it is bounded on \(x\in \Omega \) and \(\vert u \vert \in [\tau_{1},\tau_{2}]\). Thus we can choose \(\eta >0\) and \(\upsilon >p\) such that
$$ F(x,u)\leq \varepsilon \vert u \vert ^{p}+\eta \vert u \vert ^{\upsilon } $$
for all \((k,u)\in \Omega \times \mathbb{R}\). So, by (2.1), we have
$$ J(u)\leq \varepsilon k^{p}\Vert u \Vert ^{p}+ \eta k^{\upsilon }\Vert u \Vert ^{\upsilon } $$
(3.3)
for all \(u\in X\). Hence, from (3.3) we have
$$ \limsup_{\vert u \vert \rightarrow 0}\frac{J(u)}{\Phi (u)}\leq \frac{pk^{p} \varepsilon }{M^{-}}. $$
(3.4)
Moreover, by using (3.2), for each \(u\in {\mathrm{E}}\setminus \{0 \}\), we obtain
$$\begin{aligned} \frac{J(u)}{\Phi (u)} &=\frac{\int_{\vert u \vert \leq \tau_{2}}F(x,u)\,\mathrm{d}x}{ \Phi (u)} +\frac{\int_{\vert u \vert >\tau_{2}}F(x,u)\,\mathrm{d}x}{\Phi (u)} \\ &\leq \frac{\sup_{x\in \Omega ,\vert u \vert \in [0,\tau_{2}]}F(x,u)}{\Phi (u)} +\frac{\varepsilon \Vert u \Vert ^{p}}{k^{p}\Phi (u)}\leq \frac{p\sup_{x\in \Omega ,\vert u \vert \in [0,\tau_{2}]}F(x,u)}{\Vert u \Vert ^{p}}+ \frac{pk^{p}\varepsilon }{M^{-}}. \end{aligned}$$
So, we get
$$ \limsup_{\Vert u \Vert \rightarrow + \infty }\frac{J(u)}{\Phi (u)}\leq \frac{pk ^{p}\varepsilon }{M^{-}}. $$
(3.5)
In view of (3.4) and (3.5), we have
$$ \rho =\max \biggl\{ 0,\,\limsup_{\Vert u \Vert \rightarrow + \infty } \frac{J(u)}{ \Phi (u)},\, \limsup_{u\rightarrow 0}\frac{J(u)}{\Phi (u)} \biggr\} \leq \frac{pk^{p}\varepsilon }{M^{-}}. $$
(3.6)
Assumption \((\mathcal{A}_{2})\) in conjunction with (3.6) yields
$$\begin{aligned} \sigma &=\sup_{u\in \Phi^{-1}(0,+\infty )}\frac{J(u)}{\Phi (u)} \\ &= \sup_{X\setminus \{0\}}\frac{J(u)}{\Phi (u)}\geq \frac{\int_{\Omega }F(x,w(x))}{ \Phi (w(x))} \\ &= \frac{p\int_{\Omega }F(x,w(x))\,\mathrm{d}x}{K_{w}}>\frac{pk^{p}\varepsilon }{M^{-}}\geq \rho . \end{aligned}$$
Thus, all the hypotheses of Theorem 2.1 are satisfied. Clearly, \(\lambda_{1}=\frac{1}{\sigma }\) and \(\lambda_{2}=\frac{1}{ \rho }\). Then, using Theorem 2.1, for each compact interval \([c,d]\subset (\lambda_{1},\lambda_{2})\), there exists \(R> 0\) with the following property: for every \(\lambda \in [c,d]\) and every \(\mathrm{L}^{1}\)-Carathéodory function \(g:\Omega \times \mathbb{R} \to \mathbb{R}\), there exists \(\gamma >0\) such that, for each \(\mu \in [0,\gamma ]\), the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three solutions whose norms in X are less than R. □
Another announced application of Theorem 2.1 reads as follows.
Theorem 3.2
Assume that
$$ \max \biggl\{ \limsup_{u\rightarrow 0}\frac{\max_{x\in \overline{\Omega }}F(x,u(x))}{\vert u \vert ^{p}}, \,\limsup_{\vert u \vert \to +\infty } \frac{ \max_{x\in \overline{\Omega }}F(x,u(x))}{\vert u \vert ^{p}} \biggr\} \leq 0 $$
(3.7)
and
$$ \sup_{u\in {\mathrm{E}}}\frac{M^{-}\int_{\Omega }F(x,u(x))\,\mathrm{d}x}{k^{p}( \int_{\Omega }\vert \Delta u(x) \vert ^{p}\,{\mathrm{d}}x+ \widehat{M} [ \int _{\Omega }\vert \nabla u(x) \vert ^{p}\,{\mathrm{d}}x ] +\varrho \int_{\Omega }\vert u(x) \vert ^{p}\,{\mathrm{d}}x)}>0. $$
(3.8)
Then, for each compact interval
\([c,d]\subset (\lambda_{1},+\infty )\), there exists
\(R>0\)
with the following property: for every
\(\lambda \in [c,d]\)
and every
\(\mathrm{L}^{1}\)-Carathéodory function
\(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists
\(\gamma >0\)
such that for each
\(\mu \in [0,\gamma ]\), the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three weak solutions whose norms in E are less than
R.
Proof
In view of (3.7), there exist an arbitrary \(\varepsilon >0\) and \(\tau_{1}\), \(\tau_{2}\) with \(0<\tau_{1}<\tau_{2}\) such that
$$ F(x,u)\leq \varepsilon \vert u \vert ^{p} $$
for every \(x\in \Omega \) and every u with \(\vert u \vert \in [0, \tau_{1}) \cup (\tau_{2}, +\infty )\). Since \(F(x,u)\) is continuous on \(\Omega \times \mathbb{R}\), it is bounded on \(x\in \Omega \) and \(\vert u \vert \in [\tau_{1},\tau_{2}]\). Thus we can choose \(\eta >0\) and \(\upsilon >p\) in a manner that
$$ F(x,u)\leq \varepsilon \vert u \vert ^{p}+\eta \vert u \vert ^{\upsilon } $$
for all \((x,u)\in \Omega \times \mathbb{R}\). So, by the same process as that in the proof of Theorem 3.1, we have Relations (3.4) and (3.5). Since ε is arbitrary, (3.4) and (3.5) give
$$ \max \biggl\{ 0,\,\limsup_{\Vert u \Vert \rightarrow + \infty }\frac{J(u)}{ \Phi (u)},\, \limsup _{u\rightarrow 0}\frac{J(u)}{\Phi (u)} \biggr\} \leq 0. $$
Then, with the notation of Theorem 2.1, we have \(\rho =0\). By (3.8), we also have \(\sigma >0\). In this case, clearly \(\lambda_{1}=\frac{1}{\sigma }\) and \(\lambda_{2}=+\infty \). Thus, by using Theorem 2.1, the result is achieved. □
Now we formulate the following applications of Theorem 2.2 as our second main result.
