In this article, we consider the existence and asymptotic behavior of solutions for the Hénon equation −∆BN u = (d(x)) α|u|p−2u, x ∈ Ω u = 0 x ∈ ∂Ω, where ∆BN denotes the Laplace Beltrami operator on the disc model of the Hyperbolic space BN , d(x) = dBN (0, x), Ω ⊂ BN is geodesic ball with radius 1, α > 0, N ≥ 3. We study the existence of hyperbolic symmetric solutions when 2 < p < 2N+2α N−2 ....

We construct a center-stable manifold of the ground state solitons in the energy space for the critical wave equation without imposing any symmetry, as the dynamical threshold between scattering and blow-up, and also as a collection of solutions which stay close to the ground states. Up to energy slightly above the ground state, this completes the 9-set classification of the global dynamics in ...

This article concerns the Schrödinger equation −∆u+ V (x)u = f(x, u), for x ∈ R , u(x)→ 0, as |x| → ∞ . Assuming that V and f are periodic in x, and 0 is a boundary point of the spectrum σ(−∆ + V ), two types of ground state solutions are obtained with some super-quadratic conditions.

We consider the elliptic problem −Δu+ u= b(x)|u|p−2u+ h(x) in Ω, u∈H1 0 (Ω), where 2 < p < (2N/(N − 2)) (N ≥ 3), 2 < p <∞ (N = 2), Ω is a smooth unbounded domain in RN , b(x) ∈ C(Ω), and h(x) ∈ H−1(Ω). We use the shape of domain Ω to prove that the above elliptic problem has a ground-state solution if the coefficient b(x) satisfies b(x)→ b∞ > 0 as |x| →∞ and b(x)≥ c for some suitable constants ...

We study the fractional p-Kirchhoff equation $$ \Big( a+b \int_{\mathbb{R}^N}{\int_{\mathbb{R}^N}} \frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dx\, dy\Big) (-\Delta)_p^s u-\mu|u|^{p-2}u=|u|^{q-2}u, \quad x\in\mathbb{R}^N, where \((-\Delta)_p^s\) is p-Laplacian operator, a and b are strictly positive real numbers, \(s \in (0,1)\), \(1 &lt; p&lt; N/s,\) \(p&lt; q&lt; p^*_s-2\) with \(p^*_s=\frac{Np}{N-p...

‎This paper is devoted to get a ground state solution for a class of nonlinear elliptic equations with fast increasing weight‎. ‎We apply the variational methods to prove the existence of ground state solution‎.

We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schrödinger equations. We examine two constrained minimization problems, which give rise to such solutions. One yields what we call Fλ-minimizers, the other energy minimizers. We produce such ground state solution...