上海理工大学学报  2022, Vol. 44 Issue (6): 583-587 PDF

The existence of weak solutions for critical p-biharmonic equations with sign-changing weight function and Hardy term
CUI Huimin, WEI Gongming
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: The existence of weak solutions for critical p-biharmonic equations with sign-changing weight function and Hardy term was studied by using mountain pass lemma, concentration-compactness principle and Hardy’s inequality. Firstly, the geometric conditions of mountain pass lemma are verified. Next, if $0 < \mu < {\mu _0}$ , ${(PS)_c}$ condition is satisfied when mountain level $c < \dfrac{2}{N} S^{N / 2 p}-\mu^{{p^*} /\left(p^*-q\right)}G$ . Thus, it is proved that there exists at least one nontrivial weak solution for the given critical ${{p - }}$ biharmonic equations.
Key words: mountain pass lemma     concentrative-compactness principle     weak solution     Hardy’s inequality
1 问题的提出

2018年，文献[3]研究了带有临界指标和凹凸非线性项的p-双调和方程

 $\Delta _p^2u = \lambda f(x)|u{|^{q - 1}}u + |u{|^{{p^*} - 1}}u,\;\; x \in \varOmega$ (1)

2020年，文献[7]研究了带有Hardy项的p-双调和方程

 $\Delta _p^2u = \frac{{\mu |u{|^{r - 2}}u}}{{{{\left| x \right|}^s}}} + f(x,u),\;\;x \in \varOmega$ (2)

 $\Delta _p^2u - \lambda \frac{{|u{|^{p - 2}}u}}{{{{\left| x \right|}^{2p}}}} + \mu V(x)|u{|^{q - 2}}u = |u{|^{{p^ * } - 2}}u,{\text{ }}x \in {\mathbb{R}^N}$ (3)

2 预备知识

 $\int_{{\mathbb{R}^N}} {\frac{{{{\left| u \right|}^p}}}{{{{\left| x \right|}^{2p}}}}} dx \leqslant \frac{1}{{\bar \lambda }}\int_{{\mathbb{R}^N}} {{{\left| {\Delta u} \right|}^p}} {\rm{d}}x,{\text{ }}\bar \lambda = {\left( {\frac{{\left( {n - 2p} \right)(p - 1)n}}{{{p^2}}}} \right)^p}$

 $(u,v) = {\int_{{\mathbb{R}^N}} {\left| {\Delta u} \right|} ^{p - 2}}\Delta u\Delta v{\rm{d}}x - \lambda \int_{{\mathbb{R}^N}} {\frac{{{{\left| u \right|}^{p - 2}}uv}}{{{{\left| x \right|}^{2p}}}}} {\rm{d}}x$
 $\left\| u \right\| = {\left( {\int_{{\mathbb{R}^N}} \left({{\left| {\Delta u} \right|}^p} - \lambda \frac{{{{\left| u \right|}^p}}}{{{{\left| x \right|}^{2p}}}}\right){\rm{d}}x} \right)^{\frac{1}{p}}}$

$S$ 表示 $W$ 连续嵌入到 ${L^{{p^ * }}}({\mathbb{R}^N})$ 的最佳常数，即

 $S: = \inf\; \left\{ \int_{{\mathbb{R}^N}} \left({{\left| {\Delta u} \right|}^p} - \lambda \frac{{{{\left| u \right|}^p}}}{{{{\left| x \right|}^{2p}}}}\right){\rm{d}}x:{\text{ }}u \in W,{\text{ }}{\left\| u \right\|_{{p^ * }}} = 1\right\}$

 $\int_{{\mathbb{R}^N}} \left(|\Delta U{|^p} - \lambda \frac{{{{\left| U \right|}^p}}}{{{{\left| x \right|}^{2p}}}} \right){\rm{d}}x = \int_{{\mathbb{R}^N}} {|U{|^{{p^*}}}} {\rm{d}}x = {S^{\frac{N}{{2p}}}}$

