Normalized solutions for p-Laplacian equations with a $$L^{2}$$ L 2 -supercritical gr
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Annals of Functional Analysis https://doi.org/10.1007/s43034-020-00101-w ORIGINAL PAPER
Normalized solutions for p‑Laplacian equations with a L2 ‑supercritical growth Wenbo Wang1 · Quanqing Li2 · Jianwen Zhou1 · Yongkun Li1 Received: 5 June 2020 / Accepted: 21 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract We are concerned with the following p-Laplacian equation
−𝛥p u + |u|p−2 u = 𝜇u + |u|s−2 u, in ℝN , Np p, p∗ ) , p∗ = N−p is the where −𝛥p u = div(|∇u|p−2 ∇u) , 1 < p < N , 𝜇 ∈ ℝ , s ∈ ( N+2 N critical Sobolev exponent. Using constrained variational methods, we prove that the above problem has a normalized solution. Our contribution is that we can deal with p, p∗ ) by a mountain-pass argument on the prescribed the L2-supercritical case ( N+2 N 2 L -norm constraint.
Keywords p-Laplacian equations · Constrained variational methods · Mountain-pass Mathematics Subject Classification 35J92 · 35J62 · 49R05
1 Introduction and main results In the present paper, we consider the following p-Laplacian equation
Communicated by Julian Bonder. * Yongkun Li [email protected] Wenbo Wang [email protected] Quanqing Li [email protected] Jianwen Zhou [email protected] 1
School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan, P.R. China
2
Department of Mathematics, Honghe University, Mengzi, Yunnan, P.R. China
Vol.:(0123456789)
W. Wang et al.
−𝛥p u + |u|p−2 u = 𝜇u + |u|s−2 u, in ℝN ,
(1.1)
Np p < s < p∗ , p∗ = N−p where −𝛥p u = div(|∇u|p−2 ∇u) , 1 < p < N , 𝜇 ∈ ℝ , N+2 is the N critical Sobolev exponent. An natural question will be asked why the restriction p > 2 , and Remark 1.1 shall explain it. The study of this problem has received considerable attention in recent years. For 1 < p ≤ 2 , it is used to describe the image restoration in [7]. For p > 2 , an equation of the type (1.1) emerges from the mathematical description of the process of filtration of an ideal incompressible fluid through a porous medium (see [4]). (1.1) also arises in non-Newtonian fluids, pseudo-plastic fluids, nonlinear elasticity, and reaction diffusions, and please see [9] for more details of physical background. In the case N > p ≠ 2 , important contributions to p-Laplacian equation
−𝛥p u + 𝜔|u|p−2 u = f (x, u), 𝜔 ∈ ℝ, in ℝN ,
(1.2)
can be found in [16]. Soon afterwards, in [17], Li and Zhou studied the p-Laplacian problem in bounded domain with f (u) ∼ up−1 at infinity. In 2017, Gu, Zeng and Zhou [12] investigated the following problem
−𝛥p u + V(x)|u|p−2 u = 𝜆|u|p−2 u + a|u|s−2 u, in ℝN ,
where a ≥ 0 , p ∈ (1, N) , s = p + N , 𝜆 ∈ ℝ , V(x) ∈ C(ℝN ) and
(1.3)
p2
(V) infx∈ℝN V(x) = 0, lim|x|→∞ V(x) = ∞. With Gagliardo-Nirenberg inequality [1] in hand, they showed that there is a∗ > 0 , the above problem has a ground state for all a ∈ [0, a∗ ) and the blow-up behaviour N t ∗ p u(tx) , which u (x) = t also be given if . In their paper, since the rescaling a ↗ a 2 makes p + pN is the critical exponent. In 2018, Silva and Macedo [20] considered the following p-Laplacian equat
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