Existence of multiple solutions to an elliptic problem with measure data. (English) Zbl 1406.35133
Summary: In this paper we prove the existence of multiple nontrivial solutions of the following equation:
\[ \begin{aligned} -\Delta _{p}u &= \lambda |u|^{q-2}u+f(x,u)+\mu \quad \text{in }\Omega ,\\ u & = 0 \quad \text{on }\partial \Omega, \end{aligned} \]
where \(\Omega \subset \mathbb {R}^N\) is a smooth bounded domain with \(N \geq 3\), \(1< q'< q < p-1\), \(\lambda \) and \(f\) satisfy certain conditions, \(\mu >0\) is a Radon measure, \(q'=\frac{q}{q-1}\) is the conjugate of \(q\).
\[ \begin{aligned} -\Delta _{p}u &= \lambda |u|^{q-2}u+f(x,u)+\mu \quad \text{in }\Omega ,\\ u & = 0 \quad \text{on }\partial \Omega, \end{aligned} \]
where \(\Omega \subset \mathbb {R}^N\) is a smooth bounded domain with \(N \geq 3\), \(1< q'< q < p-1\), \(\lambda \) and \(f\) satisfy certain conditions, \(\mu >0\) is a Radon measure, \(q'=\frac{q}{q-1}\) is the conjugate of \(q\).
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
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