 $\begin{split}& {M_1} = \frac{2}{N} > 0,{\text{ }}{{{M}}_2} = \left(\frac{1}{p} - \frac{1}{q}\right)C\left( {\mathop {\max }\limits_{x \in \bar \Omega }\; V(x)} \right) > 0\\& \qquad G = {M_2}{\left( {\frac{{q{M_2}}}{{{p^*}{M_1}}}} \right)^{\frac{q}{{{p^*} - q}}}} > 0 \end{split}$
 $\begin{split}& {\mu _1} = {\left( {\frac{{2{S^{\frac{N}{{2p}}}}}}{{NG}}} \right)^{\frac{{{p^*} - q}}{{{p^*}}}}} > 0\\&{\text{ }}{\mu _2} = {\left( {\frac{{{t_1}^q\int_\Omega {V(x)|U{|^q}} {\rm{d}}x}}{{qG}}} \right)^{\frac{{{p^*} - q}}{{{p^*}}}}} > 0 \end{split}$

${\mu _0} = \min \;\{ {\mu _1},{\mu _2}\} > 0$ $C > 0,{{ }}{{{t}}_1} > 0$ ${t_1}$ 仅依赖于 ${\mu _1}$

 $I(u) = \frac{1}{p}\int_{} \left(|\Delta u| ^p - \lambda \frac{{{{\left| u \right|}^p}}}{{{{\left| x \right|}^{2p}}}}\right){\rm{d}}x + \frac{\mu }{q}{\int_{} {V(x)|u|} ^q}{\rm{d}}x - \frac{1}{{{p^ * }}}{\int_{} {|u|} ^{{p^ * }}}{\rm{d}}x$

 $\begin{split} \left\langle {I'(u),v} \right\rangle = &\int_{} \left({{\left| {\Delta u} \right|}^{p - 2}}\Delta u\Delta v - \lambda \frac{{{{\left| u \right|}^{p - 2}}uv}}{{{{\left| x \right|}^{2p}}}}\right){\rm{d}}x + \\&\mu {\int_{} {V(x)|u|} ^{q - 2}}uv{\rm{d}}x - {\int_{} {|u|} ^{{p^ * } - 2}}uv{\rm{d}}x \end{split}$

3 定理1的证明

a. 存在常数 $\rho ,\;\beta > 0$ ，使得 $I(u) \geqslant \beta ,\left\| u \right\| = \rho$

b. 存在 $e \in W$ $\left\| e \right\| > \rho$ ，使得 $I(e) \leqslant 0$

 $\begin{split} I(u) = & \frac{1}{p}\int_{} {\left(|\Delta u{|^p} - \lambda \frac{{{{\left| u \right|}^p}}}{{{{\left| x \right|}^{2p}}}}\right)} {\rm{d}}x + \frac{\mu }{q}\int_{} {V(x)|u{|^q}} {\rm{d}}x - \frac{1}{{{p^ * }}}\int_{} {|u{|^{{p^ * }}}} {\rm{d}}x \geqslant \\& \frac{1}{p}{\left\| u \right\|^p} + \frac{\mu }{q}\left( {\mathop {\min }\limits_{x \in \bar \varOmega } \; V(x)} \right)\int_{} {|u{|^q}} {\rm{d}}x - \frac{1}{{{p^ * }}}{S^{ - \frac{{{p^ * }}}{2}}}{\left\| u \right\|^{{p^ * }}}\geqslant \\& \frac{1}{p}{\left\| u \right\|^p} + \frac{\mu }{q}\left( {\mathop {\min }\limits_{x \in \bar \varOmega } \;V(x)} \right){C^{ - \frac{q}{2}}}{\left\| u \right\|^q} - \frac{1}{{{p^ * }}}{S^{ - \frac{{{p^ * }}}{2}}}{\left\| u \right\|^{{p^ * }}} \end{split}$

