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Three solutions for a perturbed sublinear elliptic problem in $$\mathbb{R}^N$$. (English) Zbl 1180.35190
The authors deal with the following perturbation problem $\begin{cases} -\Delta u=h(x)|u|^{s-2} u+\lambda f(x,u)\quad & \text{in }\mathbb{R}^d\\ u\in{\mathcal D}^{1,2}(\mathbb{R}^d),\end{cases} \tag{1}$ where $$s\in (1,2)$$, $$d\geq 3$$, $$\lambda\in R_+$$, $$f:\mathbb{R}^d\times \mathbb{R}\to \mathbb{R}$$ is a Caratheodory function and $$h$$ is a given function. ${\mathcal D}^{1,2}(\mathbb{R}^d)=\{u\in L^{\frac{2d}{d-2}}(\mathbb{R}^d)|\nabla u\in L^2(\mathbb{R}^d)\}$ is the completion of $C_0(\mathbb{R}^d)=\{u\in L^{\frac{2d}{d-2}}(\mathbb{R}^d)|\nabla u\in L^2(\mathbb{R}^d)\}$ is the completion of $C_0(\mathbb{R}^d)=\{u\in C(\mathbb{R}^d)|\text{supp}\,u\text{ is compact\}}$ with respect to the norm $$\| u\|=\left(\int_{\mathbb{R}^d}|\nabla u|^2\,dx\right)^{1/2}$$. Using variational methods, the authors establish a result that ensures the existence of at least three weak solutions.

MSC:
 35J20 Variational methods for second-order elliptic equations
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