Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems (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.
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Existence of weak solutions to some stationary Schrödinger equations with singular nonlinearity
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Positive Solutions for a Quasilinear Schrödinger Equation with Critical Growth
Journal of Dynamics and Differential Equations, 2011
We consider the quasilinear problem −ε p div(|∇u| p−2 ∇u) + V (z)u p−1 = f (u) + u p * −1 , u ∈ W 1,p (R N), where ε > 0 is a small parameter, 1 < p < N , p * = N p/(N − p), V is a positive potential and f is a superlinear function. Under a local condition for V we relate the number of positive solutions with the topology of the set where V attains its minimum. In the proof we apply Ljusternik-Schnirelmann theory.
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.
Multiple Positive Solutions to Nonlinear Schrödinger Equations with Competing Potential Functions
Journal of Differential Equations, 2000
We consider singularly perturbed nonlinear Schrödinger equations −ε 2 ∆u + V (x)u = f (u), u > 0, v ∈ H 1 (R N) (0.1) where V ∈ C(R N , R) and f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain Ω ⊂ R N such that m 0 ≡ inf x∈Ω V (x) < inf x∈∂Ω V (x) and we set K = {x ∈ Ω | V (x) = m 0 }. For ε > 0 small we prove the existence of at least cupl(K) + 1 solutions to (0.1) concentrating, as ε → 0 around K. We remark that, under our assumptions of f , the search of solutions to (0.1) cannot be reduced to the study of the critical points of a functional restricted to a Nehari manifold.
A note on quasilinear Schrödinger equations with singular or vanishing radial potentials
Differential and Integral Equations
In this note we complete the study of [3], where we got existence results for the quasilinear elliptic equation −∆w + V (|x|) w − w ∆w 2 = K(|x|)g(w) in R N , with singular or vanishing continuous radial potentials V (r), K(r). In [3] we assumed, for technical reasons, that K(r) was vanishing as r → 0, while in the present paper we remove this obstruction. To face the problem we apply a suitable change of variables w = f (u) and we find existence of non negative solutions by the application of variational methods. Our solutions satisfy a weak formulations of the above equation, but they are in fact classical solutions in R N \ {0}. The nonlinearity g has a double-power behavior, whose standard example is g(t) = min{t q 1 −1 , t q 2 −1 } (t > 0), recovering the usual case of a singlepower behavior when q1 = q2.
Annals of the University of Craiova - Mathematics and Computer Science Series, 2019
We study a degenerate quasilinear problem involving a singular potential and some bounded weights. The equation is perturbed by a critical nonlinear term and an indefinite sign perturbation involving a real parameter. Under suitable assumptions on the potentials and on the real parameter we use two different critical point techniques, in order to reveal two types of solutions. The proofs rely on variational arguments based on the Mountain-Pass Theorem, Ekeland's Variational Principle and energy estimates.