A note on quasilinear Schrödinger equations with singular or vanishing radial potentials (original) (raw)
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Nonlinear Analysis, 2022
Given two continuous functions V (r) ≥ 0 and K (r) > 0 (r > 0), which may be singular or vanishing at zero as well as at infinity, we study the quasilinear elliptic equation −∆w + V (|x|) w − w ∆w 2 = K(|x|)g(w) in R N , where N ≥ 3. To study this problem we apply a change of variables w = f (u), already used by several authors, and find existence results for nonnegative solutions by the application of variational methods. The main features of our results are that they do not require any compatibility between how the potentials V and K behave at the origin and at infinity, and that they essentially rely on power type estimates of the relative growth of V and K, not of the potentials separately. Our solutions satisfy a weak formulations of the above equation, but we are able to prove that they are in fact classical solutions in R N \{0}. To apply variational methods, we have to study the compactness of the embedding of a suitable function space into the sum of Lebesgue spaces L q 1 K + L q 2 K , and thus into L q K (= L q K + L q K) as a particular case. The nonlinearity g has a double-power behavior, whose standard example is g(t) = min{t q 1 −1 , t q 2 −1 }, recovering the usual case of a single-power behavior when q1 = q2.
Radial nonlinear elliptic problems with singular or vanishing potentials
arXiv (Cornell University), 2017
In this paper we prove existence of radial solutions for the nonlinear elliptic problem −div(A(|x|)∇u) + V (|x|)u = K(|x|)f (u) in R N , with suitable hypotheses on the radial potentials A, V, K. We first get compact embeddings of radial weighted Sobolev spaces into sum of weighted Lebesgue spaces, and then we apply standard variational techniques to get existence results.
Electronic Journal of Qualitative Theory of Differential Equations, 2020
We establish the existence of positive solutions for the singular quasilinear Schrödinger equation −∆u − ∆(u 2)u = h(x)u −γ + f (x, u) in Ω, u(x) = 0 on ∂Ω, where Ω ⊂ R N (N ≥ 3) is a bounded domain with smooth boundary ∂Ω, 1 < γ, h ∈ L 1 (Ω) and h > 0 almost everywhere in Ω. The function f may change sign on Ω. By using the variational method and some analysis techniques, the necessary and sufficient condition for the existence of a solution is obtained.
Non-radial normalized solutions for a nonlinear Schrodinger equation
Electronic Journal of Differential Equations
This article concerns the existence of multiple non-radial positive solutions of the L2-constrained problem \displaylines{-\Delta{u}-Q(\varepsilon x)|u|^{p-2}u=\lambda{u},\quad \text{in }\mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=1,}$$ where \(Q(x)\) is a radially symmetric function, ε>0 is a small parameter, \(N\geq 2\), and \(p \in (2, 2+4/N)\) is assumed to be mass sub-critical. We are interested in the symmetry breaking of the normalized solutions and we prove the existence of multiple non-radial positive solutions as local minimizers of the energy functional.
Existence of radial solutions for quasilinear elliptic equations with singular nonlinearities
Advanced Nonlinear Studies
We prove the existence of radial solutions of the quasilinear elliptic equation div(A(|Du|)Du) + f (u) = 0 in R n , n > 1, where f is either negative or positive for small u > 0, possibly singular at u = 0, and growths subcritically for large u. Our proofs use only elementary arguments based on a variational identity. No differentiability assumptions are made on f .
Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems
Discrete & Continuous Dynamical Systems, 2021
We prove the existence of a bounded positive solution for the following stationary Schrödinger equation −∆u + V (x)u = f (x, u), x ∈ R n , n ≥ 3, where V is a vanishing potential and f has a sublinear growth at the origin (for example if f (x, u) is a concave function near the origen). For this purpose we use a Brezis-Kamin argument included in [6]. In addition, if f has a superlinear growth at infinity, besides the first solution, we obtain a second solution. For this we introduce an auxiliar equation which is variational, however new difficulties appear when handling the compactness. For instance, our approach can be applied for nonlinearities of the type ρ(x)f (u) where f is a concave-convex function and ρ satisfies the (H) property introduced in [6]. We also note that we do not impose any integrability assumptions on the function ρ, which is imposed in most works.
Ground states of degenerate quasilinear Schrödinger equation with vanishing potentials
Nonlinear Analysis, 2019
In this paper we study the existence of nontrivial ground state solutions for the following class of p-Laplacian type equation −div (a(x, ∇u)) + V (x)|x| −αp * |u| p−2 u = K(x)|x| −αp * f (u) in R N , where 1 < p < N , N ≥ 3, −∞ < α < N −p p , α ≤ e ≤ α + 1, d = 1 + α − e, p * := p * (α, e) = N p N −dp (critical Hardy-Sobolev exponent); f has a quasicritical growth; V and K are nonnegative potentials; the function a satisfies |a(x, ∇u)| ≤ c 0 |x| −αp h 0 (x)|∇u| p−1 + c 0 (1 + |x| −αp)h 1 (x)|∇u| p−1 for any ξ ∈ R N , a.e. x ∈ R N , for any two positive functions h 1 ∈ L ∞ loc (R N), h 0 ∈ L p p−1 α (R N), with α = αp p * .
Solitary Waves for a Class of Quasilinear Schrödinger Equations Involving Vanishing Potentials
Advanced Nonlinear Studies, 2015
In this paper we study the existence of weak positive solutions for the following class of quasilinear Schrödinger equations −Δu + V(x)u − [Δ(u2)]u = h(u) in ℝN, where h satisfies some “mountain-pass” type assumptions and V is a nonnegative continuous function. We are interested specially in the case where the potential V is neither bounded away from zero, nor bounded from above. We give a special attention to the case when V may eventually vanish at infinity. Our arguments are based on penalization techniques, variational methods and Moser iteration scheme.