Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 109, pp. 1-23.
Title: Existence of bound state solutions for degenerate singular
perturbation problems with sign-changing potentials
Authors: Maria J. Alves (Univ. Federal de Minas Gerais, Brazil)
Ronaldo B. Assuncao (Univ. Federal de Minas Gerais, Brazil)
Paulo C. Carriao (Univ. Federal de Minas Gerais, Brazil)
Olimpio H. Miyagaki (Univ. Federal de Juiz de Fora, Brazil)
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.