Electron. J. Diff. Equ., Vol. 2013 (2013), No. 239, pp. 1-17.

Continuous dependence of solutions for indefinite semilinear elliptic problems

Elves A. B. Silva, Maxwell L. Silva

Abstract:
We consider the superlinear elliptic problem
$$ -\Delta u + m(x)u = a(x)u^p $$
in a bounded smooth domain under Neumann boundary conditions, where $m \in L^{\sigma}(\Omega)$, $\sigma\geq N/2$ and $a\in C(\overline{\Omega})$ is a sign changing function. Assuming that the associated first eigenvalue of the operator $-\Delta + m $ is zero, we use constrained minimization methods to study the existence of a positive solution when $\widehat{m}$ is a suitable perturbation of m.

Submitted August 12, 2011. Published October 24, 2013.
Math Subject Classifications: 35J20, 35J60, 35Q55.
Key Words: Positive solution; constrained minimization; eigenvalue problem; Neumann boundary condition; unique continuation.

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Elves A. B. Silva
Universidade de Brasília
Departamento de Matemática
Cep 70 910 900, Brasília-DF, Brazil
email: elves@unb.br
Maxwell L. Silva
Universidade Federal de Goiás
Instituto de Matemática e Estatística
Cep 74 001 970, Goiania-GO, Brazil
email: maxwell@mat.ufg.br

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