Theorem 3.3
Assume that
-
\((\mathcal{B}_{1})\)
:
-
there exist two positive functions
\(\nu ,\xi \in {\mathrm{L}}^{1}(\Omega ,\mathbb{R})\)
and
\(\alpha \in [0,p)\)
such that
$$ \bigl\vert F(x,u) \bigr\vert \leq \nu (x)\vert u \vert ^{\alpha }+ \xi (x)\quad \textit{for all } u \in \mathbb{R} \textit{ and } x\in \Omega ; $$
-
\((\mathcal{B}_{2})\)
:
-
there exist a positive constant
r
and
\(w\in {\mathrm{E}}\)
such that
\(K_{w}>r\), where
\(K_{w}\)
is as given in Assumption
\((\mathcal{A}_{2})\)
in Theorem
3.1, and
$$ \max_{x\in \overline{\Omega },\vert u \vert \leq \sqrt[p]{\frac{pr}{M^{-}}}}F(x,u)< \frac{r \int_{\Omega }F(x,w(x))\,\mathrm{d}x}{\operatorname{meas}(\Omega )K_{w}}. $$
Then there exist a nonempty open set
\(A\subset [0,+\infty )\)
and a positive number
\(R'\)
with the following property: for every
\(\lambda \in A\)
and every continuous function
\(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists
\(\delta >0\)
such that, for each
\(\mu \in [0,\delta ]\), the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three weak solutions whose norms in E are less than
\(R'\).
Proof
Take \(X=\mathrm{E}\). Let the functionals Φ and J be as given in (2.2) and (2.3), respectively. For any \(\lambda \geq 0\) and \(u\in {\mathrm{E}}\), by \((\mathcal{B}_{1})\), we have
$$\begin{aligned} \Phi (u)-\lambda J(u) &=\frac{1}{p} \int_{\Omega } \bigl\vert \Delta u(x) \bigr\vert ^{p}\,{\mathrm{d}}x+ \frac{1}{p}\widehat{M} \biggl[ \int_{\Omega } \bigl\vert \nabla u(x) \bigr\vert ^{p}\,{ \mathrm{d}}x \biggr] \\ &\quad {}+\frac{\varrho }{p} \int_{\Omega } \bigl\vert u(x) \bigr\vert ^{p}\,{\mathrm{d}}x-\lambda \int_{\Omega }F \bigl(x,u(x) \bigr)\,\mathrm{d}x \\ & \geq \frac{M^{-}}{p}\Vert u \Vert ^{p} -\lambda \int_{\Omega } \bigl( \nu (x) \bigl\vert u(x) \bigr\vert ^{ \alpha }+\xi (x) \bigr) \,\mathrm{d}x \\ &\geq \frac{M^{-}}{p}\Vert u \Vert ^{p} -\lambda k^{\alpha }\Vert u \Vert ^{\alpha }\Vert \nu \Vert _{\mathrm{L}^{1}}-\lambda \Vert \xi \Vert _{\mathrm{L}^{1}}. \end{aligned}$$
Since \(\alpha < p\), one has \(\lim_{\Vert u \Vert \to +\infty }(\Phi (u)-\lambda J(u))=+\infty \) for all \(\lambda \geq 0\). If \(\Phi (u)\leq r\), we have \(\Vert u \Vert \leq \sqrt[p]{\frac{pr}{M^{-}}}\), that is,
$$ \Phi^{-1}(-\infty ,r]\subseteq \biggl\{ u\in X:\max _{x\in \overline{\Omega }} \bigl\vert u(x) \bigr\vert \leq \sqrt[p]{ \frac{pr}{M^{-}}} \biggr\} . $$
Therefore,
$$\begin{aligned} \sup_{u\in \Phi^{-1}(-\infty ,r]}J(u) &\leq \max_{\vert u \vert \leq \sqrt[p]{\frac{pr}{M^{-}}}}J(u) \\ &=\max_{\vert u \vert \leq \sqrt[p]{\frac{pr}{M^{-}}}} \int_{\Omega }F \bigl(x,u(x) \bigr) \,\mathrm{d}x \\ & \leq {\operatorname{meas}}(\Omega ) \max_{x\in \overline{\Omega },\vert u \vert \leq \sqrt[p]{\frac{pr}{M^{-}}}}F(x,u). \end{aligned}$$
(3.9)
It is clear that \(\Phi (0)=J(0)=0\) and owing to \((\mathcal{B}_{2})\) and (3.9), \(\Phi (w)>r\) and
$$ \sup_{u\in \Phi^{-1}(-\infty ,r]}J(u)< r\frac{J(w)}{\Phi (w)}. $$
Thus we can fix ρ such that
$$ \sup_{u\in \Phi^{-1}(-\infty ,r]}J(u)< \rho < r\frac{J(w)}{\Phi (w)}. $$
Now, from Proposition 2.3, we obtain
$$ \sup_{\lambda \geq 0}\inf_{u\in {\mathrm{E}}} \bigl(\Phi (u)+\lambda \bigl(\rho -J(u) \bigr) \bigr)< \inf_{u\in {\mathrm{E}}}\sup _{\lambda \geq 0} \bigl(\Phi (u)+\lambda \bigl(\rho -J(u) \bigr) \bigr). $$
Therefore, by Theorem 2.2, for each compact interval \([a,b]\subseteq (\lambda_{1},\lambda_{2})\), there exists \(R'>0\) with the following property: for every \(\lambda \in [a,b]\) and every \(\mathrm{L}^{1}\)-Carathéodory function \(g:\Omega \times \mathbb{R} \to \mathbb{R}\), there exists \(\delta >0\) such that, for each \(\mu \in [0,\delta ]\), \(\Phi '(u)-\lambda J'(u)-\mu \Psi '(u)=0\) has at least three solutions in E. Hence, the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three weak solutions whose norms are less than \(R'\). □
Now fix \(x^{0}\in \Omega \) and pick \(s>0\) such that \(B(x^{0},s)\subset \Omega \), where \(B(x^{0},s)\) denotes the ball with center at \(x^{0}\) and radius of s. Put
