Existence and multiplicity of solutions for a class of elliptic problem with critical growth
The authors study a class of elliptic problems of the type \begin{align*} -\Delta u +\lambda V(x)u=\mu u^{q-1}+u^{2^*-1} \quad\text{in }\mathbb{R}^N\tag{P} \end{align*} where $\lambda, \mu >0$, $q\in (2,2^*)$ with $N\geq 3$ and $V: \mathbb{R}^N\to \mathbb{R}$ is a continuous function that satisfies \begin{itemize} \item[(V1)] $V(x) \geq 0$ for all $x\in \mathbb{R}^N$. \item[(V2)] There exists $M_0>0$ such that the set $\mathcal{L}=\{x\in \mathbb{R}^N\, : \, V(x) \leq M_0\}$ is non-empty and $|\mathcal{L}|<\infty,$ where $|A|$ denotes the Lebesgue measure of $A$ on $\mathbb{R}^N$. \item[(V3)] $\Omega:=\text{int\,}\left(V^{-1}(\{0\}) \right)$ is a non-empty bounded open set with smooth boundary $\partial\Omega$. \end{itemize} \par The main result in this paper, which proof is based on variational methods, is the following. \par Theorem 1.1. Assume that (V1)--(V3) hold. If \begin{align*} N\geq 4 \text{ and }2<q<2^*\quad\text{or}\quad N=3 \text{ and }4<q<6, \end{align*} then there are $\lambda^*, \mu^*>0$ such that problem (P) has at least $\text{cat\,}(\Omega)$ positive solutions for $\lambda\geq \lambda^*$ and $\mu\leq \mu^*$.
scientific article; zbMATH DE number 7031532
scientific article