The existence of infinitely many solutions for nonlinear elliptic equations involving p-Laplace type operators in \(\mathbb{R}^N\)
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Authors
Yun-Ho Kim
- Department of Mathematics Education, Sangmyung University, Seoul 03016, Republic of Korea.
Jung-Hyun Bae
- Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea.
Jongrak Lee
- Institute of Mathematical Sciences, Ewha Womans University, Seoul 03760, Republic of Korea.
Abstract
We are concerned with the following nonlinear elliptic equations
\[-div(\varphi(x,\nabla u))+b(x)|u|^{p-2}u=\lambda f(x,u) \qquad \texttt{in} \qquad\mathbb{R}^N, \]
where the function \(\varphi(x,v)\) is of type \(|v|^{p-2}v, b:\mathbb{R}^N\rightarrow (0,\infty)\) is a continuous potential function, \(\lambda\) is a real parameter, and
\(f:\mathbb{R}^N\times \mathbb{R}\rightarrow\mathbb{R}\) is a Carath´eodory function. In this paper, under suitable assumptions, we show the existence of infinitely
many weak solutions for the problem above without assuming the Ambrosetti and Rabinowitz condition, by using the fountain
theorem. Next, we give a result on the existence of a sequence of solutions for the problem above converging to zero in the
\(L^\infty\)-norm by employing the Moser iteration under appropriate conditions.
Share and Cite
ISRP Style
Yun-Ho Kim, Jung-Hyun Bae, Jongrak Lee, The existence of infinitely many solutions for nonlinear elliptic equations involving p-Laplace type operators in \(\mathbb{R}^N\) , Journal of Nonlinear Sciences and Applications, 10 (2017), no. 4, 2144--2161
AMA Style
Kim Yun-Ho, Bae Jung-Hyun, Lee Jongrak, The existence of infinitely many solutions for nonlinear elliptic equations involving p-Laplace type operators in \(\mathbb{R}^N\) . J. Nonlinear Sci. Appl. (2017); 10(4):2144--2161
Chicago/Turabian Style
Kim, Yun-Ho, Bae, Jung-Hyun, Lee, Jongrak. "The existence of infinitely many solutions for nonlinear elliptic equations involving p-Laplace type operators in \(\mathbb{R}^N\) ." Journal of Nonlinear Sciences and Applications, 10, no. 4 (2017): 2144--2161
Keywords
- p-Laplace type
- weak solution
- iteration method
- fountain theorem.
MSC
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