Positive solutions of nonlinear Robin eigenvalue problems

Existence of Positive Solutions for Nonlinear Eigenvalue Problems

Positive solutions and nonlinear multipoint conjugate eigenvalue problems ∗

Values of λ are determined for which there exist solutions in a cone of the n order nonlinear differential equation, u = λa(t)f(u), 0 < t < 1, satisfying the multipoint boundary conditions, u(ai) = 0, 0 ≤ j ≤ ni−1, 1 ≤ i ≤ k, where 0 = a1 < a2 < · · · < ak = 1, and ∑k i=1 ni = n, where a and f are nonnegative valued, and where both lim |x|→0+ f(x)/|x| and lim |x|→∞ f(x)/|x| exist.

Multiple Nontrivial Solutions for Nonlinear Eigenvalue Problems

In this paper we study a nonlinear eigenvalue problem driven by the p-Laplacian. Assuming for the right-hand side nonlinearity only unilateral and sign conditions near zero, we prove the existence of three nontrivial solutions, two of which have constant sign (one is strictly positive and the other is strictly negative), while the third one belongs to the order interval formed by the two opposi...

Bifurcation of Positive Solutions for Nonlinear Nonhomogeneous Robin and Neumann Problems with Competing Nonlinearities

In this paper we deal with Robin and Neumann parametric elliptic equations driven by a nonhomogeneous differential operator and with a reaction that exhibits competing nonlinearities (concave-convex nonlinearities). For the Robin problem and without employing the Ambrosetti-Rabinowitz condition, we prove a bifurcation theorem for the positive solutions for small values of the parameter λ > 0. F...

Positive and Nodal Solutions for Parametric Nonlinear Robin Problems with Indefinite Potential

We consider a parametric nonlinear Robin problem driven by the p−Laplacian plus an indefinite potential and a Carathéodory reaction which is (p−1)− superlinear without satisfying the Ambrosetti Rabinowitz condition. We prove a bifurcation-type result describing the dependence of the set of positive solutions on the parameter. We also prove the existence of nodal solutions. Our proofs use tools ...