$$\begin{aligned}& \vartheta_{1}:= \frac{2\pi^{\frac{N}{2}}}{\Gamma (\frac{N}{2})} \int_{\frac{s}{2}}^{s} \biggl\vert \frac{12(N+1)}{s^{3}}r- \frac{24 N}{s^{2}}+\frac{9(N-1)}{s}\frac{1}{r} \biggr\vert ^{p} r^{N-1}\,{\mathrm{d}}r , \\& \vartheta_{2}:= \int_{ B(x^{0},s)\setminus B(x^{0},\frac{s}{2})} \Biggl[ \sum_{i=1}^{N} \biggl( \frac{12(x_{i}-x^{0}_{i})}{s^{3}}-\frac{24(x_{i}-x ^{0}_{i})}{s^{2}}+\frac{9(x_{i}-x^{0}_{i})}{s\ell } \biggr)^{2} \Biggr]^{\frac{p}{2}}\,{\mathrm{d}}x, \\& \vartheta_{3}:=\frac{2\pi^{\frac{N}{2}}}{\Gamma (\frac{N}{2})} \biggl[ \frac{( \frac{s}{2})^{N}}{N} + \int_{\frac{s}{2}}^{s} \biggl\vert \frac{4}{s^{3}}r^{3} -\frac{12}{s^{2}} r^{2} +\frac{9}{s}r -1 \biggr\vert ^{p} r^{N-1}\,{\mathrm{d}}r \biggr] , \end{aligned}$$
where Γ denotes the gamma function, and
$$ L:=\vartheta_{1}+\vartheta_{2}+ \vartheta_{3}. $$
(3.10)
The next two theorems provide sufficient conditions for applying Theorems 3.1 and 3.3, which does not require to know a test function w satisfying \((\mathcal{A}_{2})\) and \((\mathcal{B} _{2})\), respectively.
Theorem 3.4
Assume that Assumption
\((\mathcal{A}_{1})\)
in Theorem
3.1
holds and there exists a positive constant
d
such that
-
\((\mathcal{A}_{3})\)
:
-
\(F(x,t)\geq 0\)
for each
\(x\in B(x^{0},s) \setminus B(x^{0},\frac{s}{2})\), \(t\in [0,d]\);
-
\((\mathcal{A}_{4})\)
:
-
\(\vartheta_{1}d^{p} +\widehat{M}(\vartheta _{2}d^{p}) +\varrho \vartheta_{3}d^{p}\neq 0\)
and
\(\varepsilon <\frac{pM ^{-}\int_{B(x^{0},\frac{s}{2})}F(x,d)\,\mathrm{d}x}{k^{p}(\vartheta_{1}d ^{p}+\widehat{M}(\vartheta_{2}d^{p})+\varrho \vartheta_{3}d^{p})}\).
Then, for each compact interval
\([c,d]\subset (\lambda_{1},\lambda _{2})\), there exists
\(R> 0\)
with the following property: for every
\(\lambda \in [c,d]\)
and every
\(\mathrm{L}^{1}\)-Carathéodory function
\(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists
\(\gamma >0\)
such that, for each
\(\mu \in [0,\gamma ]\), the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three weak solutions whose norms in E are less than
R.
Proof
We claim that all the assumptions of Theorem 3.1 are fulfilled by choosing w as follows:
$$ w(x):=\left \{\textstyle\begin{array}{l@{\quad}l} 0 &\mbox{if } x \in \overline{\Omega }\setminus B(x^{0} , s), \\ d ( \frac{4}{s^{3}}\ell^{3} -\frac{12}{s^{2}}\ell^{2} + \frac{9}{s}\ell -1 ) &\mbox{if } x\in B(x^{0},s)\setminus B(x ^{0},\frac{s}{2}), \\ d &\mbox{if } x \in B(x^{0},\frac{ s}{2}) \end{array}\displaystyle \right . $$
(3.11)
where \(\ell =\operatorname{dist}(x,x^{0})=\sqrt{\sum_{i=1}^{N} ( x_{i}-x_{i}^{0} ) ^{2}}\) (see [31, 36]). We have
$$\begin{aligned}& \frac{\partial w(x)}{\partial x_{i}} = \textstyle\begin{cases} 0&\mbox{if } x \in \overline{\Omega }\setminus B(x^{0},s)\cup B(x^{0},\frac{s}{2}), \\ d ( \frac{12 \ell (x_{i}-x_{i}^{0})}{s^{3}}-\frac{24(x_{i}-x_{i} ^{0})}{s^{2}}+\frac{9}{s}\frac{(x_{i}-x_{i}^{0})}{\ell } ) & \mbox{if } x\in B(x^{0},s)\setminus B(x^{0},\frac{s}{2}) \end{cases}\displaystyle \end{aligned}$$
and
$$\begin{aligned}& \frac{\partial^{2} w(x)}{\partial x_{i}^{2}}= \textstyle\begin{cases} 0&\mbox{if } x \in \overline{\Omega }\setminus B(x^{0},s)\cup B(x^{0},\frac{s}{2}), \\ d ( \frac{12}{s^{3}}\frac{(x_{i}-x_{i}^{0})^{2}+\ell^{2}}{\ell }-\frac{24}{s ^{2}}+\frac{9}{s}\frac{\ell^{2}-(x_{i}-x_{i}^{0})^{2}}{\ell^{3}} ) & \mbox{if } x\in B(x^{0},s)\setminus B(x^{0},\frac{s}{2}), \end{cases}\displaystyle \end{aligned}$$
and so that
$$ \sum_{i=1}^{N}\frac{\partial^{2} w(x)}{\partial x_{i}^{2}}= \textstyle\begin{cases} 0&\mbox{if } x \in \overline{\Omega }\setminus B(x^{0},s)\cup B(x^{0},\frac{s}{2}), \\ d ( \frac{12l(N+1)}{s^{3}}-\frac{24N}{s^{2}}+\frac{9}{s}\frac{N-1}{ \ell } ) & \mbox{if } x\in B(x^{0},s)\setminus B(x^{0},\frac{s}{2}). \end{cases} $$
In particular, since
$$\begin{aligned}& \int_{\Omega } \bigl\vert \Delta w(x) \bigr\vert ^{p}\,{\mathrm{d}}x=d^{p}\frac{2\pi^{\frac{N}{2}}}{\Gamma (\frac{N}{2})} \int_{\frac{s}{2}}^{s} \biggl\vert \frac{12(N+1)}{s^{3}}r- \frac{24N}{s^{2}}+\frac{9(N-1)}{s}\frac{1}{r} \biggr\vert ^{p}r^{N-1} \,{\mathrm{d}}r, \\& \int_{\Omega } \bigl\vert \nabla w(x) \bigr\vert ^{p}\,{\mathrm{d}}x \\& \quad =\int_{B(x^{0},s)\backslash B(x^{0},\frac{s}{2})} \Biggl[ \sum_{i=1}^{N}d ^{2} \biggl( \frac{12l(x_{i}-x^{0}_{i})}{s^{3}}- \frac{24(x_{i}-x^{0} _{i})}{s^{2}}+\frac{9}{s} \frac{(x_{i}-x^{0}_{i})}{l} \biggr) ^{2} \Biggr] ^{\frac{p}{2}}\,{\mathrm{d}}x \\& \quad = d^{p} \int_{B(x^{0},s)\backslash B(x^{0},\frac{s}{2})} \Biggl[ \sum_{i=1} ^{N} \biggl( \frac{12l(x_{i}-x^{0}_{i})}{s^{3}}- \frac{24(x_{i}-x^{0} _{i})}{s^{2}}+\frac{9}{s} \frac{(x_{i}-x^{0}_{i})}{l} \biggr) ^{2} \Biggr] ^{\frac{p}{2}}\,{\mathrm{d}}x \end{aligned}$$
