The Schrödinger–Poisson System with Positive Potential

Mugnai, Dimitri

doi:10.1080/03605302.2011.558551

Cited by 74 publications

(54 citation statements)

References 29 publications

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“…In particular for γ < 0, b > 0, 2 < p < 6, we quote the results [3,4,16,30,6]. Conversely, for γ > 0 and b < 0, the corresponding system has been studied in [8] and it represents a Hartree model for crystals (see also [26]). …”

Section: Introductionmentioning

confidence: 94%

On the planar Schrödinger–Poisson system

Cingolani

Weth

2016

Ann. Inst. H. Poincaré C Anal. Non Linéaire

116

165

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We develop a variational framework to detect high energy solutions of the planar Schrödingerwith a positive function a ∈ L ∞ (R 2 ) and γ > 0. In particular, we deal with the periodic setting where the corresponding functional is invariant under Z 2 -translations and therefore fails to satisfy a global Palais-Smale condition. The key tool is a surprisingly strong compactness condition for Cerami sequences which is not available for the corresponding problem in higher space dimensions. In the case where the external potential a is a positive constant, we also derive, as a special case of a more general result, the existence of nonradial solutions (u, w) such that u has arbitrarily many nodal domains. Finally, in the case where a is constant, we also show that solutions of the above problem with u > 0 in R 2 and w(x) → −∞ as |x| → ∞ are radially symmetric up to translation. Our results are also valid for a variant of the above system containing a local nonlinear term in u in the first equation.

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Section: Introductionmentioning

confidence: 94%

On the planar Schrödinger–Poisson system

Cingolani

Weth

2016

Ann. Inst. H. Poincaré C Anal. Non Linéaire

116

165

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“…In fact, the system can arise in many mathematical physics contexts. For instance, in Abelian Gauge Theories, (SP ) ε provides a model to describe the interaction of a nonlinear Schrödinger field with the electromagnetic field (see [10,11,20]). In quantum electrodynamics, (SP ) ε describes the interaction between a charge particle interacting with the electromagnetic field.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Later, the papers [3,4,33] generalized the nonlinearity c(x)h(u) with c(x) u p p replaced by a more general function g(x, u)(or u p p replaced by g(u)) were considered. Recently, in [20], the authors proved that (SP ) 1 has positive radially symmetric solutions for λ < 0 and g(x, u) ≤ 0. Also, the paper [37] proved the existence of multi-bump solutions for (1.1) with subcritical growth nonlinearity.…”

Section: Casementioning

confidence: 99%

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Existence of multiple positive solutions for Schrödinger–Poisson systems with critical growth

Wang

et al. 2015

Z. Angew. Math. Phys.

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In this paper, we are concerned with the existence, multiplicity and concentration of positive ground state solutions for the semilinear Schrödinger-Poisson systemwhere ε > 0 is a small parameter, f is a continuous, superlinear and subcritical nonlinearity, and λ = 0 is a real parameter. Suppose that a(x) has at least one global minimum and b(x) has at least one global maximum. We prove that there are two families of positive solutions for sufficiently small ε > 0, of which one is concentrating on the set of minimal points of a and the other on the sets of maximal points of b. Moreover, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.Mathematics Subject Classification. 35J61 · 35J50 · 35Q55 · 49J40.

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“…Finally, let us recall that reversed AR-conditions like the one in Hypothesis 13 have already been used also to treat superlinear problems, for instance see [28][29][30].…”

Section: Remarkmentioning

confidence: 98%

Superlinear Neumann problems with the p-Laplacian plus an indefinite potential

Fragnelli

Mugnai

Papageorgiou³

2016

Annali di Matematica

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We consider nonlinear Neumann problems driven by the p-Laplacian plus an indefinite potential and with a superlinear reaction which need not satisfy the Ambrosetti–Rabinowitz condition. First, we prove an existence theorem, and then, under stronger conditions on the reaction, we prove a multiplicity theorem producing three nontrivial solutions. Then, we examine parametric problems with competing nonlinearities (concave and convex terms). We show that for all small values of the parameter (Formula presented.), the problem has five nontrivial solutions and if (Formula presented.) (semilinear equation), there are six nontrivial solutions. Finally, we prove a bifurcation result describing the set of positive solutions as the parameter (Formula presented.) varies

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The Schrödinger–Poisson System with Positive Potential

Cited by 74 publications

References 29 publications

On the planar Schrödinger–Poisson system

On the planar Schrödinger–Poisson system

Existence of multiple positive solutions for Schrödinger–Poisson systems with critical growth

Superlinear Neumann problems with the p-Laplacian plus an indefinite potential

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