Electron. J. Diff. Equ., Vol. 2010(2010), No. 05, pp. 1-16.

Existence of positive and sign-changing solutions for p-laplace equations with potentials in R^N

Mingzhu Wu, Zuodong Yang

Abstract:
We study the perturbed equation
$$\displaylines{ -\varepsilon^{p}\hbox{div}(|\nabla u|^{p-2}\nabla u)+V(x)|u|^{p-2}u=h(x,u)+K(x)|u|^{p^*-2}u,\quad x\in \mathbb{R}^N\cr u(x)\to 0\quad \hbox{as } |x|\to\infty\,. }$$
where $2\leq p<N$, $p^*={\frac{pN}{N-p}}$, $p<q<p^*$. Under proper conditions on $V(x)$ and $h(x,u)$, we obtain the existence and multiplicity of solutions. We also study the existence of solutions which change sign.

Submitted July 23, 2009. Published January 13, 2010.
Math Subject Classifications: 35J25, 35J60.
Key Words: Potential; critical point theory; p-Laplace; sign changing solution; multiplicity of solutions; concentration-compactness.

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  Mingzhu Wu
Institute of Mathematics, School of Mathematical Science
Nanjing Normal University, Jiangsu Nanjing 210046, China
email: wumingzhu_2010@163.com
Zuodong Yang
Institute of Mathematics, School of Mathematical Science
Nanjing Normal University, Jiangsu Nanjing 210046, China.
College of Zhongbei, Nanjing Normal University
Jiangsu Nanjing 210046, China
email: zdyang_jin@263.net

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