Search results for "Nehari manifold"
showing 6 items of 6 documents
Positive solutions for singular double phase problems
2021
Abstract We study the existence of positive solutions for a class of double phase Dirichlet equations which have the combined effects of a singular term and of a parametric superlinear term. The differential operator of the equation is the sum of a p-Laplacian and of a weighted q-Laplacian ( q p ) with discontinuous weight. Using the Nehari method, we show that for all small values of the parameter λ > 0 , the equation has at least two positive solutions.
Least energy solutions to the Dirichlet problem for the equation −D(u) = f (x, u)
2017
Let be a bounded smooth domain in RN. We prove a general existence result of least energy solutions and least energy nodal ones for the problem −u = f(x, u) in u = 0 on ∂ (P) where f is a Carathéodory function. Our result includes some previous results related to special cases of f . Finally, we propose some open questions concerning the global minima of the restriction on the Nehari manifold of the energy functional associated with (P) when the nonlinearity is of the type f(x, u) = λ|u| s−2u − μ|u| r−2u, with s, r ∈ (1, 2) and λ,μ > 0.
Continuous spectrum for a two phase eigenvalue problem with an indefinite and unbounded potential
2020
Abstract We consider a two phase eigenvalue problem driven by the ( p , q ) -Laplacian plus an indefinite and unbounded potential, and Robin boundary condition. Using a modification of the Nehari manifold method, we show that there exists a nontrivial open interval I ⊆ R such that every λ ∈ I is an eigenvalue with positive eigenfunctions. When we impose additional regularity conditions on the potential function and the boundary coefficient, we show that we have smooth eigenfunctions.
Least Energy Solutions with Sign Information for Parametric Double Phase Problems
2022
We consider a parametric double phase Dirichlet problem. In the reaction there is a superlinear perturbation term which satisfies a weak Nehari-type monotonicity condition. Using the Nehari manifold method, we show that for all parameters below a critical value, the problem has at least three nontrivial solutions all with sign information. The critical parameter value is precisely identified in terms of the spectrum of the lower exponent part of the differential operator.
Nonlinear concave-convex problems with indefinite weight
2021
We consider a parametric nonlinear Robin problem driven by the p-Laplacian and with a reaction having the competing effects of two terms. One is a parametric (Formula presented.) -sublinear term (concave nonlinearity) and the other is a (Formula presented.) -superlinear term (convex nonlinearity). We assume that the weight of the concave term is indefinite (that is, sign-changing). Using the Nehari method, we show that for all small values of the parameter (Formula presented.), the problem has at least two positive solutions and also we provide information about their regularity.
Hitchhiker's guide to the fractional Sobolev spaces
2012
AbstractThis paper deals with the fractional Sobolev spaces Ws,p. We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results.Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.