On quasilinear elliptic problems with finite or infinite potential wells

Shibo Liu

doi:10.1515/math-2021-0053

Abstract

Abstract We consider quasilinear elliptic problems of the form − div ( ϕ ( ∣ ∇ u ∣ ) ∇ u ) + V ( x ) ϕ ( ∣ u ∣ ) u = f ( u ) , u ∈ W 1 , Φ ( R N ) , -{\rm{div}}\hspace{0.33em}(\phi \left(| \nabla u| )\nabla u)+V\left(x)\phi \left(| u| )u=f\left(u),\hspace{1.0em}u\in {W}^{1,\Phi }\left({{\mathbb{R}}}^{N}), where ϕ \phi and f f satisfy suitable conditions. The positive potential V ∈ C ( R N ) V\in C\left({{\mathbb{R}}}^{N}) exhibits a finite or infinite potential well in the sense that V ( x ) V\left(x) tends to its supremum V ∞ ≤ + ∞ {V}_{\infty }\le +\infty as ∣ x ∣ → ∞ | x| \to \infty . Nontrivial solutions are obtained by variational methods. When V ∞ = + ∞ {V}_{\infty }=+\infty , a compact embedding from a suitable subspace of W 1 , Φ ( R N ) {W}^{1,\Phi }\left({{\mathbb{R}}}^{N}) into L Φ ( R N ) {L}^{\Phi }\left({{\mathbb{R}}}^{N}) is established, which enables us to get infinitely many solutions for the case that f f is odd. For the case that V ( x ) = λ a ( x ) + 1 V\left(x)=\lambda a\left(x)+1 exhibits a steep potential well controlled by a positive parameter λ \lambda , we get nontrivial solutions for large λ \lambda .

Highlights

In this paper, we consider the following quasilinear elliptic problem in N,−div(φ(∣∇u∣)∇u) + V (x)φ(∣u∣)u = f (u), u ∈ W 1,Φ( N). (1.1)where φ : [0, ∞) → [0, ∞) is a C1-function satisfying the following assumptions: (φ1) the function t ↦ φ(t)t is increasing in (0, ∞), (φ2) there exist l, m ∈ (1, N ) such that l ≤ φ(∣t∣)t2 ≤ m for all t ≠ 0, (1.2) Φ(t )where l ≤ m < l∗ (note that for p ∈ (1, N ) we set p∗ = Np/(N − p)), ∣t∣ Φ(t) = ∫φ(s)sds.Nonlinear elliptic problems in N like (1.1) have been extensively studied
Where φ : [0, ∞) → [0, ∞) is a C1-function satisfying the following assumptions: (φ1) the function t ↦ φ(t)t is increasing in (0, ∞), (φ2) there exist l, m ∈ (1, N ) such that l ≤ φ(∣t∣)t2 ≤ m for all t ≠ 0, (1.2) Φ(t )
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Summary

Introduction

We consider the case that V satisfies the following condition due to Bartsch and Wang [26] in their study of (1.3):. For problem (1.3), these conditions are introduced by Bartsch and Wang [26] and characterize V as possessing a steep potential well whose height is controlled by the positive parameter λ Our result for this case is the following theorem. Even for the semilinear case that φ(t) ≡ 1, our Theorem 1.4 is slightly general than the corresponding result in [26], because in ( f1) we only require f to be asymptotically subcritical, that is the second limit in (1.4) holds, while in [26, Theorem 2.4] the nonlinearity f is strictly subcritical, meaning that the growth of f at infinity is controlled by a subcritical power function ∣t ∣q−2t for some q ∈ (2, 2∗).

Orlicz-Sobolev spaces

Nontrivial solutions

Multiple solutions