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GROUND STATE SOLUTIONS FOR $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$-SUPERLINEAR $p$-LAPLACIAN EQUATIONS

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  15 May 2014

YI CHEN  and
X. H. TANG
YI CHEN
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, PR China email mathcyt@163.com
X. H. TANG*
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, PR China email tangxh@csu.edu.cn
*
✉tangxh@csu.edu.cn
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Abstract

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In this paper, we deduce new conditions for the existence of ground state solutions for the $p$-Laplacian equation

$$\begin{equation*}\left \{ \begin{array}{@{}ll} -\mathrm {div}(|\nabla u|^{p-2}\nabla u)+V(x)|u|^{p-2}u=f(x, u), \quad x\in {\mathbb {R}}^{N},\\[5pt] u\in W^{1, p}({\mathbb {R}}^{N}), \end{array} \right . \end{equation*}$$
which weaken the Ambrosetti–Rabinowitz type condition and the monotonicity condition for the function $t\mapsto f(x, t)/|t|^{p-1}$. In particular, both $tf(x, t)$ and $tf(x, t)-pF(x, t)$ are allowed to be sign-changing in our assumptions.

Keywords

p-Laplacian equationp-superlinearground state solutions

MSC classification

Secondary: 35J30: Higher-order elliptic equations

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 97 , Issue 1 , August 2014 , pp. 48 - 62
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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