Biharmonic-Kirchhoff type equation involving critical Sobolev exponent with singular term
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Using variational methods, we show the existence of a unique weak solution of the following singular biharmonic problems of Kirchhoff type involving critical Sobolev exponent: \begin{equation*} (\mathcal{P}_{\lambda })\left\{ \begin{array}{ll} \!\!\!\!\Delta ^{2}u\!-\!(a\int_{\Omega }|\nabla u|^{2}dx\!+\!b)\Delta u\!+\!cu\!=\!f\left( x\right) \left\vert u\right\vert ^{-\gamma }\!-\!\lambda \left\vert u\right\vert ^{p-2}u & \!\!\text{in }\Omega , \\ \!\!\!\!\Delta u=u=0 & \!\!\text{on }\partial \Omega , \end{array} \right. \end{equation*} where $\Omega $ is a smooth bounded domain of $ \mathbb{R} ^{n}$ $\left( n\geq 5\right) $, $\Delta ^{2}$ is the biharmonic operator, and $\nabla u$ denotes the spatial gradient of $u$ and $0<\gamma <1,$ $ \lambda >0$, $0<p\leq 2^{\sharp }$ and $a,b,c$ are three positive constants with $a+b>0$ and $f$ belongs to a given Lebesgue space.
Abstract
Using variational methods, we show the existence of a unique weak solution of the following singular biharmonic problems of Kirchhoff type involving critical Sobolev exponent: \begin{equation*} (\mathcal{P}_{\lambda })\left\{ \begin{array}{ll} \!\!\!\!\Delta ^{2}u\!-\!(a\int_{\Omega }|\nabla u|^{2}dx\!+\!b)\Delta u\!+\!cu\!=\!f\left( x\right) \left\vert u\right\vert ^{-\gamma }\!-\!\lambda \left\vert u\right\vert ^{p-2}u & \!\!\text{in }\Omega , \\ \!\!\!\!\Delta u=u=0 & \!\!\text{on }\partial \Omega , \end{array} \right. \end{equation*} where $\Omega $ is a smooth bounded domain of $ \mathbb{R} ^{n}$ $\left( n\geq 5\right) $, $\Delta ^{2}$ is the biharmonic operator, and $\nabla u$ denotes the spatial gradient of $u$ and $0<\gamma <1,$ $ \lambda >0$, $0<p\leq 2^{\sharp }$ and $a,b,c$ are three positive constants with $a+b>0$ and $f$ belongs to a given Lebesgue space.
참고문헌 (15)
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