Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part

Zhang, Hui; Xu, Ji‐Qing; Zhang, Fubao

doi:10.1002/mma.3054

Cited by 12 publications

(13 citation statements)

References 36 publications

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“…Now, we prove that

I_{p} (ū) \leq c .

By , we obtain

I_{p} (ū) = I_{p} (ū) - \frac{1}{2} 〈 I_{p}^{'} (ū), ū 〉 = \int G_{p} (x, ū),

where G p is defined in . By and Fatou lemma, we have that

\int G_{p} (x, ū) \leq \int G_{p} (x, ū_{n}) + o_{n} (1) .

Then

\begin{matrix} I_{p} (ū) & \leq \int G_{p} (x, ū_{n}) + o_{n} (1) = I_{p} (ū_{n}) - \frac{1}{2} 〈 I_{p}^{'} (ū_{n}), ū_{n} 〉 + o_{n} (1) \\ = I_{p} (u_{n}) - \frac{1}{2} 〈 I_{p}^{'} (u_{n}), u_{n} 〉 + o_{n} (1) . \end{matrix}

By the properties of f and , Lemma 3.6], we obtain that

\begin{matrix} \int (V (x) - V_{p} (x)) f^{2} ( \end{matrix}

…”

Section: Proof Of Theorems 11 and 12mentioning

confidence: 92%

“…If ( V 1 ) and ( K 1 ) are satisfied, then I ′ and

I_{p}^{'}

are weakly sequentially continuous.Proof Set

\tilde{f} (x, v) = V (x) v - V (x) f^{'} (v) f (v) + K (x) f^{3} (v) f^{'} (v) .

Then the equation (EQ) is turned into

- Δ v + V (x) v = \tilde{f} (x, v), v \in H^{1} (R^{3}) .

We easilyhave that

\overset{f}{true}

and V satisfy the conditions in , Lemma 2.2]. So, lemma 2.2 in implies that I ′ is weakly sequentially continuous. In the same way, we easily infer that

I_{p}^{'}

is also weakly sequentially continuous.…”

Section: Variational Settingmentioning

confidence: 95%

See 1 more Smart Citation

Existence and nonexistence of ground states for a class of quasilinear Schrödinger equations

Zhang

2016

Math Methods in App Sciences

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“…Now, we prove that

I_{p} (ū) \leq c .

By , we obtain

I_{p} (ū) = I_{p} (ū) - \frac{1}{2} 〈 I_{p}^{'} (ū), ū 〉 = \int G_{p} (x, ū),

where G p is defined in . By and Fatou lemma, we have that

\int G_{p} (x, ū) \leq \int G_{p} (x, ū_{n}) + o_{n} (1) .

Then

\begin{matrix} I_{p} (ū) & \leq \int G_{p} (x, ū_{n}) + o_{n} (1) = I_{p} (ū_{n}) - \frac{1}{2} 〈 I_{p}^{'} (ū_{n}), ū_{n} 〉 + o_{n} (1) \\ = I_{p} (u_{n}) - \frac{1}{2} 〈 I_{p}^{'} (u_{n}), u_{n} 〉 + o_{n} (1) . \end{matrix}

By the properties of f and , Lemma 3.6], we obtain that

\begin{matrix} \int (V (x) - V_{p} (x)) f^{2} ( \end{matrix}

…”

Section: Proof Of Theorems 11 and 12mentioning

confidence: 92%

“…If ( V 1 ) and ( K 1 ) are satisfied, then I ′ and

I_{p}^{'}

are weakly sequentially continuous.Proof Set

\tilde{f} (x, v) = V (x) v - V (x) f^{'} (v) f (v) + K (x) f^{3} (v) f^{'} (v) .

Then the equation (EQ) is turned into

- Δ v + V (x) v = \tilde{f} (x, v), v \in H^{1} (R^{3}) .

We easilyhave that

\overset{f}{true}

and V satisfy the conditions in , Lemma 2.2]. So, lemma 2.2 in implies that I ′ is weakly sequentially continuous. In the same way, we easily infer that

I_{p}^{'}

is also weakly sequentially continuous.…”

Section: Variational Settingmentioning

confidence: 95%

Existence and nonexistence of ground states for a class of quasilinear Schrödinger equations

Zhang

2016

Math Methods in App Sciences

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“…It is worth pointing out that the related semilinear equation with the asymptotically periodic condition has been extensively studied, see [13,17,18,34,37,38] and their references. In [13,34,37,38], they discussed the existence of solutions for problem (3) without the second order derivatives ∆(u 2 )u, when the problem is strongly indefinite, that is, 0 lies in a spectral gap of − + V . We would like to point out that in a recent paper [17] and [18], Liu et al have given reformative conditions which unify the asymptotic processes of V, g at infinity.…”

Section: Yanfang Xue and Chunlei Tangmentioning

confidence: 99%

“…We would like to point out that in a recent paper [17] and [18], Liu et al have given reformative conditions which unify the asymptotic processes of V, g at infinity. The asymptotic processes is weaker than those in [13,34,37,38]. We borrow an idea from [17] and [18] to obtain the ground state solution for problem (3).…”

Section: Yanfang Xue and Chunlei Tangmentioning

confidence: 99%

Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth

Xue¹,

Tang²

2018

Communications on Pure &Amp; Applied Analysis

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In this paper, we are concerned with the existence of ground state solutions for the following quasilinear Schrödinger equation: − ∆u + V (x)u − ∆(u 2)u = K(x)|u| 22 * −2 u + g(x, u), x ∈ R N (1) where N ≥ 3, V, g are asymptotically periodic functions in x. By combining variational methods and the concentration-compactness principle, we obtain a ground state solution for equation (1) under a new reformative condition which unify the asymptotic processes of V, g at infinity.

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“…In literature, B Sirakov improved the class of potentials contained in Bartsch and Wang and preserve the compactness of the energy functional associated to Equation . For existence of ground states for strongly indefinite case, when f ( x , u ) is superlinear as

false| u false| \to + \infty

and 0 lies in a gap of the spectrum of the operator −Δ+ V , we cite Zhang et al, where the authors used the method of generalized Nehari manifold (see Szulkin and Weth), and Lion's compactness Lemma to overcome the difficulties. Concerning to problems defined in two‐dimensional domains and involving nonlinearities with exponential growth, we refer the readers to other studies …”

Section: Introductionmentioning

confidence: 99%

Positive ground state of coupled systems of Schrödinger equations in R2 involving critical exponential growth

Albuquerque

2017

Math Methods in App Sciences

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In this paper, we study the existence of a positive ground state solution to the following coupled system of nonlinear Schrödinger equations:where the nonlinearities f 1 (x, s) and f 2 (x, s) are superlinear at infinity and have exponential critical growth of the Trudinger-Moser type. The potentials V 1 (x) and V 2 (x)are nonnegative and satisfy a condition involving the coupling term (x), namely,For this purpose, we use the minimization technique over the Nehari manifold and strong maximum principle to get a positive ground state solution. Moreover, by using a bootstrap argument and L q -estimates, we get regularity and asymptotic behavior.

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Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part

Abstract: Communicated by P. SacksUsing the method of the Nehari manifold, we study the existence of ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part and superlinear nonlinearity.

Cited by 12 publications

References 36 publications

Existence and nonexistence of ground states for a class of quasilinear Schrödinger equations

Existence and nonexistence of ground states for a class of quasilinear Schrödinger equations

Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth

Positive ground state of coupled systems of Schrödinger equations in R2 involving critical exponential growth

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