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In this article, we study Lie symmetries to fundamental solutions to the Leutwiler-Weinstein equation $$ Lu:={\Delta} u+\frac{k}{x^{n}}\frac{\partial u}{\partial x^{n}}+\frac{\ell}{(x^{n})^{2}}u=0 $$ L u : = Δ u + k x n ∂ u ∂ x n + ℓ ( x n ) 2 u = 0 in the upper half-space $\mathbb {R}^{n}_{+}$ ℝ + n . Starting from the infinitesimal generators of the equation Lu = 0, we deduce symmetries of the equation Lu = δ(x − x0), and using its invariant solutions, we construct a fundamental solution. As an application, we study a Green functions of the operator in the hyperbolic unit ball.
In this article, we study Lie symmetries to fundamental solutions to the Leutwiler-Weinstein equation $$ Lu:={\Delta} u+\frac{k}{x^{n}}\frac{\partial u}{\partial x^{n}}+\frac{\ell}{(x^{n})^{2}}u=0 $$ L u : = Δ u + k x n ∂ u ∂ x n + ℓ ( x n ) 2 u = 0 in the upper half-space $\mathbb {R}^{n}_{+}$ ℝ + n . Starting from the infinitesimal generators of the equation Lu = 0, we deduce symmetries of the equation Lu = δ(x − x0), and using its invariant solutions, we construct a fundamental solution. As an application, we study a Green functions of the operator in the hyperbolic unit ball.
<p style='text-indent:20px;'>In this paper, we investigate the nonlinear Schrödinger-Poisson equation with magnetic field. By combining non-Nehari manifold method and some new energy estimate inequalities, we obtain the existence of a ground state solution, where the strict monotonicity condition and the Ambrosetti-Rabinowitz growth condition are not needed. Moreover, when both the potential and the nonlinearity are sign-changing, by applying the Fountain Theorem and some analytical techniques, we prove the existence of infinitely many solutions. Our results extend and improve the present ones in the literature.</p>
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