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# Communications in Mathematical Sciences

## Volume 21 (2023)

### Number 7

### Existence of positive solutions for a class of quasilinear elliptic equations with parameters

Pages: 1989 – 2012

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n7.a11

#### Authors

#### Abstract

This paper is devoted to investigating the existence of positive solutions for a class of parameter-dependent quasilinear elliptic equations\[-\Delta u+V(x)u-\frac{\gamma u}{2\sqrt{1+u^2}}\Delta\sqrt{1+u^2}= \lambda |u|^{p-2}u,\ \ u\in H^1(\mathbb{R}^N),\]where $\gamma,\lambda$ are positive parameters, $N\ge 3$. For a trapping potential $V(x)$ and $p\in (2,2^\ast)$, by controlling the range of $\gamma$ and $\lambda$, we establish the existence of positive solutions $u_{\gamma,\lambda}$ for the above problem, where $2^\ast=\frac{2N}{N-2}$ is critical exponent. For super-critical case, we find a constant $p^\ast\in [2^\ast,\,\min\{\frac{9+2\gamma}{8+2\gamma},\frac{2\gamma+4-2\sqrt{4+2\gamma}}{\gamma}\}2^\ast)$ such that Equation (0.1) has no positive solution for all $\gamma,\lambda>0$ if $p\geq p^\ast$ and $\nabla V(x)\cdot x\geq 0$ in $ \mathbb{R}^N$. Furthermore, for fixed $\lambda >0$, the asymptotic behavior of positive solutions $u_{\gamma,\lambda}$ is also obtained when $V(x)$ is a positive constant as $\gamma \rightarrow 0$.

#### Keywords

quasilinear elliptic equations, positive solutions, asymptotic behavior

#### 2010 Mathematics Subject Classification

35J20, 35J60

The authors’ research was supported by the Natural Science Foundation of China (No.12271196) and the Natural Science Foundation of Guangdong (No.2023A1515012812).

Received 19 June 2022

Accepted 27 January 2023

Published 9 October 2023