Geodesic
Existence of bound state solutions for degenerate singular perturbation problems with sign-changing potentials
Electronic journal of differential equations, Tome 2012 (2012)
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.
Classification : 35J20, 35J61, 35J70, 35J75, 35P10, 35P30
Keywords: semilinear degenerate elliptic equation, singular perturbation, variational method, sign-changing potential, nonlinear Schrödinger equation, bound state solution 
@article{EJDE_2012__2012__a69,
     author = {Alves,  Maria J. and Assuncao,  Ronaldo B. and Carriao,  Paulo C. and Miyagaki,  Olimpio H.},
     title = {Existence of bound state solutions for degenerate singular perturbation problems with sign-changing potentials},
     journal = {Electronic journal of differential equations},
     year = {2012},
     volume = {2012},
     zbl = {1259.35075},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a69/}
}
TY  - JOUR
AU  - Alves,  Maria J.
AU  - Assuncao,  Ronaldo B.
AU  - Carriao,  Paulo C.
AU  - Miyagaki,  Olimpio H.
TI  - Existence of bound state solutions for degenerate singular perturbation problems with sign-changing potentials
JO  - Electronic journal of differential equations
PY  - 2012
VL  - 2012
UR  - http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a69/
LA  - en
ID  - EJDE_2012__2012__a69
ER  - 
%0 Journal Article
%A Alves,  Maria J.
%A Assuncao,  Ronaldo B.
%A Carriao,  Paulo C.
%A Miyagaki,  Olimpio H.
%T Existence of bound state solutions for degenerate singular perturbation problems with sign-changing potentials
%J Electronic journal of differential equations
%D 2012
%V 2012
%U http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a69/
%G en
%F EJDE_2012__2012__a69
Alves,  Maria J.; Assuncao,  Ronaldo B.; Carriao,  Paulo C.; Miyagaki,  Olimpio H. Existence of bound state solutions for degenerate singular perturbation problems with sign-changing potentials. Electronic journal of differential equations, Tome 2012 (2012). http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a69/