Abstract
This paper is concerned with the following semilinear elliptic systems: $$\left \{ \textstyle\begin{array}{@{}l} -\Delta u+V(x)u=H(x)F_{u}(x, u, v), \quad x\in\mathbb{R}^{N}, -\Delta v+V(x)v=H(x)F_{v}(x, u, v), \quad x\in\mathbb{R}^{N}, u(x)\rightarrow0,\qquad v(x)\rightarrow0\quad \mbox{as } |x|\rightarrow\infty, \end{array}\displaystyle \right . $$ where $V(x)$ , $H(x)$ are nonnegative continuous functions. Under some appropriate assumptions on $V(x)$ , $H(x)$ , and $F(x, u, v)$ , we prove the existence of infinitely many small negative-energy solutions by using the fountain theorem established by Zou. Recent results from the literature are extended.
Highlights
In this paper, we consider the existence and multiplicity of solutions to the following semilinear elliptic systems:⎧ ⎪⎨– u + V (x)u = H(x)Fu(x, u, v), x ∈ RN, ⎪⎩–u(x)v + V (x)v →, H (x)Fv v(x) →(x, u, v), as |x| x ∈ RN → ∞, ( . )where V (x), H(x) are nonnegative continuous functions, we assume that the functions V (x), H(x), and F(x, u, v) satisfy the following hypotheses:(H ) V ∈ C(RN, R) satisfies infx∈RN V (x) ≥ a >, where a > is a constant
Under some appropriate assumptions on V(x), H(x), and F(x, u, v), we prove the existence of infinitely many small negative-energy solutions by using the fountain theorem established by Zou
Where V (x), H(x) are nonnegative continuous functions, we assume that the functions V (x), H(x), and F(x, u, v) satisfy the following hypotheses: (H ) V ∈ C(RN, R) satisfies infx∈RN V (x) ≥ a >, where a > is a constant
Summary
Under some appropriate assumptions on V(x), H(x), and F(x, u, v), we prove the existence of infinitely many small negative-energy solutions by using the fountain theorem established by Zou. Recent results from the literature are extended. We consider the existence and multiplicity of solutions to the following semilinear elliptic systems: For the results on existence, multiple solutions, and positive solutions to problem
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