On ground state solutions for some non-autonomous Schrödinger-Poisson systems
The authors consider the Schrödinger-Poisson system \[ \begin{cases} -\Delta u + u + K(x) \phi (x) u = a(x) f(u) & \mathrm{in}\,\, {\mathbb R}^3,\\ -\Delta \phi = K(x) u^2 & \mathrm{in} \,\, {\mathbb R}^3 . \end{cases} \tag{SP} \] Briefly, the Poisson equation is solved by using the Lax-Milgram theorem, so, for all \(u \in H^1({\mathbb R}^3)\), there exists a unique solution \(\phi _u \in D^{1,2}({\mathbb R}^3)\) of \(-\Delta \phi = K(x) u^2\), where \(D^{1,2}({\mathbb R}^3)\) is the completion of \(C_0^{\infty }({\mathbb R}^3)\) with respect to the norm \[ \| u \| _{D^{1,2}}:= \biggl( \int _{{\mathbb R}^3}| \nabla u | ^2 dx \biggr)^{1/2}. \] If one inserts the solution into the first equation of (SP), one gets the equation \[ -\Delta u + u + K(x) \phi _u (x) u = a(x) f(u). \tag{1} \] (1) is variational and its solutions are the critical points of the functional defined in \(H^1({\mathbb R}^3)\) by \[ I(u) = \frac12 \| u \| ^2 + \frac14 \int _{{\mathbb R}^3} K(x) \phi _u (x) u^2 dx - \int _{{\mathbb R}^3}a(x) F(u) dx, \tag{2} \] where \(\| u \| ^2= \int _{{\mathbb R}^3} (| \nabla u | ^2 + u^2 )dx\) and \(F(u)= \int _0^u f(s)ds \). The authors prove that \(u\in H^1({\mathbb R}^3)\) is a ground state of (SP) if \(u\) is a solution of (SP) which has the least energy among all solution of (SP), that is \(I'(u)=0\) and \(I(u)= \inf \{ I(v); v \in H^1({\mathbb R}^3) \setminus \{0\}, \mathrm{and}\,\, I'(v)=0 \}\). The main theorem is as follows. \textbf{Theorem 1.1} Assume that the following conditions hold: \((F_1)\) \(f \in C({\mathbb R}, {\mathbb R}^+)\), \(f(s) \equiv 0\) for all \(s<0\) and \(f(s)/s \to 0\) as \(s \to 0^+\). \((F_2)\) There exists \(l \in (0,+\infty )\) such that \(f(s)/s \to l\) as \(s\to + \infty \). \((A_1)\) \(a(x)\) is a positive continuous function and there exists \(R_0>0\) such that \[ \sup \{ f(s)/s ;s>0 \} \leq \inf \{ 1/a(x); | x | \geq R_0 \}. \] \((A_2)\) There exists a constant \(\beta \in (0,1)\) such that \[ \begin{aligned} (1-\beta ) l > \mu ^*:&= \inf \biggl\{ \int _{{\mathbb R}^3} (| \nabla u | ^2 + u^2 dx; u \in H^1({\mathbb R}^3), \\ & \quad \int _{{\mathbb R}^3}a(x) F(u) dx \geq 1/2, \, \, \mathrm{and} \, \, \int _{{\mathbb R}^3} K(x) \phi _u u^2 dx < 2\beta l \biggr\}. \end{aligned} \] \((K_1)\) \(K\in L^2({\mathbb R}^3), K(x)\geq 0\) for all \(x\in {\mathbb R}^3\), but \(K(x) \not\equiv 0\). Then the system (SP) possesses a ground state solution in \(H^1({\mathbb R}^3)\). For the proof, they use the variant version of the mountain pass theorem.
non-autonomous Schrödinger-Poisson system
scientific article; zbMATH DE number 6011345
scientific article