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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]

A sufficient condition for blowup of solutions to a class of pseudo-parabolic equations with a nonlocal term is established in this paper. In virtue of the potential wells method, we first extend the results obtained by Xu and Su in [J. Funct. Anal., 264 (12): 2732-2763, 2013] to the nonlocal case and describe successfully the behavior of solutions by using the energy functional, Nehari functional, and the ground state energy of the stationary equation. Sequently, we study the boundedness and convergency of any global solution. Finally, we achieve a criterion to guarantee the blowup of solutions without any limit of the initial energy.Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]