and
$$ \int_{\Omega } \bigl\vert w(x) \bigr\vert ^{p}\,{\mathrm{d}}x=d^{p}\frac{2\pi^{\frac{N}{2}}}{ \Gamma (\frac{N}{2})} \biggl( \frac{(\frac{s}{2})^{N}}{N}+ \int_{ \frac{s}{2}}^{s} \biggl\vert \frac{4}{s^{3}}r^{3}- \frac{12}{s^{2}}r^{2}+\frac{9}{s}r-1 \biggr\vert ^{p}r^{N-1}\,{\mathrm{d}}r \biggr) . $$
It is easy to see that \(w\in {\mathrm{E}}\), and one has
$$\begin{aligned} \frac{d^{p}}{p}M^{-}L &\leq \frac{1}{p} \bigl( \vartheta_{1}d^{p} +m_{0}^{p-1} \vartheta_{2}d^{p}+ \varrho \vartheta_{3}d^{p} \bigr) \\ &\leq \Phi (w) \\ &= \frac{1}{p} \bigl(\vartheta_{1}d^{p} +\widehat{M} \bigl(\vartheta_{2}d^{p} \bigr) + \varrho \vartheta_{3}d^{p} \bigr) \\ & \leq \frac{1}{p} \bigl(\vartheta_{1}d^{p}+m_{1}^{p-1} \vartheta_{2}d^{p}+ \varrho \vartheta_{3}d^{p} \bigr) \\ &\leq \frac{d^{p}}{p}M^{+}L. \end{aligned}$$
(3.12)
From Assumptions \((\mathcal{A}_{3})\) and \((\mathcal{A}_{4})\) we observe that Assumption \((\mathcal{A}_{2})\) in Theorem 3.1 is satisfied. Hence, Theorem 3.1 follows the result. □
Theorem 3.5
Assume that Assumption
\((\mathcal{B}_{1})\)
in Theorem
3.3
and Assumption
\((\mathcal{A}_{3})\)
in Theorem
3.4
hold and there exist three positive constants
c, d
and
α
with
\(\sqrt[p]{L}d< c\), where
L
is given as in (3.10), and
\(\alpha \in [0,p)\)
such that
-
\((\mathcal{B}_{3})\)
:
-
\(\max_{x\in \overline{\Omega },\vert u \vert \leq c}F(x,u)<\frac{M ^{-}}{M^{+}{\operatorname{meas}}(\Omega )}\int_{B(x^{0},\frac{s}{2})}F(x,d)\,\mathrm{d}x\).
Then, there exist a nonempty open set
\(A\subset [0,+\infty )\)
and a positive number
\(R'\)
with the following property: for every
\(\lambda \in A\)
and every continuous function
\(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists
\(\delta >0\)
such that, for each
\(\mu \in [0,\delta ]\), the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three weak solutions whose norms in E are less than
\(R'\).
Proof
We claim that all the hypotheses of Theorem 3.1 are satisfied by choosing w as given in (3.11) and \(r<\frac{LM^{-}}{p}d ^{p}\). We observe that
$$ K_{w}=\frac{1}{p} \bigl(\vartheta_{1}d^{p} +\widehat{M} \bigl(\vartheta_{2}d^{p} \bigr)+ \varrho \vartheta_{3}d^{p} \bigr)\geq \frac{LM^{-}}{p}d^{p}>r, $$
where \(K_{w}\) is as given in Assumption \((\mathcal{A}_{1})\). Owing to \((\mathcal{B}_{3})\) and \(F(t,0)=0\), one has \(\int_{B(x^{0}, \frac{s}{2})}F(x,w(x))\,\mathrm{d}x>0\). So, by \((\mathcal{A}_{3})\), \((\mathcal{B}_{3})\) and (3.12), we have
$$\begin{aligned} r\frac{\int_{\Omega }F(x,w(x))\,\mathrm{d}x}{\operatorname{meas}(\Omega )K_{w}} &=\frac{LM ^{-}d^{p}}{p}\frac{p\int_{\Omega }F(x,w(x))\,\mathrm{d}x}{\operatorname{meas}( \Omega )(\theta_{1}d^{p} +\widehat{M}(\theta_{2}d^{p}) + \varrho \theta_{3}d^{p})} \\ &> \frac{M^{-}}{\operatorname{meas}(\Omega )M^{+}} \int_{B(x^{0},\frac{s}{2})}F(x,d) \,\mathrm{d}x \\ &> \max_{x\in \overline{\Omega },\vert u \vert \leq c}F(x,u) \\ &> \max_{x\in \overline{\Omega },\vert u \vert \leq \sqrt[p]{\frac{pr}{M^{-}}}}F(x,u). \end{aligned}$$
Thus, Assumption \((\mathcal{B}_{2})\) in Theorem 3.3 holds. Therefore, by Theorem 3.3, for each compact interval \([a,b]\subseteq (\lambda_{1},\lambda_{2})\), there exists \(R'>0\) with the following property: for every \(\lambda \in [a,b]\) and every \(\mathrm{L}^{1}\)-Carathéodory function \(g:\Omega \times \mathbb{R} \to \mathbb{R}\), there exists \(\delta >0\) such that, for each \(\mu \in [0,\delta ]\), \(\Phi '(u)-\lambda J'(u)-\mu \Psi '(u)=0\) has at least three solutions in E. Hence, the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three weak solutions whose norms are less than \(R'\). □
Remark 3.1
Clearly, Theorem 3.5 gives the result of at least three solutions for the problem (\(P^{f,g}_{\lambda ,\mu}\)) with \(F(x,u)\) being of subquadratic growth.
