Ground states and multiple solutions for Hamiltonian elliptic system with gradient term
Wen Zhang, Jian Zhang, Heilong Mi
Abstract
Abstract This paper is concerned with the following nonlinear Hamiltonian elliptic system with gradient term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:mfenced open="{" close=""> <m:mrow> <m:mspace width="thinmathspace"/> <m:mspace width="thinmathspace"/> <m:mtable columnalign="left left" rowspacing="0.1em 0.4em" columnspacing="1em"> <m:mtr> <m:mtd> <m:mo>−</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-mathit" mathvariant="italic">Δ</m:mi> </m:mrow> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mover> <m:mi>b</m:mi> <m:mo stretchy="false">→</m:mo> </m:mover> </m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>⋅</m:mo> <m:mi mathvariant="normal">∇</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>V</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi>H</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>v</m:mi> </m:mrow> </m:msub> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> <m:mo stretchy="false">)</m:mo> <m:mspace width="thinmathspace"/> <m:mspace width="thinmathspace"/> <m:mtext>in</m:mtext> <m:mspace width="thinmathspace"/> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mo>−</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-mathit" mathvariant="italic">Δ</m:mi> </m:mrow> <m:mi>v</m:mi> <m:mo>−</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mover> <m:mi>b</m:mi> <m:mo stretchy="false">→</m:mo> </m:mover> </m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>⋅</m:mo> <m:mi mathvariant="normal">∇</m:mi> <m:mi>v</m:mi> <m:mo>+</m:mo> <m:mi>V</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mi>v</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi>H</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>u</m:mi> </m:mrow> </m:msub> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> <m:mo stretchy="false">)</m:mo> <m:mspace width="thinmathspace"/> <m:mspace width="thinmathspace"/> <m:mtext>in</m:mtext> <m:mspace width="thinmathspace"/> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>.</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $$\begin{array}{} \displaystyle \left\{\,\, \begin{array}{ll} -{\it\Delta} u +\vec{b}(x)\cdot \nabla u+V(x)u = H_{v}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N},\\[-0.3em] -{\it\Delta} v -\vec{b}(x)\cdot \nabla v +V(x)v = H_{u}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N}.\\ \end{array} \right. \end{array}$$ Compared with some existing issues, the most interesting feature of this paper is that we assume that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition. This case allows the nonlinearity to be super-quadratic on some domains and asymptotically quadratic on other domains. Furthermore, by using variational method, we obtain new existence results of ground state solutions and infinitely many geometrically distinct solutions under local super-quadratic condition. Since we are without more global information on the nonlinearity, in the proofs we apply a perturbation approach and some special techniques.