In this paper, we consider the following fractional Schrödinger equation
ε2s(−Δ)su+V(x)u=P(x)f(u)+Q(x)|u|2s∗−2uinℝN,$$ {\varepsilon}^{2s}{\left(-\Delta ...\right)}^su+V(x)u=P(x)f(u)+Q(x){\left|u\right|}^{2_s^{\ast }-2}u\kern0.30em \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}^N, $$
where
ε>0$$ \varepsilon >0 $$ is a parameter,
s∈(0,1),2s∗=2NN−2s,N>2s,(−Δ)s$$ s\in \left(0,1\right),{2}_s^{\ast }=\frac{2N}{N-2s},N>2s,{\left(-\Delta \right)}^s $$ is the fractional Laplacian and
f$$ f $$ is a superlinear and subcritical nonlinearity. Under a local condition imposed on the potential function, combining the penalization method and the concentration‐compactness principle, we prove the existence of a positive solution for the above equation.
This paper is concerned with the existence and multiplicity of solutions for a class of problems involving the Φ‐Laplacian operator with general assumptions on the nonlinearities, which include both ...semipositone cases and critical concave convex problems. The research is based on the subsupersolution technique combined with a truncation argument and an application of the Mountain Pass Theorem. The results in this paper improve and complement some recent contributions to this field.
In this paper, we are concerned with the system of Schrödinger–Poisson equations
(*)
{
−
Δ
u
+
V
(
x
)
u
+
ϕ
u
=
f
(
x
,
u
)
,
in
R
3
,
−
Δ
ϕ
=
u
2
,
in
R
3
.
Under certain assumptions on
V and
f, ...the existence and multiplicity of solutions for (*) are established via variational methods.
We consider the multiplicity of positive solutions for a Schrödinger-Poisson-Slater equation of the type−△u+(u2⁎1|4πx|)u=μ|u|p−1u+g(x)|u|4u,inR3, where μ>0, 3<p<5, and g∈C(R3,R+). Using Ekeland's ...variational principle and the well-known arguments of the concentration-compactness principle of Lions (1984) 24,25, when g has one local maximum point, we obtain a positive ground-state solution for all μ>0, while for g with k strict local maximum points, we prove that the equation has at least k distinct positive solutions for μ>0 small. The proof uses a minimization procedure on the Nehari manifold.
In this paper, we focus on the existence of positive solutions to the following planar Schrödinger-Newton system with general subcritical growth{−Δu+u+ϕu=f(u)inR2,Δϕ=u2inR2, where f is a smooth ...reaction. We introduce a new variational approach, which enables us to study the above problem in the Sobolev space H1(R2). The analysis developed in this paper also allows to investigate the relationship between a Schrödinger-Newton system of Riesz-type and a Schrödinger-Newton system of logarithmic-type. Furthermore, this new approach can provide a new look at the planar Schrödinger-Newton system and may it have some potential applications in various related problems.
In this paper we study the following type of the Schrödinger–Poisson–Slater equation with critical growth−△u+(u2⋆1|4πx|)u=μ|u|p−1u+|u|4u,inR3, where μ>0 and p∈(11/7,5). For the case of p∈(2,5). We ...develop a novel perturbation approach, together with the well-known Mountion–Pass theorem, to prove the existence of positive ground states. For the case of p=2, we obtain the nonexistence of nontrivial solutions by restricting the range of μ and also study the existence of positive solutions by the constrained minimization method. For the case of p∈(11/7,2), we use a truncation technique developed by Brezis and Oswald 9 together with a measure representation concentration-compactness principle due to Lions 27 to prove the existence of radial symmetrical positive solutions for μ∈(0,μ⁎) with some μ⁎>0. The above results nontrivially extend some theorems on the subcritical case obtained by Ianni and Ruiz 18 to the critical case.
In this paper we study the following nonlinear Choquard equation −Δu+u=ln1|x|∗F(u)f(u),inR2,where f∈C1(R,R) and F is the primitive of the nonlinearity f vanishing at zero. We use an asymptotic ...approximation approach to establish the existence of positive solutions to the above problem in the standard Sobolev space H1(R2). We give a new proof and at the same time extend part of the results established in (Cassani-Tarsi, Calc.Var.PDE, 2021) 11.
In this paper, we study the following fractional Schrödinger‐Poisson system involving competing potential functions
ϵ2s(−Δ)su+V(x)u+φu=K(x)f(u)+Q(x)|u|2s∗−2u,inR3,ϵ2t(−Δ)tφ=u2,inR3,
where ϵ > 0 is a ...small parameter, f is a function of C1 class, superlinear and subcritical nonlinearity,
2s∗=63−2s,
s>34, t ∈ (0,1), V(x), K(x), and Q(x) are positive continuous functions. Under some suitable assumptions on V, K, and Q, we prove that there is a family of positive ground state solutions with polynomial growth for sufficiently small ϵ > 0, of which it is concentrating on the set of minimal points of V(x) and the sets of maximal points of K(x) and Q(x).
In general, the existence of nodal solution for Schrödinger–Poisson systems with the nonlinearity f(x)|u|p−2u(4≤p<6) in R3 can be established by using the nodal Nehari manifold method. However, for ...the case where 2<p<4, such an approach is not applicable because Palais-Smale sequences restricted on the nodal Nehari manifold can be not bounded. In this paper, we introduce a novel constraint method to prove the existence of nodal solution to a class of non-autonomous Schrödinger–Poisson systems in the case where 2<p<4. We conclude that such solution changes sign exactly once in R3 and is bounded in H1(R3)×D1,2(R3). Moreover, the existence of least energy nodal solution is obtained in the case where 1+733<p<4, which remains unsolved in the existing literature.
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