Remark 3.2
The statements of Theorems 3.4 and 3.5 depend upon the test function w defined by (3.11). If we take the other choices of w, we have another statement. For example, if \(x^{0} \in \Omega \) and we pick \(s>0\) such that \(B(x^{0},s)\subset \Omega \), where \(B(x^{0}, s)\) denotes the ball with center at \(x^{0}\) and radius of s, and
$$\begin{aligned}& \vartheta_{1}':=\frac{2^{5P+1}\pi^{N/2}d^{p}}{s^{4p}\Gamma (N/2)} \int_{s/2}^{s} \bigl\vert 2(N+2)r^{2}-3(N+1)sr+Nr^{2} \bigr\vert ^{p}r^{N+1}\,{\mathrm{d}}r, \\& \vartheta_{2}':= \biggl( \frac{32}{s^{4}} \biggr) ^{p} \int_{B(x^{0},s)\backslash B(x^{0},\frac{s}{2})} \Biggl[ \sum_{i=1}^{N} \bigl( (s-\ell ) (s-3\ell ) \bigl(x_{i}-x_{i}^{0} \bigr) \bigr) ^{2} \Biggr] ^{ \frac{p}{2}}\,{\mathrm{d}}x, \\& \vartheta_{3}':= \biggl( \frac{16}{s^{4}} \biggr) ^{p}\frac{2\pi^{ \frac{N}{2}}}{\Gamma (\frac{N}{2})} \biggl( \frac{(\frac{s}{2})^{N}}{N}+ \int_{\frac{s}{2}}^{s} \bigl\vert r^{2}(s-r) \bigr\vert ^{p}r^{N-1}\,{\mathrm{d}}r \biggr) \end{aligned}$$
and where Γ denotes the gamma function,
$$ L':=\vartheta_{1}'+\vartheta_{2}'+ \vartheta_{3}', $$
and we take
$$ w(x):=\left \{ \textstyle\begin{array}{l@{\quad}l} 0 &\mbox{if } x \in \overline{\Omega }\setminus B(x^{0} , s), \\ 16\frac{\ell^{2}}{s^{4}}(s- \ell )^{2}d &\mbox{if } x\in B(x^{0},s) \setminus B(x^{0},\frac{s}{2}), \\ d &\mbox{if } x \in B(x^{0},\frac{ s}{2}) \end{array}\displaystyle \right . $$
(3.13)
with \(\ell =\operatorname{dist}(x,x^{0})=\sqrt{\sum_{i=1}^{N} ( x _{i}-x_{i}^{0} ) ^{2}}\) (see [37]), then we have
$$\begin{aligned}& \frac{\partial w(x)}{\partial x_{i}}=\left \{ \textstyle\begin{array}{l@{\quad}l} 0&\mbox{if } x \in \overline{\Omega }\setminus B(x^{0},s)\cup B(x^{0},\frac{s}{2}), \\ \frac{32d}{s^{4}}(s-\ell )(s-3\ell )(x_{i}-x_{i}^{0})& \mbox{if } x\in B(x^{0},s)\setminus B(x^{0},\frac{s}{2}), \end{array}\displaystyle \right . \\& \frac{\partial^{2} w(x)}{\partial x_{i}^{2}}= \left \{ \textstyle\begin{array}{l@{\quad}l} 0&\mbox{if } x \in \overline{\Omega }\setminus B(x^{0},s)\cup B(x^{0},\frac{s}{2}), \\ \frac{32(x_{i}-x_{i}^{0})d}{s^{4}} ( \frac{6(x_{i}-x_{i}^{0})( \ell -4s)}{\ell }+s^{2}-4ls+3l^{2} ) & \mbox{if } x\in B(x^{0},s)\setminus B(x^{0},\frac{s}{2}) \end{array}\displaystyle \right . \end{aligned}$$
and
$$ \sum_{i=1}^{N}\frac{\partial^{2} w(x)}{\partial x_{i}^{2}}=\left \{ \textstyle\begin{array}{l@{\quad}l} 0&\mbox{if } x \in \overline{\Omega }\setminus B(x^{0},s)\cup B(x^{0},\frac{s}{2}), \\ 32d ( \frac{2(N+2)\ell^{2}-3s(N+1)\ell +Ns^{2}}{s^{4}} ) & \mbox{if } x\in B(x^{0},s)\setminus B(x^{0},\frac{s}{2}). \end{array}\displaystyle \right . $$
It is easy to see that \(w\in {\mathrm{E}}\) and, in particular, since
$$\begin{aligned} &\int_{\Omega } \bigl\vert \Delta w(x) \bigr\vert ^{p}\,{\mathrm{d}}x =\frac{2^{5P+1}\pi^{N/2}d^{p}}{s^{4p}\Gamma (N/2)}\int_{s/2}^{s} \bigl\vert 2(N+2)r^{2}-3(N+1)rs+Nr^{2}\bigr\vert ^{p}r^{N+1}\,{\mathrm{d}}r, \\ &\int_{\Omega } \bigl\vert \nabla w(x) \bigr\vert ^{p}\,{\mathrm{d}}x \\ &\quad = \int_{B(x^{0},s)\backslash B(x^{0},\frac{s}{2})} \Biggl[ \sum_{i=1}^{N} \biggl( \frac{32d}{s^{4}}(s-\ell ) (s-3\ell ) \bigl(x_{i}-x_{i}^{0} \bigr) \biggr) ^{2} \Biggr] ^{\frac{p}{2}}\,{\mathrm{d}}x \\ &\quad = \biggl( \frac{32d}{s^{4}} \biggr) ^{p} \int_{B(x^{0},s)\backslash B(x^{0},\frac{s}{2})} \Biggl[ \sum_{i=1}^{N} \bigl( (s-\ell ) (s-3\ell ) \bigl(x_{i}-x_{i}^{0} \bigr) \bigr) ^{2} \Biggr] ^{ \frac{p}{2}}\,{\mathrm{d}}x \end{aligned}$$
and