b. $u \in W\backslash \{ 0\}$ ，则

 $\begin{split} I(tu) =& \frac{{{t^p}}}{p}\int_{} \left(|\Delta u{|^p} - \lambda \frac{{{{\left| u \right|}^p}}}{{{{\left| x \right|}^{2p}}}}\right){\rm{d}}x + \frac{{\mu {t^q}}}{q}\int_{} {V(x)|u{|^q}} {\rm{d}}x -\\& \frac{{{t^{{p^ * }}}}}{{{p^ * }}}\int_{} {|u{|^{{p^ * }}}}{\rm{d}}x \end{split}$

 $I({u_n}) \to c,{\text{ }}I'({u_n}) \to 0,\;\; c = {\inf _{h \in \Gamma }}{\sup _{t \in [0,1]}}I\left( {h\left( t \right)} \right)$
 $\varGamma = \left\{ {h \in C\left( {[0,1],W} \right):{\text{ }}h(0) = 0,{\text{ }}h(1) = e} \right\}$

 $|I({u_n})| \leqslant M,{\text{ |}}I'({u_n}){u_n}| \leqslant M\left\| {{u_n}} \right\|$

 $\begin{split} M(1 + \left\| {{u_n}} \right\|) \geqslant & I({u_n}) - \frac{1}{q}I'({u_n}){u_n} = \left(\frac{1}{p} - \frac{1}{q}\right){\left\| {{u_n}} \right\|^p} + \\&\left(\frac{1}{q} - \frac{1}{{{p^ * }}}\right)\int_{} {|{u_n}{|^{{p^ * }}}} {\rm{d}}x \geqslant \left(\frac{1}{p} - \frac{1}{q}\right){\left\| {{u_n}} \right\|^p} \end{split}$

$\{ {u_n}\} \subset W$ 是一个有界列，此时可以假设存在一个 $u \in W$ ，使得在 $W$ ${u_n} \rightharpoonup u$ ，在 $L_{loc}^q({\mathbb{R}^N})(1 < q < {2^*})$ 中有 ${u_n} \to u$ ，利用集中紧性原理[14]，有

 $\begin{split}& |\Delta {{{u}}_n}{{\text{|}}^p}{\text{dx}} \rightharpoonup {\text{d}}\gamma \geqslant {\left| {\Delta {{u}}} \right|^p}dx + \sum\limits_{k \in J} {{\gamma _k}{\delta _{{x_k}}}} + {\gamma _0}{\delta _0}\\& |{{{u}}_n}{{\text{|}}^{{p^ * }}}{\text{dx}} \rightharpoonup {\text{d}}\nu = {\left| u \right|^{{p^ * }}}dx + \sum\limits_{k \in J} {{\nu _k}{\delta _{{x_k}}}} + {\nu _0}{\delta _0} \\& \frac{{\left|{u}_{n}\right|}^{p}}{{\left|x\right|}^{2p}}\text{dx}\rightharpoonup d\sigma =\frac{{\left|u\right|}^{p}}{{\left|x\right|}^{2p}}\text{dx+}{\sigma }_{0}{\delta }_{0} \end{split}$ (4)

 $\begin{split}& \mathop {\lim }\limits_{n \to \infty } \int_{} {|\Delta {u_n}{|^{p - 2}}\Delta {u_n}} \Delta (\phi {u_n}){\rm{d}}x - \lambda \int_{} \phi {\rm{d}}\sigma {\text{ = }} \\&\qquad- \mu \int_{} {V(x)|u} {|^q}\phi {\rm{d}}x + \int_{} \phi {\rm{d}}\nu \end{split}$ (5)

