In this paper we study the existence of nontrivial solutions for a nonlinear boundary value problem posed on the half-line. Our approach is based on Ekeland’s variational principle.
Let \(\mathbb{D}\) denote the unit disk \(\{z:|z|<1\}\) in the complex plane \(\mathbb{C}\). In this paper, we study a family of polynomials \(P\) with only one zero lying outside \(\overline{\mathbb{D}}\). We establish criteria for \(P\) to satisfy implying that each of \(P\) and \(P'\) has exactly one critical point outside \(\overline{\mathbb{D}}\).
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In this paper we study a quasilinear resonant problem with discontinuous right hand side. To develop an existence theory we pass to a multivalued version of the problem, by filling in the gaps at the discontinuity points. We prove the existence of a nontrivial solution using a variational approach based on the critical point theory of nonsmooth locally Lipschitz functionals.
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We consider a quasilinear elliptic eigenvalue problem with a discontinuous right hand side. To be able to have an existence theory, we pass to a multivalued problem (elliptic inclusion). Using a variational approach based on the critical point theory for locally Lipschitz functions, we show that we have at least three nontrivial solutions when $λ → λ_1$ from the left, $λ_1$ being the principal eigenvalue of the p-Laplacian with the Dirichlet boundary conditions.
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