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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
##### Keywords:
multiplicity; bubble solutions; mountain pass solution
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##### References:
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