 $\begin{split}& \int_{} {|\Delta {u_n}{|^{p - 2}}\Delta {u_n}\Delta (\phi {u_n})} {\rm{d}}x{\text{ = }}\int_{} {\Delta {u_n}{|^{p - 2}}\Delta {u_n}} \left( (\Delta \phi ){u_n} +\right.\\&\qquad \left.2\left\langle {\nabla \phi ,\nabla {u_n}} \right\rangle + \phi \Delta {u_n} \right){\rm{d}}x {{ = }}{\int_{} {\phi |\Delta {u_n}|} ^p}{\rm{d}}x + \\& \qquad \int_{} {|\Delta {u_n}{|^{p - 2}}\Delta } {u_n}\left( {2\left\langle {\nabla \phi ,\nabla {u_n}} \right\rangle + {u_n}\Delta \phi } \right){\rm{d}}x{\text{ }} \\[-5pt] \end{split}$ (6)

 $\begin{split}& \mathop {\lim }\limits_{n \to \infty } \int_{} {|\Delta {u_n}{|^{p - 2}}\Delta {u_n}\Delta (\phi {u_n})} {\rm{d}}x{\text{ = }}\int_{} {\phi {\rm{d}}\gamma } + \\&\qquad \mathop {\lim }\limits_{n \to \infty } \int_{} {|\Delta {u_n}{|^{p - 2}}\Delta } {u_n}\left( {2\left\langle {\nabla \phi ,\nabla {u_n}} \right\rangle + {u_n}\Delta \phi } \right){\rm{d}}x \\[-6pt] \end{split}$ (7)

$\varepsilon \to 0$ 时，

 $\begin{split}& 0 \leqslant \mathop {\lim }\limits_{n \to \infty } \;\left| {\int_{} {|\Delta {u_n}{|^{p - 2}}\Delta {u_n}} \left\langle {\nabla \phi ,\nabla {u_n}} \right\rangle {\rm{d}}x} \right|\leqslant \\& \mathop {\lim }\limits_{n \to \infty } {\;\left({\int_{} {|\Delta {u_n}|} ^p}{\rm{d}}x\right)^{\frac{{p - 1}}{p}}}{\left(\int_{} {|\nabla \phi {|^p}|\nabla {u_n}{|^p}} {\rm{d}}x\right)^{\frac{1}{p}}} \leqslant \\& C\mathop {\lim }\limits_{n \to \infty } {\;\left(\int_{} {|\nabla \phi {|^p}|\nabla {u_n}{|^p}} {\rm{d}}x\right)^{\frac{1}{p}}} \leqslant \\& C{\left(\int_{\mathbb{R}^N} \cap B(0,2\varepsilon ) {|\nabla \phi {|^p}|\nabla u{|^p}} {\rm{d}}x\right)^{\frac{1}{p}}}\leqslant \\& {\text{ }} C{\left( \int_{\mathbb{R}^N} \cap B(0,2\varepsilon ) |\nabla \phi {|^N}{\rm{d}}x{)^{\frac{p}{N}}}\int_\mathbb{R}^N \cap B(0,2\varepsilon \right) {|\nabla u{|^{\frac{{Np}}{{N - p}}}}} {\rm{d}}x)^{\frac{{N - p}}{N}}} \leqslant \\& C\int_{B(0,2\varepsilon )} {|\nabla u} {|^{\frac{{Np}}{{N - p}}}}{\rm{d}}x{)^{\frac{{N - p}}{N}}} \to 0{\text{ }} \\[-15pt] \end{split}$ (8)

