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

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

Electronic Journal of Differential Equations, Tome 2012 (2012).

Résumé

Summary: 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.

Détail

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