Multiple small solutions for Schrödinger equations involving positive quasilinear term

We consider the multiplicity of solutions of a class of quasilinear Schrödinger equations involving the $p$-Laplacian: \begin{equation*} -\Delta_{p} u+V(x)|u|^{p-2}u+\Delta_{p}(u^{2})u=K(x)f(x,u),\qquad x\in \mathbb{R}^{N}, \end{equation*} where $\Delta_{p} u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$, $1<p<N$, $N\geq3$, $V$, $K$ belong to $C(\mathbb{R}^{N})$ and $f$ is an odd continuous function without any growth restrictions at large. Our method is based on a direct modification of the indefinite variational problem to a definite one. Even for the case $p=2$, the approach also yields new multiplicity results

Multiple and least energy sign-changing solutions for Schrodinger-Poisson equations in R3 with restraint

In this paper, we study the existence of multiple sign-changing solutions with a prescribed Lp+1−norm and theexistence of least energy sign-changing restrained solutions for the following nonlinear Schr¨odinger-Poisson system:By choosing a proper functional restricted on some appropriate subset to using a method of invariant sets of descending flow,we prove that this system has infinitely many sign-changing solutions with the prescribed Lp+1−norm and has a least energy forsuch sign-changing restrained solution for p ∈ (3, 5). Few existence results of multiple sign-changing restrained solutions areavailable in the literature. Our work generalize some results in literature

Multiple solutions to elliptic equations on RN\mathbb{R}^N with combined nonlinearities

In this paper, we are concerned with the multiplicity of nontrivial radial solutions for the following elliptic equation \begin{equation*} \begin{cases} - \Delta u +V(x)u = -\lambda Q(x)|u|^{q-2}u+ Q(x)f(u),\quad x\in\mathbb{R}^N,\\ u(x)\rightarrow 0,\quad \hbox{as}\ |x|\rightarrow +\infty,\end{cases} \tag*{(P)$_\lambda$} \end{equation*} where $1<q<2,\ \lambda\in \mathbb{R}^+,\ N\geq 3$, $V$ and $Q$ are radial positive functions, which can be vanishing or coercive at infinity, $f$ is asymptotically linear or superlinear at infinity

Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains

We consider a semilinear elliptic problem with a nonlinear term which is the product of a power and the Riesz potential of a power. This family of equations includes the Choquard or nonlinear Schroedinger--Newton equation. We show that for some values of the parameters the equation does not have nontrivial nonnegative supersolutions in exterior domains. The same techniques yield optimal decay rates when supersolutions exists.Comment: 47 pages, 8 figure

A positive solution of asymptotically periodic Schrödinger equations with local superlinear nonlinearities

In this paper, we investigate the following Schrödinger equation −∆u + V(x)u = λ f(u) in R N, where N ≥ 3, λ > 0, V is an asymptotically periodic potential and the nonlinearity term f(u) is only locally defined for |u| small and satisfies some mild conditions. By using Nehari manifold and Moser iteration, we obtain the existence of positive solutions for the equation with sufficiently large λ

International Conference on Nonlinear Differential Equations and Applications

Dear Participants, Colleagues and Friends It is a great honour and a privilege to give you all a warmest welcome to the first Portugal-Italy Conference on Nonlinear Differential Equations and Applications (PICNDEA). This conference takes place at the Colégio Espírito Santo, University of Évora, located in the beautiful city of Évora, Portugal. The host institution, as well the associated scientific research centres, are committed to the event, hoping that it will be a benchmark for scientific collaboration between the two countries in the area of mathematics. The main scientific topics of the conference are Ordinary and Partial Differential Equations, with particular regard to non-linear problems originating in applications, and its treatment with the methods of Numerical Analysis. The fundamental main purpose is to bring together Italian and Portuguese researchers in the above fields, to create new, and amplify previous collaboration, and to follow and discuss new topics in the area

Multiplicity and concentration of solutions for a fractional Kirchhoff equation with magnetic field and critical growth

We investigate the existence, multiplicity and concentration of nontrivial solutions for the following fractional magnetic Kirchhoff equation with critical growth: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3} [u]_{A/\varepsilon}^{2}\right)(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u+|u|^{\2-2}u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon$ is a small positive parameter, $a, b>0$ are fixed constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential verifying the global condition due to Rabinowitz \cite{Rab}, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ subcritical nonlinearity. Due to the presence of the magnetic field and the critical growth of the nonlinearity, several difficulties arise in the study of our problem and a careful analysis will be needed. The main results presented here are established by using minimax methods, concentration compactness principle of Lions \cite{Lions}, a fractional Kato's type inequality and the Ljusternik-Schnirelmann theory of critical points.Comment: arXiv admin note: text overlap with arXiv:1808.0929

Finding Multiple Saddle Points for G-differential Functionals and Defocused Nonlinear Problems

We study computational theory and numerical methods for finding multiple unstable solutions (saddle points) for two types of nonlinear variational functionals. The first type consists of Gateaux differentiable (G-differentiable) M-type (focused) problems. Motivated by quasilinear elliptic problems from physical applications, where energy functionals are at most lower semi-continuous with blow-up singularities in the whole space and G-differntiable in a subspace, and mathematical results and numerical methods for C1 or nonsmooth/Lipschitz saddle points existing in the literature are not applicable, we establish a new mathematical frame-work for a local minimax method and its numerical implementation for finding multiple G-saddle points with a new strong-weak topology approach. Numerical implementation in a weak form of the algorithm is presented. Numerical examples are carried out to illustrate the method. The second type consists of C^1 W-type (defocused) problems. In many applications, finding saddles for W-type functionals is desirable, but no mathematically validated numerical method for finding multiple solutions exists in literature so far. In this dissertation, a new mathematical numerical method called a local minmaxmin method (LMMM) is proposed and numerical examples are carried out to illustrate the efficiency of this method. We also establish computational theory and present the convergence results of LMMM under much weaker conditions. Furthermore, we study this algorithm in depth for a typical W-type problem and analyze the instability performances of saddles by LMMM as well