 $\begin{split} 0 \leqslant& \mathop {\lim }\limits_{n \to \infty } \;\left| {\int_{} {|\Delta {u_n}{|^{p - 2}}\Delta {u_n}} ({u_n}\Delta \phi ){\rm{d}}x} \right| \leqslant\\& {\lim _{n \to \infty }}{\int_{} {|\Delta {u_n}|} ^{p - 1}}|{u_n}||\Delta \phi |{\rm{d}}x \leqslant \\& \mathop {\lim }\limits_{n \to \infty } {\;\left(\int_{} {|\Delta {u_n}{|^p}} {\rm{d}}x\right)^{\frac{{p - 1}}{p}}}{\left(\int_{} {|{u_n}{|^p}|\Delta \phi {|^p}} {\rm{d}}x\right)^{\frac{1}{p}}}\leqslant \\& {{C}}{\left(\int_{{\mathbb{R}^N} \cap B(0,2\varepsilon )} {|\Delta \phi {|^{\frac{N}{2}}}} {\rm{d}}x\right)^{\frac{{2p}}{N}}}{\left(\int_{{\mathbb{R}^N} \cap B(0,2\varepsilon )} {|u{|^{{p^ * }}}} {\rm{d}}x\right)^{\frac{1}{{{p^ * }}}}} \leqslant \\& {\text{ }} {{C}}\left(\int_{B(0,2\varepsilon )} {|\Delta u{|^{{p^*}}}} {\rm{d}}x\right)^{\frac{1}{{{p^*}}}} \to 0\\[-12pt] \end{split}$ (9)

 $\begin{split} 0{\text{ = }}& \mathop {\lim }\limits_{\varepsilon \to 0} \left\{ - \int {\lambda \phi {\rm{d}}\sigma } + \int_{} {\phi {\rm{d}}\gamma } + \mu \int_{} {V(x)|u{|^q}{\rm{d}}x - \int_{} {\phi {\rm{d}}\nu } } \right\} \geqslant\\& - \lambda {\sigma _0} + {\gamma _0} - {\nu _0} \end{split}$

$V(x) \in {C_c}({\mathbb{R}^N})$ 可得， $\displaystyle \int_{} {V(x)} |{u_n}{|^q}{\rm{d}}x \to \displaystyle \int_{} {V(x)} |u{|^q}{\rm{d}}x$ 。故有

 $\begin{split} {c}=&\underset{n\to \infty }{\mathrm{lim}}\;I({u}_{n})=\underset{n\to \infty }{\mathrm{lim}}\;\left(I({u}_{n})-\frac{1}{p}\langle {I}^{\prime }({u}_{n})，{u}_{n}\rangle \right)=\\ &\underset{n\to \infty }{\mathrm{lim}}\left(\left(\frac{1}{q}-\frac{1}{p}\right)\mu {\displaystyle {\int }_{}V(x)|{u}_{n}{|}^{q}{\rm{d}}x+\left(\frac{1}{p}-\frac{1}{{p}^{\ast }}\right)}{{\displaystyle {\int }_{}\left|{u}_{n}\right|}}^{{p}^{\ast }}{\rm{d}}x\right)=\\ &\left(\frac{1}{q}-\frac{1}{p}\right)\mu {\displaystyle {\int }_{}V(x)|{u}_{n}{|}^{q}{\rm{d}}x+\underset{n\to \infty }{\mathrm{lim}}\left(\frac{1}{p}-\frac{1}{{p}^{\ast }}\right)}{{\displaystyle {\int }_{}\left|{u}_{n}\right|}}^{{p}^{\ast }}{\rm{d}}x\geqslant\\& \left(\frac{1}{q}-\frac{1}{p}\right)\mu {\displaystyle {\int }_{}V(x)|u{|}^{q}{\rm{d}}x+\left(\frac{1}{p}-\frac{1}{{p}^{\ast }}\right)}\left({\displaystyle {\int }_{}|u{|}^{{p}^{\ast }}{\rm{d}}x+{\nu }_{0}}\right)\geqslant\\ & \left(\frac{1}{q}-\frac{1}{p}\right)\mu C\left(\underset{x\in \overline{\varOmega }}{\mathrm{max}}V(x)\right)\left(\displaystyle {\int }_{}|u{|}^{{p}^{\ast }}{\rm{d}}x\right)^{\frac{q}{{p}^{\ast }}}+\\ &\frac{2}{N}{\displaystyle {\int }_{}|u{|}^{{p}^{\ast }}{\rm{d}}x+\frac{2}{N}{S}^{\frac{N}{2p}}} \end{split}$

