This paper studies the existence of multiple solutions of the second-order difference boundary value problem Δ2u n − 1 V ′ u n 0, n ∈ Z 1, T , u 0 0 u T 1 . By applying Morse theory, critical groups, and the mountain pass theorem, we prove that the previous equation has at least three nontrivial solutions when the problem is resonant at the eigenvalue λk k ≥ 2 of linear difference problem Δ2u n...

In this paper, we study a kind of anisotropic p-Laplacian equations in R n. Nontrivial solutions are obtained using mountain pass theorem given by Ambrosetti-Rabinowitz [1].

Given a Riemannian manifold (M,g) and group G of isometries (M,g), we investigate the existence G-invariant positive solutions u:M→R to nonlinear equation Δgu+au=u2⋆(k,s)−1dg(x,Gx0)s+huq−1 where Δg=−divg(∇). The singularity nonlinearity are such that problem is critical for Hardy-Sobolev embeddings. We prove by using Aubin minimization Mountain-Pass lemma Ambrosetti-Rabinowitz. As product our a...

We consider the following nonlinear singular elliptic equation −div (|x| −2a ∇u) = K(x)|x| −bp |u| p−2 u + λg(x) in R N , where g belongs to an appropriate weighted Sobolev space, and p denotes the Caffarelli–Kohn– Nirenberg critical exponent associated to a, b, and N. Under some natural assumptions on the positive potential K(x) we establish the existence of some λ 0 > 0 such that the above pr...

We establish the existence of an entire solution for a class of stationary Schrödinger equations with subcritical discontinuous nonlinearity and lower bounded potential that blows-up at infinity. The abstract framework is related to Lebesgue–Sobolev spaces with variable exponent. The proof is based on the critical point theory in the sense of Clarke and we apply Chang’s version of the Mountain ...

In this paper we consider two elliptic problems. The first one is a Dirichlet problem while the second is Neumann. We extend all the known results concerning Landesman-Laser conditions by using the Mountain-Pass theorem with the Cerami (PS) condition.

Let Ω be a bounded subset of Rn with smooth boundary. We investigate the solvability for a class of the system of the nonlinear elliptic equations with Dirichlet boundary condition. Using the mountain pass theorem we prove that the system has at least one nontrivial solution.

In this paper, we study the existence and multiplicity of periodic solutions of the following second-order Hamiltonian systems ẍ(t) + V ′(t, x(t)) = 0, where t ∈ R, x ∈ R and V ∈ C(R × R ,R). By using a symmetric mountain pass theorem, we obtain a new criterion to guarantee that second-order Hamiltonian systems has infinitely many periodic solutions. We generalize and improve recent results fro...

This paper is concerned with the existence of solutions for quasilinear elliptic equations −Δpu−Δp(|u|2α)|u|2α−2u+V(x)|u|p−2u=|u|q−2u,x∈RN, where α≥1, 10 a continuous function. In this work, we mainly focus on nontrivial solutions. When 2αp<q<p∗, establish by using Mountain-Pass lemma; when q≥2αp∗, Pohozaev type variational identity, p...

‎we study the existence of soliton solutions for a class of‎ ‎quasilinear elliptic equation in $mathbb{textbf{r}}^2$ with critical exponential growth‎. ‎this model has been proposed in the self-channeling of a‎ ‎high-power ultra short laser in matter‎.