Remarks on least energy solutions for quasilinear elliptic problems in $R^N$

Abstract

In this work we establish some properties of the solutions to the quasilinear second-order problem $$ -Delta_p w=g(w)quad hbox{in } mathbb{R}^N $$ where $Delta_p u=mathop{ m div}(| abla u|^{p-2} abla u)$ is the $p$-Laplacian operator and $ 1<pleq N $. We study a mountain pass characterization of least energy solutions of this problem. Without assuming the monotonicity of the function $t^{1-p}g(t)$, we show that the Mountain-Pass value gives the least energy level. We also prove the exponential decay of the derivatives of the solutions.

Keywords

• Variational methods
• minimax methods
• superlinear elliptic problems
• p-Laplacian
• ground-states
• moutain-pass solutions