Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator

Alves, Claudianor O.; Nóbrega, Alânnio B.

doi:10.1007/s00605-016-0967-0

Cited by 10 publications

(6 citation statements)

References 15 publications

(13 reference statements)

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“…Moreover, these solutions found in [4] and [10] have the same characteristics of those found in [19]. For the further research about the nonlinear Schrödinger equations with the deepening potential well, we refer to [1], [8], [9], [11], [12], [18], [21] and their references.…”

Section: Introductionmentioning

confidence: 56%

Multi-bump solutions for Choquard equation with deepening potential well

2016

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In this paper we study the existence of multi-bump solutions for the following Choquard equationwhere µ ∈ (0, 3), p ∈ (2, 6 − µ), λ is a positive parameter and the nonnegative continuous function a(x) has a potential well Ω := int(a −1 (0)) which possesses k disjoint bounded components Ω := ∪ k j=1 Ωj . We prove that if the parameter λ is large enough, then the equation has at least 2 k − 1 multi-bump solutions.Mathematics Subject Classifications (2010): 35J20, 35J65

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Section: Introductionmentioning

confidence: 56%

Multi-bump solutions for Choquard equation with deepening potential well

2016

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“…Some related results can be found, for example, in Arosio and Panizzi [7], Autuori, Colasuonno and Pucci [8], Autuori, Pucci and Salvatori [12], Counsin, Frota, Lar'kin and Medeiros [19], and D'Ancona and Spagnolo [20]. We also point out that Arosio and Panizzi [7] studied the Cauchy-Dirichlet problem related to (1) [16] pointed out that this kind of problems models several physical and biological systems, where u describes a process depending on the average of itself (for example, population density). We also refer to Colasuonno and Pucci [18], which covers for the first time in the literature the degenerate case of the Kirchhoff equation for the existence of solutions in the elliptic case.…”

Section: Introductionmentioning

confidence: 99%

“…We refer the reader, for instance, to Alves and Corrêa [1], Alves and Figueiredo [3], Andrade and Ma [5], Chipot and Lovat [16], Chipot and Rodrigues [17], and Vasconcellos [28]. Problem (D where ∆ is the usual Laplace operator.…”

Section: Introductionmentioning

confidence: 99%

Mountain pass solutions for nonlocal equations

Bisci¹,

Rădulescu²

2014

Ann. Acad. Sci. Fenn. Math.

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Abstract. This work is devoted to study the existence of solutions to nonlocal equations involving the p-Laplacian. More precisely, we prove the existence of at least one nontrivial weak solution, and under additional assumptions, the existence of infinitely many weak solutions. In order to apply mountain pass results, we require rather general assumptions on on the local operator. Finally, a concrete application is presented.

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“…Assume also that g : R × C → C is a continuous nonlinearity. Hence the standing waves to the (N LS) equation given in (2)…”

Section: Introductionmentioning

confidence: 99%

Periodic and asymptotically periodic fourth-order Schrödinger equations with critical and subcritical growth

Silva¹,

Carvalho²,

Goulart³

2022

DCDS

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<p style='text-indent:20px;'>It is established existence of solutions for subcritical and critical nonlinearities considering a fourth-order elliptic problem defined in the whole space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>. The work is devoted to study a class of potentials and nonlinearities which can be periodic or asymptotically periodic. Here we consider a general fourth-order elliptic problem where the principal part is given by <inline-formula><tex-math id="M2">\begin{document}$ \alpha \Delta^2 u + \beta \Delta u + V(x)u $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ \alpha > 0, \beta \in \mathbb{R} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ V: \mathbb{R}^N \rightarrow \mathbb{R} $\end{document}</tex-math></inline-formula> is a continuous potential. Hence our main contribution is to consider general fourth-order elliptic problems taking into account the cases where <inline-formula><tex-math id="M5">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> is negative, zero or positive. In order to do that we employ some fine estimates proving the compactness for the associated energy functional.</p>

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Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator

Cited by 10 publications

References 15 publications

Multi-bump solutions for Choquard equation with deepening potential well

Multi-bump solutions for Choquard equation with deepening potential well

Mountain pass solutions for nonlocal equations

Periodic and asymptotically periodic fourth-order Schrödinger equations with critical and subcritical growth

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