Abstract
In this paper we are going to study a class of Schrodinger-Poisson system $$ \left\{ \begin{array}{ll} - \Delta u + (\lambda a(x)+1)u+ \phi u = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3},\\ -\Delta \phi=4\pi u^2 \mbox{ in } \,\,\, \mathbb{R}^{3}.\\ \end{array} \right. $$ Assuming that the nonnegative function $a(x)$ has a potential well $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint bounded components $\Omega_1, \Omega_2, ....., \Omega_k$ and the nonlinearity $f(t)$ has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by variational methods.
Highlights
This paper was motivated by some recent works concerning the nonlinear SchrödingerPoisson system
3- When we apply variational methods to prove the existence of solution to (SP )λ, we are led to study a nonlocal, see problem (P )λ above
For this class of problem, it is necessary to make a careful revision in the sets used in the deformation lemma found in [1] and [15] to get multi-bump solution, since they don’t work well for this class of system, see Sections 6 and 7 for more details
Summary
This paper was motivated by some recent works concerning the nonlinear SchrödingerPoisson system. 3- When we apply variational methods to prove the existence of solution to (SP )λ, we are led to study a nonlocal, see problem (P )λ above For this class of problem, it is necessary to make a careful revision in the sets used in the deformation lemma found in [1] and [15] to get multi-bump solution, since they don’t work well for this class of system, see Sections 6 and 7 for more details. If we fix the subset Υ, for any sequence λn → ∞ we can extract a subsequence (λni ) such that (uλni ) converges strongly in H1(R3) to a function u, which satisfies u = 0 outside ΩΥ = ∪j∈ΥΩj, and u|ΩΥ is a least energy solution for the nonlocal problem. Since we want to look for least energy for (P )∞,Υ, our goal is to prove the existence of a critical point for J in the set M
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