$$ \int_{\Omega } \bigl\vert w(x) \bigr\vert ^{p}\,{ \mathrm{d}}x= \biggl( \frac{16d}{s^{4}} \biggr) ^{p}\frac{2 \pi^{\frac{N}{2}}}{\Gamma (\frac{N}{2})} \biggl( \frac{(\frac{s}{2})^{N}}{N}+ \int_{\frac{s}{2}}^{s} \bigl\vert r^{2}(s-r) \bigr\vert ^{p}r^{N-1}\,{\mathrm{d}}r \biggr) , $$
one has
$$\begin{aligned} \frac{d^{p}}{p}M^{-}L'&\leq \frac{1}{p} \bigl( \vartheta_{1}'d^{p} +m_{0}^{p-1} \vartheta_{2}'d^{p}+ \varrho \vartheta_{3}'d^{p} \bigr) \\ &\leq \Phi (w) \\ &= \frac{1}{p} \bigl(\vartheta_{1}'d^{p} + \widehat{M} \bigl(\vartheta_{2}'d^{p} \bigr) + \varrho \vartheta_{3}'d^{p} \bigr) \\ & \leq \frac{1}{p} \bigl(\vartheta_{1}'d^{p}+m_{1}^{p-1} \vartheta_{2}'d^{p}+ \varrho \vartheta_{3}'d^{p} \bigr) \\ &\leq \frac{d^{p}}{p}M^{+}L'. \end{aligned}$$
Therefore, condition \((\mathcal{A}_{4})\) in Theorem 3.4 takes the following form:
-
\(({\mathcal{A}_{5}})\)
:
-
there exists a positive constant d such that
$$ \vartheta_{1}'d^{p} +\widehat{M} \bigl( \vartheta_{2}'d^{p} \bigr) +\varrho \vartheta_{3}'d^{p}\neq 0\quad\mbox{and}\quad \varepsilon < \frac{pM ^{-}\int_{\Omega }F(x,w(x))\,\mathrm{d}x}{k^{p}(\vartheta_{1}'d^{p}+ \widehat{M}(\vartheta_{2}'d^{p})+\varrho \vartheta_{3}'d^{p})}, $$
where w is given by (3.13).
Moreover, the condition \(\sqrt[p]{L}d< c\) in Theorem 3.5 becomes the condition \(\sqrt[p]{L'}d< c\). Also, by choosing w as given in [37], Remark 3.4, which is as follows:
$$ w(x):=\left \{ \textstyle\begin{array}{l@{\quad}l} 0 &\mbox{if } x \in \overline{\Omega }\setminus B(x^{0},r_{2}), \\ \frac{3(\ell^{4}-r^{4})-4(r_{1}+r_{2})(\ell^{3}-r_{2}^{3})+6r_{1}r _{2}(\ell^{2}-r_{2}^{2})}{(r_{2}-r_{1})^{3}(r_{1}+r_{2})}d & \mbox{if } x\in B(x^{0},r_{2})\setminus B(x^{0},r_{1}), \\ d &\mbox{if } x \in B(x^{0},r_{1}), \end{array}\displaystyle \right . $$
where \(\ell =\operatorname{dist}(x,x^{0})=\sqrt{\sum_{i=1}^{N} ( x_{i}-x_{i}^{0} ) ^{2}}\) and \(r_{1}, r_{2}\in \mathbb{R}\) with \(r_{2}>r_{1}>0\) (see [33, 34]), we have other forms of conditions \((\mathcal{A}_{4})\) and \(\sqrt[p]{L}d< c\).
Now, we point out some results in which the function f has separated variables. To be precise, consider the following problem:
$$ \left \{ \textstyle\begin{array}{l@{\quad}l} T(u)= \lambda \theta (x)f(u)+\mu g(x,u),&\textrm{in }\Omega , \\ u=\Delta u=0,&\textrm{on } \partial \Omega , \end{array}\displaystyle \right . \hspace{30pt} (P^{f,\theta ,g}_{\lambda ,\mu }) $$
where \(\theta :\Omega \to \mathbb{R}\) is a nonnegative and nonzero function, \(\theta \in {\mathrm{L}}^{1}(\Omega )\), \(f:\mathbb{R}\to \mathbb{R}\) is a continuous function and \(g:\Omega \times \mathbb{R} \to \mathbb{R}\) is as introduced in the problem (\(P^{f,g}_{\lambda ,\mu}\)) in Introduction.
Set \(F(x,t)= \theta (x)F(t)\) for every \((x,t)\in \Omega \times \mathbb{R}\), where
$$ F(t)= \int_{0}^{t} f(\xi )\,\mathrm{d}\xi $$
for all \(t\in \mathbb{R}\). The following existence results are consequences of Theorem 3.4.
Theorem 3.6
Assume that
-
\((\mathcal{A}_{6})\)
:
-
there exists a constant
\(\varepsilon >0\)
such that
$$ \sup_{x\in \Omega }\theta (x)\cdot\max \biggl\{ \limsup _{u\rightarrow 0} \frac{F(u)}{\vert u \vert ^{p}} ,\,\limsup_{\vert u \vert \rightarrow \infty } \frac{F(u)}{\vert u \vert ^{p}} \biggr\} < \varepsilon ; $$
-
\((\mathcal{A}_{7})\)
:
-
there exists a positive constant
d
such that
$$ \vartheta_{1}d^{p} +\widehat{M} \bigl(\vartheta_{2}d^{p} \bigr) +\varrho \vartheta _{3}d^{p}\neq 0\quad \textit{and }\quad \varepsilon < \frac{pM^{-} \int_{\Omega }F(x,w(x))\,\mathrm{d}x}{k^{p}(\vartheta_{1}d^{p}+\widehat{M}( \vartheta_{2}d^{p})+\varrho \vartheta_{3}d^{p})}, $$
where
w
is given by (3.11).
