Ground state solutions for magnetic Schrödinger equations with polynomial growth

Abstract

In this article, we investigate the following nonlinear magnetic Schrödinger equations: (−i∇+A(x))2u+V(x)u=f1(x,∣v∣2)v,(−i∇+A(x))2v+V(x)v=f2(x,∣u∣2)u,\left\{\begin{array}{l}{\left(-i\nabla +A\left(x))}^{2}u+V\left(x)u={f}_{1}\left(x,{| v| }^{2})v,\\ {\left(-i\nabla +A\left(x))}^{2}v+V\left(x)v={f}_{2}\left(x,{| u| }^{2})u,\end{array}\right. where VV is the electric potential and AA is the magnetic potential. Assuming that the nonlinear function fi(i=1,2){f}_{i}\left(i=1,2) satisfies three types of polynomial growth assumptions: super-quadratic, asymptotically quadratic, and local super-quadratic at ∣x∣→∞| x| \to \infty , we prove the existence of the Nehari-Pankov type ground state solutions using critical point theory together with the non-Nehari manifold method. The resulting problem engages two major difficulties: the first one is that the associated functional is strongly indefinite, and the second lies in verifying the link geometry and showing the boundedness of Cerami sequences. Our results extend and complement the present ones in the literature.

Keywords

• schrödinger equations
• magnetic field
• ground state solutions
• non-nehari manifold method
• 35j20
• 35j60
• 35q55