Multiple solution for a fourth-order nonlinear eigenvalue problem with singular and sublinear potential

Csaba Farkas, Ildikó-Ilona Mezei, Zsuzsánna Tímea Nagy


Let \((M,g)\) be a Cartan-Hadamard manifold. For certain positive numbers \(\mu\) and \(\lambda\), we establish the multiplicity of solutions to the problem $$\Delta_g^2 u-\Delta_g u+u=\mu \frac{u}{d_g(x_0,x)^4}+\lambda \alpha(x)f(u),\ \mbox{ in } M,$$ where \(x_0\in M\), while \(f:\mathbb{R}\to\mathbb{R}\) is a continuous function, superlinear at zero and sublinear at infinity.


singular potential; sublinearity at infinity; multiple solutions; Rellich inequality; Hadamard manifolds

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