Nonradial solutions for semilinear Schrödinger equations with sign-changing potential

Multiple Solutions for Semilinear Elliptic Equations with Sign-changing Potential and Nonlinearity

In this article, we study the multiplicity of solutions for the semilinear elliptic equation −∆u + a(x)u = f(x, u), x ∈ Ω, u = 0, x ∈ ∂Ω, where Ω ⊂ RN (N ≥ 3), the potential a(x) satisfies suitable integrability conditions, and the primitive of the nonlinearity f is of super-quadratic growth near infinity and is allowed to change sign. Our super-quadratic conditions are weaker the usual super-q...

Multiple Positive Solutions for Semilinear Elliptic Equations with Sign - Changing Weight Functions

and Applied Analysis 3 In order to describe our main result, we need to define Λ0 ( 2 − q ( p − q‖a‖L∞ ) 2−q / p−2 ( p − 2 ( p − q)‖b ‖Lq∗ ) S p 2−q /2 p−2 q/2 p > 0, 1.3 where ‖a‖L∞ supx∈RNa x , ‖b ‖Lq∗ ∫ RN |b x |qdx 1/q∗ and Sp is the best Sobolev constant for the imbedding of H1 R into L R . Theorem 1.1. Assume that a1 , b1 b2 hold. If λ ∈ 0, q/2 Λ0 , Ea,λb admits at least two positive solu...

Infinitely Many Solutions for a Semilinear Elliptic Equation with Sign-Changing Potential

Nondegeneracy of Nonradial Sign-changing Solutions to the Nonlinear Schrödinger Equations

We prove the non-degeneracy of the non-radial sign-changing solutions to the nonlinear Schrödinger equation ∆u− u+ |u|p−1u = 0 in R constructed by Musso, Pacard and Wei [19].

Sign-changing Multi-bump Solutions for Nonlinear Schrödinger Equations with Steep Potential Wells

We study the nonlinear Schrödinger equations: (Pλ) −∆u+(λa(x)+1)u = |u|p−1u, u ∈ H(R ), where p > 1 is a subcritical exponent, a(x) is a continuous function satisfying a(x) ≥ 0, 0 < lim inf |x|→∞ a(x) ≤ lim sup|x|→∞ a(x) < ∞ and a−1(0) consists of 2 connected bounded smooth components Ω1 and Ω2. We study the existence of solutions (uλ) of (Pλ) which converge to 0 in RN \ (Ω1 ∪Ω2) and to a presc...