We consider nonlocal problems in which the leading operator contains a sign‐changing weight which can be unbounded. We begin studying the existence and the properties of the first eigenvalue. Then we study a nonlinear problem in which the nonlinearity does not satisfy the usual Ambrosetti‐Rabinowitz condition. Finally, we study a problem with general concave‐convex nonlinearities.
We consider a smooth, complete and non-compact Riemannian manifold (M, g) of dimension d ≥ 3, and we look for solutions to the semilinear elliptic equationThe potential V : M → R is a continuous function which is coercive in a suitable sense, while the nonlinearity f has a subcritical growth in the sense of Sobolev embeddings. By means of ∇-theorems introduced by Marino and Saccon, we prove that at least three nontrivial solutions exist as soon as the parameter λ is sufficiently close to an eigenvalue of the operator −Δ g .
Keywords Schrödinger equations • Riemannian manifolds • Variational methods • ∇-theoremsThe first and third authors are supported by GNAMPA, project "Equazioni alle derivate parziali: problemi e modelli.".
We consider a smooth, complete and non-compact Riemannian manifold (M, ) of dimension ≥ 3, and we look for positive solutions to the semilinear elliptic equationThe potential : M → ℝ is a continuous function which is coercive in a suitable sense, while the nonlinearity has a subcritical growth in the sense of Sobolev embeddings. By means of ∇-Theorems introduced by Marino and Saccon, we prove that at least three solution exists as soon as the parameter is sufficiently close to an eigenvalue of the operator − .
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