In this article, we study the existence and multiplicity of positive solutions for the quasi-linear elliptic problems involving critical Sobolev exponent and a Hardy term. The main tools adopted in our proofs are the concentration compactness principle and Nehari manifold.

In this article, we show the existence of multiple positive solutions to a class of degenerate elliptic equations involving critical cone Sobolev exponent and sign-changing weight function on singular manifolds with the help of category theory and the Nehari manifold method.

We employ Nehari manifold methods and critical point theory to study the existence of nontrivial homoclinic solutions of discrete p-Laplacian equations with a coercive weight function and superlinear nonlinearity. Without assuming the classical Ambrosetti-Rabinowitz condition and without any periodicity assumptions, we prove the existence and multiplicity results of the equations.

In this work we derive oscillation and nonoscillation criteria for the one dimensional p-laplacian in terms of an eigenvalue inequality for a mixed problem. We generalize the results obtained in the linear case by Nehari and Willet, and the proof is based on a Picone type identity.

and Applied Analysis 3 Let Kλ,μ : E → R be the functional defined by Kλ,μ (z) = ∫ Ω (λf (x) |u| q + μg (x) |V| q ) dx ∀z = (u, V) ∈ E. (11) We know that Iλ,μ is not bounded below on E. From the following lemma, we have that Iλ,μ is bounded from below on the Nehari manifoldNλ,μ defined in (9). Lemma 3. The energy functional Iλ,μ is coercive and bounded below onNλ,μ. Proof. If z = (u, V) ∈ Nλ,μ, ...

In this article, we study the semilinear elliptic equation −∆u = |u|p(x)−2u, x ∈ R u ∈ D(R ), where N ≥ 3, p(x) = ( p, x ∈ Ω 2∗, x 6∈ Ω, with 2 < p < 2∗ := 2N/(N − 2), Ω ⊂ RN is a bounded set with nonempty interior. By using the Nehari manifold, we obtain a positive ground state solution.

where Ω ⊂ R is a smooth domain with smooth boundary ∂Ω such that 0 Î Ω, Δpu = div(|∇u|∇u), 1 < p < N, μ < μ̄ = ( N−p p ), l >0, 1 < q < p, sign-changing weight functions f and g are continuous functions on ̄, μ̄ = ( N−p p ) p is the best Hardy constant and p∗ = Np N−p is the critical Sobolev exponent. By extracting the Palais-Smale sequence in the Nehari manifold, the multiplicity of positive solu...

Recall that the rank of a finitely generated group is the minimal number of elements needed to generate it. In [Whi02], M. White proved that the injectivity radius of a closed hyperbolic 3-manifold M is bounded above by some function of rank(π1(M)). Building on a technique that he introduced, we determine the ranks of the fundamental groups of a large class of hyperbolic 3-manifolds fibering ov...

In the last decade or so, variational gluing methods have been widely used to construct homoclinic and heteroclinic type solutions of nonlinear elliptic equations and Hamiltonian systems. This note is concerned with the procedure of gluing mountain-pass type solutions. The rst procedure to glue mountain-pass type solutions was developed through the work of S er e, and Coti Zelati-Rabinowitz. Th...