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Multiple solutions for a problem with resonance involving the \(p\)-Laplacian. (English) Zbl 0968.35047

Summary: We will investigate the existence of multiple solutions for the equation \[ -\Delta_p u+ g(x,u)= \lambda_1 h(x)|u|^{p-2} u,\quad\text{in }\Omega,\quad u\in H^{1,p}_0(\Omega),\tag{P} \] where \(\Delta_p u= \text{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian operator, \(\Omega\subseteq \mathbb{R}^N\) is a bounded domain with smooth boundary, \(h\) and \(g\) are bounded functions, \(N\geq 1\) and \(1< p<\infty\). Using the mountain pass theorem and the Ekeland variational principle, we show the existence of at least three solutions for (P).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A15 Variational methods applied to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J25 Boundary value problems for second-order elliptic equations
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