Abstract

This article concerns the following class of system {−Δu+V(x)u+ℓ(x)ϕu=f(u)+λ∣u∣q−2uin R3,−Δϕ=ℓ(x)u2in R3,u,ϕ∈D1,2(R3), u,ϕ≥0in R3,\left\{ \begin{array}{lr} -\Delta u +V(x)u+\ell(x)\phi u = f(u) + \lambda|u|^{q-2}u & \text{in } \mathbb{R}^3,\\[2mm] -\Delta \phi = \ell(x)u^{2} & \text{in } \mathbb{R}^3,\\[2mm] u,\phi\in D^{1,2}(\mathbb{R}^3), \ u,\phi\geq0 & \text{in } \mathbb{R}^3, \end{array} \right. where λ≥0\lambda\geq0 and q≥2∗=6q\geq2^*=6 is the critical Sobolev exponent in dimension 3, the nonlinearity f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} is superlinear and has subcritical growth, V,ℓ:R3→RV,\ell: \mathbb{R}^3\rightarrow \mathbb{R} are measurable functions with ℓ∈L2(R3)\ell\in L^2(\mathbb{R}^3), the potential VV can change sign in R3\mathbb{R}^3 and vanish at infinity, that is, V(x)→0V (x) \rightarrow 0 as ∣x∣→∞|x|\rightarrow\infty. Our approach is based on variational method combined with Benci-Fortunato's reduction argument [ Topol. Methods Nonlinear Anal. 11 (1998) 283–293], Del Pino-Felmer's penalization technique [ Calc. Var. Partial Diff. Equations 4 (1996) 121–137] and L∞L^\infty-estimate.

Author information

Contact details are reproduced from the original publication and may be historical.

Genivaldo P. Corrêa

Faculty of Exact and Technological Sciences, Federal University of Pará, Abaetetuba, Brazil

genivaldo@ufpa.br

Gelson C. G. dos Santos

Institute of Exact and Natural Sciences, Federal University of Pará, Belém, Brazil

gelsonsantos@ufpa.br

Keywords

  • Schrödinger-Poisson system
  • variational methods
  • Mountain Pass Theorem
  • nontrivial solution
  • supercritical exponents
  • vanishing potential
  • sign-changing potential

Mathematics Subject Classification

35A15, 35Q61, 35B38, 35B09

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Suggested citation

G. P. Corrêa, G. C. G. dos Santos. “Schrödinger-Poisson System Involving Potential Vanishing at Infinity and Unbounded Below.” Journal of Convex Analysis 32 (2025), No. 4, 1117–1134. https://doi.org/10.68381/jca32059

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