Multiple solutions for nonhomogeneous Schrödinger–Maxwell and Klein– Gordon–Maxwell equations on R 3

In this paper we study the following nonhomogeneous Schrödinger–Maxwell equations $\left\{\begin{array}{ll} {-\triangle u+V(x)u+ \phi u=f(x,u)+h(x),} \quad {\rm in}\,\,\,{\mathbf{R}}^3,\\ {-\triangle \phi=u^2, \qquad\qquad\qquad\qquad\qquad\qquad\,\,\, {\rm in} \,\,{\mathbf{R}}^3,} \end{array} \right.$ where f satisfies the Ambrosetti–Rabinowitz type condition. Under appropriate assumptions on V, f and h, the existence of multiple solutions is proved by using the Ekeland’s variational principle and the Mountain Pass Theorem in critical point theory. Similar results for the nonhomogeneous Klein–Gordon–Maxwell equations $\left\{\begin{array}{ll} {-\triangle u+[m^2-(\omega+\phi)^2]u=|u|^{q-2}u+h(x), \quad {\rm in} \,\,\,{\mathbf{R}}^3,}\\ {-\triangle \phi+ \phi u^2=-\omega u^2, \qquad\qquad\qquad\qquad\qquad\,\,\, {\rm in} \,\,\,{\mathbf{R}}^3,} \end{array} \right.$ are also obtained when 2 < q < 6.     

Multiple solutions for a Schrödinger–Bopp–Podolsky system with positive potentials

In this paper, we prove existence of solutions for a Schrödinger–Bopp–Podolsky system under positive potentials. We use the Ljusternick–Schnirelmann and Morse Theories to get multiple solutions with a priori given “interaction energy.”     

Multiple positive solutions for a quasilinear system of Schrödinger equations

We consider the quasilinear system

Multiple soliton solutions for a quasilinear Schrödinger equation

Using Morse theory, truncation arguments and an abstract critical point theorem, we obtain the existence of at least three or infinitely many nontrivial solutions for the following quasilinear Schrödinger equation in a bounded smooth domain

$$\left\{ {\begin{array}{*{20}{c}} { - {\Delta _p}u - \frac{p}{{{2^{p - 1}}}}u{\Delta _p}\left( {{u^2}} \right) = f\left( {x,u} \right)\;in\;\Omega } \\ {u = 0\;on\;\partial \Omega .} \end{array}} \right.$$

(0.1)

Our main results can be viewed as a partial extension of the results of Zhang et al. in [28] and Zhou and Wu in [29] concerning the the existence of solutions to (0.1) in the case of p = 2 and a recent result of Liu and Zhao in [21] two solutions are obtained for problem 0.1.

     

Multiplicity of solutions for a class of Schrödinger–Maxwell systems

In this paper, we study existence and multiplicity of nontrivial solutions for a class of Schrödinger–Maxwell systems via variational methods. Some new existence results of nontrivial solutions are obtained.     

Infinitely many positive solutions for a Schrödinger–Poisson system

We are interested in the existence of infinitely many positive non-radial solutions of a Schrödinger–Poisson system with a positive radial bounded external potential decaying at infinity.     

Infinitely many solutions for a class of sublinear Schrödinger–Maxwell equations

In this paper we study the existence of infinitely many solutions for a class of sublinear Schrödinger–Maxwell equations. The proof is based on the variant fountain theorem established by Zou. Recent results from the literature are extended.     

Multiple positive solutions of a class of Schrödinger–Poisson equation involving indefinite nonlinearity in R3

In this paper, we study the existence of multiple positive solutions of Schrödinger–Poisson type equations with indefinite nonlinearity. Our main tool is the mountain pass theorem.     

Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations

We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations

$$\begin{aligned} L^{\hbar }_{A,V} u = f(|u|^2)u \quad \hbox {in}\quad \mathbb {R}^N \end{aligned}$$

(0.1)

where \(N \ge 3\), \(L^{\hbar }_{A,V}\) is the Schrödinger operator with a magnetic field having source in a \(C^1\) vector potential A and a scalar continuous (electric) potential V defined by

$$\begin{aligned} L^{\hbar }_{A,V}= -\hbar ^2 \Delta -\frac{2\hbar }{i} A \cdot \nabla + |A|^2- \frac{\hbar }{i}\mathrm{div}A + V(x). \end{aligned}$$

(0.2)

Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain \(\Omega \subset \mathbb {R}^N\) such that

$$\begin{aligned} m_0 \equiv \inf _{x \in \Omega } V(x) < \inf _{x \in \partial \Omega } V(x) \end{aligned}$$

and we set \(K = \{ x \in \Omega \ | \ V(x) = m_0\}\). For \(\hbar >0\) small we prove the existence of at least \({\mathrm{cupl}}(K) + 1\) geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as \(\hbar \rightarrow 0\).

     

Multiple solutions for non-homogeneous degenerate Schrödinger equations in cone Sobolev spaces

The present paper deals with the study of semilinear and non-homogeneous Schrödinger equations on a manifold with conical singularity. We provide a suitable constant by Sobolev embedding constant and for p ∈ (2, 2?) with respect to non-homogeneous term g(x) ∈ L ₂ ^n/2 (B), which helps to find multiple solutions of our problem. More precisely, we prove the existence of two solutions to the problem 1.1 with negative and positive energy in cone Sobolev space H _2,0 ^1,n/2 (B). Finally, we consider p = 2 and we prove the existence and uniqueness of Fuchsian-Poisson problem.     

Multiple positive solutions for a nonlinear Schr?dinger equation

We are interested in positive entire solutions of the nonlinear Schrödinger equation -Du+(la(x)+1)u = u^p-\Delta u+(\lambda a(x)+1)u = u^p where a ≤ 0 has a potential well and p > 1 is subcritical. Using variational methods we prove the existence of multiple positive solutions which localize near the potential well int(a^-1(0)) for l\lambda large.