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).


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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