QUASILINEAR DOUBLE PHASE PROBLEMS ON THE ENTIRE SPACE $\mathbb{R}^N$

Guo-Feng Feng; Qing-Hai Cao; Bin Ge

doi:10.11948/20240512

2025 Volume 15 Issue 4

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Article navigation > Journal of Applied Analysis & Computation > 2025 > 15(4): 2440-2460

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Guo-Feng Feng, Qing-Hai Cao, Bin Ge. QUASILINEAR DOUBLE PHASE PROBLEMS ON THE ENTIRE SPACE $\mathbb{R}^N$[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2440-2460. doi: 10.11948/20240512

Citation:

Guo-Feng Feng, Qing-Hai Cao, Bin Ge. QUASILINEAR DOUBLE PHASE PROBLEMS ON THE ENTIRE SPACE $\mathbb{R}^N$[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2440-2460. doi: 10.11948/20240512

QUASILINEAR DOUBLE PHASE PROBLEMS ON THE ENTIRE SPACE $\mathbb{R}^N$

1.
School of Mathematical Science, Harbin Engineering University, Harbin 150001, China

Author Bio: Email: Fengguofeng@hrbeu.edu.cn(G.-F. Feng); Email: caoqinghai1104@hrbeu.edu.cn(Q.-H. Cao)
Corresponding author: Email: gebin1025@aliyun.com(B. Ge)

Abstract

Abstract

This study is concerned with the following double phase problem

$ \begin{array}{ll} \quad-{\rm div}(|\nabla u|^{^{p-2}}\nabla u+\mu(x)|\nabla u|^{^{q-2}}\nabla u) +V(x)(|u|^{^{p-2}}u+\mu(x)|u|^{^{q-2}}u) \\ = \lambda f(x, u), \; x\in \mathbb{R}^N, \end{array} $

where $1<p<q<N$, $\frac{q}{p}\leq 1+\frac{\alpha}{N}$, $\lambda$ is a real parameter, $0\leq\mu\in C{^{0,\alpha}}(\mathbb{R}^N)$ with $\alpha\in(0,1]$, $V(x)$ is an unbounded potential function and $f(x,u)$ is the reaction term. The aim is to determine the precise positive interval of $\lambda$ for which the problem admits at least one or two nontrivial solutions by applying abstract critical point results.
- Double phase operator /
- entire space /
- multiple solutions /
- critical point theorems /
- variational methods
MSC: 35D30, 35J20, 35J60, 35J70, 35J92

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References

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