The existence of nontrivial solution for biharmonic equation with sign‐changing potential

Abstract

In this paper, we study the following biharmonic equation: {arraynormalΔ2u+Vλ(x)u=α(x)f(u)+νK(x)|ufalse|q−2uindouble-struckRN,arrayu∈H2(double-struckRN), where N⩾5,ν ∈ (0,ν0],1 < q < 2,Δ2u = Δ(Δu) and Vλ(x) = λa(x) − b(x) with λ > 0. Firstly, we prove the bipolar Rellich inequality. Secondly, by using bipolar Rellich inequality, Gigliardo‐Nirenberg inequality, and Ekeland variational principle, we prove the existence of nontrivial solution for problem Bν.

Existence of nontrivial solutions for a class of biharmonic equations with singular potential in R N $\mathbb{R}^{N}$

“…Zhang and Costa [11] investigated the existence of a nontrivial solution by applying the mountain pass theorem. For other interesting results of biharmonic equations, we refer to [12][13][14][15][16][17] and references therein. Motivated by the papers mentioned, especially by [6,11], the aim of this paper is to revisit the existence and multiplicity of nontrivial solutions for problem (1.1) with singular potential V (x) satisfying the following condition:…”

Radial solutions of a biharmonic equation with vanishing or singular radial potentials

“…Bilaplacian equations arise in describing different physical phenomena, such as the propagation of laser beams in Kerr media or nonlinear oscillations in suspension bridges (see some references in [5,13]), and have been extensively studied in the last decades (see e.g. [5][6][7]9,13,14] and the references therein). In spite of that, equations of type (1.1), namely with radial potentials possibly singular at the origin and vanishing at infinity, has been treated only in [6,7] (at least to our knowledge), where the authors essentially consider power type potentials.…”

Schrödinger‐Poisson system with Hardy‐Littlewood‐Sobolev critical exponent

“…In another research, 5 Gao and Yang investigated Equation 1 with upper critical exponent N+ N−2 in a bounded domain with Lipschitz boundary, and established the existence, multiplicity, and nonexistence of nontrivial solutions to the critical Choquard equation. For details and recent works, we refer to other literature [6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein.…”