Existence solution for weighted \(p(x)\)-Laplacian equation. (English) Zbl 1387.35248
Summary: This paper deals with the existence solution for the following type of boundary value problems:
\[ \begin{cases} \Delta(|x|^{p(x)} |\Delta u|^{p(x)-2}\Delta u) = \lambda|u|^{q(x)-2} u,&\text{ in }\; \Omega,\\ u = \Delta u = 0,&\text{ on }\; \partial\Omega,\end{cases} \]
where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\). It is established for a negative \(\lambda\), there exists at least one weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces and a variant of the mountain pass theorem.
\[ \begin{cases} \Delta(|x|^{p(x)} |\Delta u|^{p(x)-2}\Delta u) = \lambda|u|^{q(x)-2} u,&\text{ in }\; \Omega,\\ u = \Delta u = 0,&\text{ on }\; \partial\Omega,\end{cases} \]
where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\). It is established for a negative \(\lambda\), there exists at least one weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces and a variant of the mountain pass theorem.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35J40 | Boundary value problems for higher-order elliptic equations |