 ${M}_{1}\text=\frac{2}{N}\text{，}\text{ }{M}_{2}=\left(\frac{1}{p}-\frac{1}{q}\right)C\left(\underset{x\in \overline{\varOmega }}{\mathrm{max}}\;V(x)\right)$

$g'(x) = 0$ 很容易求得 $g(x)$ 的临界点为 ${x_0} = {\left( {\dfrac{{q{M_2}\mu }}{{{p^ * }{M_1}}}} \right)^{\frac{1}{{{p^ * } - q}}}}$ 。通过计算可知 $g(x)$ ${x_0} = {\left( {\dfrac{{q{M_2}\mu }}{{{p^ * }{M_1}}}} \right)^{\frac{1}{{{p^ * } - q}}}}$ 上取得极小值 $g({x_0}){\text{ = }}{M_3} - \mu {M_4}$ ，其中，

 ${M_3}{\text{ = }}{M_1}{\left(\frac{{q{M_2}\mu }}{{{p^*}{M_1}}}\right)^{\frac{{{p^*}}}{{{p^*} - q}}}}\text{，}{M_4}{\text{ = }}{M_2}{\left(\frac{{q{M_2}\mu }}{{{p^*}{M_1}}}\right)^{\frac{q}{{{p^*} - q}}}}$

 $\begin{split} & c \geqslant \frac{2}{N}{S^{\frac{N}{{2p}}}}{\text{ + }}{M_3} - \mu {M_4} = \frac{2}{N}{S^{\frac{N}{{2p}}}} - {\mu ^{\frac{{{p^*}}}{{{p^*} - q}}}}G\\&G = {M_2}{\left(\frac{{q{M_2}}}{{{p^*}{M_1}}}\right)^{\frac{q}{{{p^*} - q}}}} \end{split}$

 $\begin{split} o(1) = &\left\langle {I'({u_n}) - I'(u),{u_n} - u} \right\rangle {\text{ = }}\\ & \int_{} {\left( {|\Delta {u_n}{|^{p - 2}}{u_n} - |\Delta u{|^{p - 2}}u} \right)} \left( {{u_n} - u} \right){\rm{d}}x +\\ & \mu \int_{} {V(x)} (|{u_n}{|^{q - 2}}{u_n} - |u{|^{q - 2}}u)({u_n} - u){\rm{d}}x - \\ & \int_{} ( |{u_n}{|^{{p^*} - 2}}{u_n} - |u{|^{{p^*} - 2}}u)({u_n} - u){\rm{d}}x\geqslant \\ & {{C}}{\left\| {{u_n} - u} \right\|^p} + o(1) \end{split}$

 $h(t) = \frac{1}{p}{t^p}{\left\| U \right\|^p} - \frac{{{t^{{p^*}}}}}{{{p^*}}}\int_{} {|U{|^{{p^*}}}} {\rm{d}}x = :{C_1}{t^p} - {C_2}{t^{{p^*}}},\;\;t \geqslant 0$

 $\begin{split} I(tu) =& h(t) - \frac{{\mu {t^q}}}{q}\int_{} V(x)|U{|^q}{\rm{d}}x \leqslant \\& \frac{2}{N} {S^{\frac{N}{{2p}}}} - \frac{{\mu {t^q}}}{q}\int_{} {V(x} )|U{|^q}{\rm{d}}x \end{split}$ (10)

 $\mathop {\max }\limits_{0 \leqslant t \leqslant {t_1}}\; I(tu) < \frac{2}{N}{S^{\frac{N}{{2p}}}} - {\mu ^{\frac{{{p^*}}}{{{p^*} - q}}}}G$

 $\mathop {\max }\limits_{t > {t_1}} \;I(tu) \leqslant \frac{2}{N}{S^{\frac{N}{{2p}}}} - \frac{{\mu {t_1}^q}}{q}\int_{} {V(x} )|U{|^q}{\rm{d}}x$

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