Electron. J. Diff. Equ., Vol. 2012 (2012), No. 109, pp. 1-23.

Existence of bound state solutions for degenerate singular perturbation problems with sign-changing potentials

Maria J. Alves, Ronaldo B. Assuncao, Paulo C. Carriao, Olimpio H. Miyagaki

Abstract:
In this article, we study the degenerate singular perturbation problems
$$\displaylines{ -\varepsilon^2\hbox{div}(|x|^{-2a}\nabla u)+|x|^{-2(a+1)}V(x)u = |x|^{-b2^*(a,b)}g(x,u),\cr -\hbox{div}(|x|^{-2a}\nabla u)+ \lambda |x|^{-2(a+1)}V(x)u = |x|^{-b2^*(a,b)}g(x,u), }$$
for $\varepsilon$ small and $\lambda$ large positive, where $x \in \mathbb{R}^N$ with $N \geq 3$. We search for solutions that decay to zero as $|x| \to +\infty$, when g is superlinear in the potential function changes signs. We prove the existence of bound state solutions for degenerate, singular, semilinear elliptic problems. Additionally, when the nonlinearity g(x,u) is an odd function of u, we obtain infinitely many geometrically distinct solutions.

Submitted September 22, 2011. Published June 27, 2012.
Math Subject Classifications: 35J20, 35J61, 35J70, 35J75, 35P10, 35P30.
Key Words: Semilinear degenerate elliptic equation; singular perturbation; variational method; sign-changing potential; nonlinear Schrodinger equation; bound state solution.

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Maria J. Alves
Departamento de Matemática
Universidade Federal de Minas Gerais, UFMG
Av. Antônio Carlos, 6627, Caixa Postal 702
CEP 30161-970, Belo Horizonte, MG, Brasil
email: mariajose@ufmg.br
Ronaldo B. Assunção
Departamento de Matemática
Universidade Federal de Minas Gerais, UFMG
Av. Antônio Carlos, 6627, Caixa Postal 702
CEP 30161-970, Belo Horizonte, MG, Brasil
email: ronaldo@mat.ufmg.br
Paulo C. Carrião
Departamento de Matemática
Universidade Federal de Minas Gerais, UFMG
Av. Antônio Carlos, 6627, Caixa Postal 702
CEP 30161-970, Belo Horizonte, MG, Brasil
email: carrion@mat.ufmg.br
Olímpio H. Miyagaki
Departamento de Matemática
Universidade Federal de Juiz de Fora, UFJF
Cidade Universitária, Bairro Martelos
CEP 36036-330, Juiz de Fora, MG, Brasil
email: ohmiyagaki@gmail.com

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