Then, for each compact interval
\([c,d]\subset (\lambda_{3},\lambda _{4})\), where
\(\lambda_{3}\)
and
\(\lambda_{4}\)
are the same as
\(\lambda_{1}\)
and
\(\lambda_{2}\), but
\(\int_{\Omega }F(x,u(x))\,\mathrm{d}x\)
is replaced by
\(\int_{\Omega } \theta (x)F(u(x))\,\mathrm{d}x\), respectively, there exists
\(R> 0\)
with the following property: for every
\(\lambda \in [c,d]\)
and every
\(\mathrm{L}^{1}\)-Carathéodory function
\(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists
\(\gamma >0\)
such that for each
\(\mu \in [0,\gamma ]\), the problem (\(P^{f,\theta ,g}_{\lambda ,\mu }\)) has at least three weak solutions whose norms in E are less than
R.
Theorem 3.7
Assume that there exists a positive constant
d
such that
$$ \vartheta_{1}d^{p} +\widehat{M} \bigl( \vartheta_{2}d^{p} \bigr)+\varrho \vartheta _{3}d^{p}>0\quad \textit{and}\quad F(d)>0. $$
(3.14)
Moreover, suppose that
$$ \limsup_{u\rightarrow 0}\frac{f(u)}{\vert u \vert ^{p-1}}= \limsup _{\vert u \vert \rightarrow \infty }\frac{f(u)}{\vert u \vert ^{p-1}}=0. $$
(3.15)
Then, for each compact interval
\([c,d]\subset (\lambda_{3},\infty )\), where
\(\lambda_{3}\)
is the same as
\(\lambda_{1}\)
but
\(\int_{\Omega }F(x, u(x)) \,\mathrm{d}x\)
is replaced by
\(\int_{\Omega } \theta (x)F(u(x))\,\mathrm{d}x\), there exists
\(R>0\)
with the following property: for every
\(\lambda \in [c,d]\)
and every
\(\mathrm{L}^{1}\)-Carathéodory function
\(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists
\(\gamma >0\)
such that for each
\(\mu \in [0,\gamma ]\), the problem (\(P^{f,\theta ,g}_{\lambda ,\mu }\)) has at least three weak solutions whose norms in E are less than R.
Proof
We easily observe that from (3.15) Assumption \((\mathcal{A} _{6})\) is satisfied for every \(\varepsilon >0\). Moreover, using (3.14), by choosing \(\varepsilon >0\) small enough, one can derive Assumption \((\mathcal{A}_{7})\). Hence, the conclusion follows from Theorem 3.6. □
Remark 3.3
Our results show that no asymptotic conditions on f and g are required, and merely the algebraic conditions on f are supposed to guarantee the existence of solutions.
Now, we present the following example to illustrate Theorem 3.7.
Example 3.1
Let \(N=2\), \(p=6\), \(\varrho =2\), \(\Omega =\{(x_{1},x_{2})\in \mathbb{R}^{2};\, x_{1}^{2}+x_{2}^{2}<5\}\subset \mathbb{R}^{2}\), \(x^{0}=(0,0)\), \(s=2\),
$$ M(t)=1+\frac{1}{\cosh t}\quad \mbox{for all } t\in \mathbb{R}, $$
\(\vartheta (x)=1\) for all \(x\in \Omega \) and
$$ f(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} 2(t+\sin t)^{2},& \text{if} \,\, t< \pi , \\ 2\pi^{2}+\tanh (t-\pi ),& \text{if} \,\, t\geq \pi . \end{array}\displaystyle \right . $$
Thus,
$$ S \bigl(x^{0},s \bigr)=S \bigl((0,0),2 \bigr)= \bigl\{ (x_{1},x_{2}) \in \mathbb{R}^{2};\, x_{1}^{2}+x _{2}^{2}< 4 \bigr\} \subset \Omega , $$
\(m_{0}=1\), \(m_{1}=2\) and f is a nonnegative and continuous function. By choosing \(d=1\), we have
$$ w(x_{1},x_{2})= \textstyle\begin{cases} 0&\mbox{if } (x_{1},x_{2})\in \overline{\Omega }\setminus S ((0,0),2 ), \\ 1&\mbox{if } (x_{1},x_{2})\in S ((0,0),1 ), \\ \frac{1}{2}\sqrt{(x_{1}^{2}+x_{2}^{2})^{3}} \\ \quad {}-3(x_{1}^{2}+x_{2}^{2}) + \frac{9}{2}\sqrt{x_{1}^{2}+x_{2}^{2}}-1&\mbox{if } (x_{1},x_{2})\in S ((0,0),2 )\setminus S ((0,0),1 ). \end{cases} $$
Therefore,
$$\begin{aligned}& \vartheta_{1}=2\pi \int_{1}^{2} \biggl( \frac{9}{2}\xi -12+ \frac{9}{2\xi ^{3}} \biggr) ^{6}\xi \,{\mathrm{d}}\xi , \\& \vartheta_{2}= \iint_{S ((0,0),2 )\setminus S ((0,0),1 )} \Biggl[\sum_{i=1}^{2} \biggl(\frac{3}{2}x_{i}\sqrt{x_{1}^{2}+x_{2}^{2}}-6x _{i}+\frac{9x_{i}}{2\sqrt{x_{1}^{2}+x_{2}^{2}}} \biggr)^{2} \Biggr]^{4} { \mathrm{d}}x_{1}\,{\mathrm{d}}x_{2} \end{aligned}$$
and
$$ \vartheta_{3}=2\pi \biggl( \frac{1}{2}+ \int_{1}^{2} \biggl( \frac{1}{2}\xi ^{3}-3\xi^{2}+\frac{9}{2}\xi -1 \biggr) ^{6}\xi \,{\mathrm{d}}\xi \biggr) $$
are positive. So,
$$ \vartheta_{1}d^{p} +\widehat{M} \bigl(\vartheta_{2}d^{p} \bigr) +\varrho \vartheta _{3}d^{p}>0. $$
Moreover, we have
$$\begin{aligned} F(d)=F(1)= \int_{0}^{1} 2(t+\sin t)^{2}\,{\mathrm{d}}t>0 \end{aligned}$$
and
$$ \lim_{u\to 0}\frac{f(u)}{\vert u \vert }=\lim_{u\rightarrow \infty } \frac{f(u)}{\vert u \vert }=0. $$
Hence, by applying Theorem 3.7 for each compact interval \([c,d]\subset (0,\infty )\), there exists \(R>0\) with the following property: for every \(\lambda \in [c,d]\) and every continuous function \(g:\mathbb{R}\to \mathbb{R}\), there exists \(\gamma >0\) such that, for each \(\mu \in [0,\gamma ]\), the problem
$$ \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta (\vert \Delta u \vert ^{4}\Delta u)- (1+\frac{1}{\cosh (\int_{\Omega }\vert \nabla u \vert ^{6}\,{\mathrm{d}}x)} )^{5}\Delta_{6}u+2\vert u \vert ^{4}u=\lambda f(u)+ \mu g(u)&\mbox{in } \Omega , \\ u=\Delta u=0 &\mbox{on } \partial \Omega \end{array}\displaystyle \right . $$
has at least three weak solutions whose norms in the space \(\mathrm{W} ^{2,6}(\Omega )\cap {\mathrm{W}}^{1,6}_{0}(\Omega )\) are less than R.
