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A semilinear elliptic problem involving nonlinear boundary condition and sign-changing potential. (English) Zbl 1128.35350
Summary: In this paper, we study the multiplicity of nontrivial nonnegative solutions for a semilinear elliptic equation involving nonlinear boundary condition and sign-changing potential. With the help of the Nehari manifold, we prove that the semilinear elliptic equation: \[ \displaylines{ -\Delta u+u=\lambda f(x)|u|^{q-2}u \quad \text{in }\Omega , \cr \frac{\partial u}{\partial \nu }=g(x)|u| ^{p-2}u \quad \text{on }\partial \Omega , } \] has at least two nontrivial nonnegative solutions for \(\lambda \) is sufficiently small.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35J60 Nonlinear elliptic equations
47J30 Variational methods involving nonlinear operators
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