Multiplicity and concentration results for a $(p,q)$-Laplacian problem in ${\mathbb{R}}^N$

Abstract

In this paper we study the multiplicity and concentration of positive solutions for the following $(p,q)$-Laplacian problem: \begin{equation*} \left\{ \begin{array}{ll} -Δ_{p} u -Δ_{q} u +V(\varepsilon x) \left(|u|^{p-2}u + |u|^{q-2}u\right) = f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in W^{1, p}(\mathbb{R}^{N})\cap W^{1, q}(\mathbb{R}^{N}), \quad u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\epsilon >0$ is a small parameter, $1<p<q<N$, $Δ_{r}u=\mbox{div}(|\nabla u|^{r-2}\nabla u)$, with $r\in \{p,q\}$, is the $r$-Laplacian operator, $V:{\mathbb{R}}^N\to \mathbb{R}$ is a continuous function satisfying the global Rabinowitz condition, and $f:\mathbb{R}\to \mathbb{R}$ is a continuous function with subcritical growth. Using suitable variational arguments and Ljusternik-Schnirelmann category theory, we investigate the relation between the number of positive solutions and the topology of the set where $V$ attains its minimum for small $\epsilon$.

arXiv admin note: text overlap with arXiv:1901.11016, arXiv:1810.03171

References in corpus (1)

• Nonlinear nonhomogeneous singular problems

Cited by in corpus (1)

• On singularly perturbed $(p,N)$-Laplace Schrödinger equation with logarithmic nonlinearity