The following existence result is a consequence of Theorem 3.5.
Theorem 3.8
Assume that there exist five positive constants
c, d, α, ν
and
ξ
with
\(c>\sqrt[p]{L}d\), where
L
is given by (3.10), and
\(\alpha \in [0,p)\)
such that
-
\((\mathcal{B}_{4})\)
:
-
\(\max_{\vert u \vert \leq c}F(u)<\frac{M^{-}F(d)\pi ^{N/2}}{M^{+}{\operatorname{meas}}(\Omega )\Gamma (\frac{N}{2})}\);
-
\((\mathcal{B}_{5})\)
:
-
\(F(u)>0\)
for each
\(u\in \mathbb{R}\);
-
\((\mathcal{B}_{6})\)
:
-
\(\vert F(u) \vert \leq \nu \vert u \vert ^{\alpha }+\xi \)
for all
\(u\in \mathbb{R}\).
Then there exist a nonempty open set
\(A\subset [0,\infty )\)
and a positive number
\(R'\)
with the following property: for every
\(\lambda \in A\)
and every continuous function
\(g:\mathbb{R}\to \mathbb{R}\), there exists
\(\delta >0\)
such that, for each
\(\mu \in [0, \delta ]\), the problem (\(P^{f,\theta ,g}_{\lambda ,\mu }\)) has at least three weak solutions whose norms in E are less than
\(R'\).
Finally, we present the following example to illustrate Theorem 3.8.
Example 3.2
Let \(N=2\), \(p=3\), \(\varrho =1\), \(\Omega =\{(x_{1},x_{2})\in \mathbb{R}^{2};\, x_{1}^{2}+x_{2}^{2}<\frac{1}{100}\}\subset \mathbb{R}^{2}\), \(x^{0}=(0,0)\), \(s=\frac{1}{20}\),
$$ M(t)=2+\tanh t\quad \mbox{for all } t\in [0,+\infty [ $$
and
$$ f(t)=\frac{1}{1+t^{2}}\quad \mbox{for all } \, t\in \mathbb{R}. $$
Thus,
$$ S \bigl(x^{0},s \bigr)=S \biggl((0,0),\frac{1}{20} \biggr)= \biggl\{ (x_{1},x_{2})\in \mathbb{R}^{2};\, x_{1}^{2}+x_{2}^{2}< \frac{1}{400} \biggr\} \subset \Omega , $$
\(m_{0}=1\), \(m_{1}=3\), and f is a nonnegative and continuous function. By choosing \(c=100\), \(d=1\), \(\alpha =2\), \(\nu =1\) and \(\xi =\pi \), we have \(\alpha =2\in [0,3)=[0,p)\), \(M^{-}=1\), \(M^{+}=3\) and \(L=58.18309 \pi \). Therefore,
$$\begin{aligned}& c=100>\sqrt[3]{58.18309\pi }>\sqrt[p]{L}d, \\& \max_{\vert u \vert \leq c}F(u)=\max_{\vert u \vert \leq 100}F(u)=\arctan (100)< \frac{ \pi }{2}< \frac{100\pi }{36}=\frac{M^{-}F(d)\pi }{M^{+}{\operatorname{meas}}( \Omega )}, \\& F(d)=F(1)=\arctan (1)=\frac{\pi }{4}>0 \end{aligned}$$
and
$$ \bigl\vert F(u) \bigr\vert \leq \vert u \vert ^{2}+\pi \quad \mbox{for all } u\in \mathbb{R}. $$
Hence, by applying Theorem 3.8, there exist a nonempty open set \(A\subset [0,+\infty )\) and a positive number \(R'\) with the following property: for every \(\lambda \in A\) and every nonnegative continuous function \(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists \(\delta >0\) such that, for each \(\mu \in [0,\delta ]\), the problem
$$ \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta (\vert \Delta u \vert \Delta u)- (2+\tanh (\int_{\Omega }\vert \nabla u \vert ^{3} \,{\mathrm{d}}x) )^{2}\Delta_{3}u+\vert u \vert u=\lambda \frac{1}{1+u^{2}}+\mu g(u)& \mbox{in }\Omega , \\ u=\Delta u=0 &\mbox{on } \partial \Omega \end{array}\displaystyle \right . $$
has at least three weak solutions whose norms in the space \(\mathrm{W} ^{2,3}(\Omega )\cap {\mathrm{W}}^{1,3}_{0}(\Omega )\) are less than R.
Remark 3.4
We point out that the same statements of the above given results can be obtained by considering the special case
$$ M(t)= b_{1}+b_{2}t \quad \mbox{for } t\in [\iota ,\kappa ], $$
where \(b_{1}\), \(b_{2}\), ι and κ are positive numbers. In fact, we have
$$\begin{aligned}& \widehat{M}(t)= \int_{0}^{t} (b_{1}+b_{2}\xi ) \,\mathrm{d}\xi = \frac{(b _{1}+b_{2}t)^{2}}{2b_{2}}-\frac{b_{1}^{2}}{2b_{2}} \quad \textrm{for } t\in [\iota ,\kappa ], \\& m_{1}=b_{1}+b_{2}\iota \quad \mbox{and}\quad m_{2}=b_{1}+b_{2}\kappa . \end{aligned}$$
Arguing as in the proof of Theorems 3.1 and 3.5, three weak solutions can be obtained.