Positive Ground State Solution Research Articles

Positive and Nodal Solutions For a Nonlinear Schrödinger Equation with Indefinite Potential

Abstract We deal with the nonlinear Schrödinger equation -Δu + V(x)u = f(u) in ℝN, where V is a (possible) sign changing potential satisfying mild assumptions and the nonlinearity f ∈ C1(ℝ, ℝ) is a subcritical and superlinear function. By combining variational techniques and the concentration-compactness principle we obtain a positive ground state solution and also a nodal solution. The proofs rely in localizing the infimum of the associated functional constrained to Nehari type sets.

Multiple solutions for some quasilinear Neumann problems in exterior domains

In this paper, we show that if $K(x)$ satisfies suitable conditions, then the quasilinear Neumann problem $-\Delta_pu+|u|^{p-2}u=K(x)|u|^{q-2}u$ in an exterior domain $\varOmega$ has at least two solutions, of which one is a positive ground-state solution and one is a nodal solution.

Symmetry of Ground States of p -Laplace Equations via the Moving Plane Method

. In this paper we use the moving plane method to get the radial symmetry about a point \(x_0 \in {\mathbb R}^N\) of the positive ground state solutions of the equation \(-{\rm div}\; (|D u|^{p - 2} Du) = f(u)\) in \({\mathbb R}^N\), in the case \(1 < p < 2\). We assume f to be locally Lipschitz continuous in \((0, + \infty)\) and nonincreasing near zero but we do not require any hypothesis on the critical set of the solution. To apply the moving plane method we first prove a weak comparison theorem for solutions of differential inequalities in unbounded domains.

On concentration of positive bound states of nonlinear Schrödinger equations

We study the concentration behavior of positive bound states of the nonlinear Schrödinger equation $$ih\frac{{\partial \psi }}{{\partial t}} = \frac{{ - h^2 }}{{2m}}\Delta \psi + V\left( x \right)\psi - \gamma \left| \psi \right|^{p - 1} \psi .$$ Under certain condition ofV, we show that positive ground state solutions must concentrate at global minimum points ofV ash→0+; moreover, a point at which a sequence of positive bound states concentrates must be a critical point ofV. In cases thatV is radial, we prove that the positive radial solutions with least energy among all nontrivial radial solutions must concentrate at the origin ash→0+.