Electron. J. Diff. Eqns., Vol. 2003(2003), No. 83, pp. 1-14.

Remarks on least energy solutions for quasilinear elliptic problems in $\mathbb{R}^N$

Joao Marcos do O & Everaldo S. Medeiros

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 $\Delta_p u=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator and $ 1 lesss than p\leq 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.

Submitted June 3, 2003. Published August 11, 2003.
Math Subject Classifications: 35J20, 35J60.
Key Words: Variational methods, minimax methods, superlinear elliptic problems, p-Laplacian, ground-states, moutain-pass solutions

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Joao Marcos do O
Departamento de Matematica,
Univ. Fed. Paraiba
58059-900 Joao Pessoa, PB, Brazil
email: jmbo@mat.ufpb.br
Everaldo S. Medeiros
Departamento de Matematica,
Univ. Fed. Paraiba
58059-900 Joao Pessoa, PB, Brazil
email: everaldo@mat.ufpb.br

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