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Multiplicity of solutions to nearly critical elliptic equation in the bounded domain of \(\mathbb{R}^3\). (English) Zbl 1308.35081
Summary: We consider the following Dirichlet boundary value problem \[ \begin{cases} -\Delta u=u^{5-\varepsilon}+\lambda u^q,\quad u > 0\quad &\text{in }\Omega; \\ u=0\quad & \text{on } \partial \Omega, \end{cases}\eqno{(0.1)} \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^3\), \(1 < q < 3\), the parameters \(\lambda > 0\) and \(\varepsilon > 0\). By the Lyapunov-Schmidt reduction method and the Mountain Pass Theorem, we prove that in suitable ranges for the parameters \(\lambda\) and \(\varepsilon\), problem (0.1) has at least two solutions. Additionally if \(2 \leq q < 3\), we prove the existence of at least three solutions. Consequently, we prove a non-uniqueness result for a subcritical problem with an increasing nonlinearity.

MSC:
35J25 Boundary value problems for second-order elliptic equations
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