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In this paper, we study the uniqueness of nodal radial solutions to nonlinear elliptic equations in the unit ball in ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$. Under suitable conditions, we prove that, for any given positive integer k$$ k $$, the problem we considered has at most one solution possessing exactly k−1$$ k-1 $$ nodes. Together with the results presented by Nagasaki [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (2): 211–232, 1989] and Tanaka [Proc. Roy. Soc. Edinburgh Sect. A. 138 (6): 1331–1343, 2008], we can prove that more types of nonlinear elliptic equations have the uniqueness of nodal radial solutions. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear Schrödinger‐type equation −(∇+iA)2u+u+λIα∗K|u|2Ku=af(|u|)|u|uinℝ3$$ -{\left(\nabla + iA\right)}^2u+u+\lambda \left[{I}_{\alpha}\ast \left(K{\left|u\right|}^2\right)\right] Ku=a\frac{f\left(|u|\right)}{\mid u\mid }u\kern.5em \mathrm{in}\kern.5em {\mathrm{\mathbb{R}}}^3 $$is considered in the presence of magnetic field, where A∈C1(ℝ3,ℝ3)$$ A\in {C}^1\left({\mathrm{\mathbb{R}}}^3,{\mathrm{\mathbb{R}}}^3\right) $$, α∈(0,3)$$ \alpha \in \left(0,3\right) $$, Iα$$ {I}_{\alpha } $$ denotes the Riesz potential, K∈Lp(ℝ3)$$ K\in {L}^p\left({\mathrm{\mathbb{R}}}^3\right) $$ is a positive potential for some p∈(6/(1+α),∞]$$ p\in \left(6/\left(1+\alpha \right),\infty \right] $$, a∈Lq(ℝ3)\{0}$$ a\in {L}^q\left({\mathrm{\mathbb{R}}}^3\right)\backslash \left\{0\right\} $$ is a nonnegative potential for some q∈(3/2,∞]$$ q\in \left(3/2,\infty \right] $$, and f∈C(ℝ,[0,∞))$$ f\in C\left(\mathrm{\mathbb{R}},\right[0,\infty \left)\right) $$ is assumed to be asymptotically linear at infinity. Under suitable assumptions regarding A$$ A $$, K$$ K $$, a$$ a $$, and f$$ f $$, variational methods are used to establish the existence of ground‐state solutions of the above equation for sufficiently small values of the parameter λ$$ \lambda $$. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of solutions to a quadratically coupled Schrödinger systems − Δ u = μ 1 u + 2 α u v i n R N , − Δ v = μ 2 v + α u 2 i n R N under the condition ∫ R N u 2 = a , ∫ R N v 2 = b. Here N=1, 2, α > 0 and a, b>0 are fixed. In the systems, μ 1 and μ 2 are unknown. We prove that there exists a solution (μ 1 , μ 2 , u ~ , v ~) to the systems with μ 1 < 0 , μ 2 < 0 , and u ~ , v ~ ∈ C 2 (R N) being positive, radially symmetric, and decreasing with r = | x | . Our argument is heavily based on the rearrangement techniques. [ABSTRACT FROM AUTHOR]