Existence of nontrivial positive solutions for generalized quasilinear elliptic equations with critical exponent

Shulin Zhang; Shulin Zhang

doi:10.3934/math.2022543

AIMS Mathematics

2022, Volume 7, Issue 6: 9748-9766. doi: 10.3934/math.2022543

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Research article

Existence of nontrivial positive solutions for generalized quasilinear elliptic equations with critical exponent

Shulin Zhang ^{1,2
,
,}

1.
School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
2.
School of Mathematics, Xuzhou Vocational Technology Academy of Finance and Economics, Xuzhou 221116, China

Received: 23 January 2022 Revised: 05 March 2022 Accepted: 08 March 2022 Published: 17 March 2022
MSC : 35J20, 35J60, 35J62

In this paper, we are concerned with the existence of nontrivial positive solutions for the following generalized quasilinear elliptic equations with critical growth

$ \begin{equation*} -{\rm{div}}(g^{p}(u)|\nabla u|^{p-2}\nabla u)+ g^{p-1}(u)g'(u)|\nabla u|^{p}+ V(x)|u|^{p-2}u = h(x, u), \; \; x\in \mathbb{R}^{N}, \end{equation*} $

where $ N\geq3 $, $ 1 < p < N $. Under some suitable conditions, we prove that the above equation has a nontrivial positive solution by variational methods. To some extent, our results improve and supplement some existing relevant results.
- generalized quasilinear elliptic equations,
- variational methods,
- nontrivial positive solutions
Citation: Shulin Zhang. Existence of nontrivial positive solutions for generalized quasilinear elliptic equations with critical exponent[J]. AIMS Mathematics, 2022, 7(6): 9748-9766. doi: 10.3934/math.2022543

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Abstract

In this paper, we are concerned with the existence of nontrivial positive solutions for the following generalized quasilinear elliptic equations with critical growth

$ \begin{equation*} -{\rm{div}}(g^{p}(u)|\nabla u|^{p-2}\nabla u)+ g^{p-1}(u)g'(u)|\nabla u|^{p}+ V(x)|u|^{p-2}u = h(x, u), \; \; x\in \mathbb{R}^{N}, \end{equation*} $

where $ N\geq3 $, $ 1 < p < N $. Under some suitable conditions, we prove that the above equation has a nontrivial positive solution by variational methods. To some extent, our results improve and supplement some existing relevant results.

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