Related papers: Multiple solutions to a magnetic nonlinear Choquar…

In this paper we study the normalized solutions of the following critical growth Choquard equation with mixed local and non-local operators: \begin{equation*} \begin{array}{rcl} -\Delta_p u +(-\Delta_p)^s u & = & \lambda |u|^{p-2}u +\mu…

We propose an existence result for the semirelativistic Choquard equation with a local nonlinearity in $\mathbb{R}^N$ \begin{equation*} \sqrt{\strut -\Delta + m^2} u - mu + V(x)u = \left( \int_{\mathbb{R}^N}…

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ \Delta})^{\frac{\alpha}{2}}u- \gamma \frac{u}{|x|^{\alpha}}=…

We discuss the existence of ground state solutions for the Choquard equation $$-\Delta u=(I_\alpha*F(u))F'(u)\quad\quad\quad\text{in }\mathbb R^N.$$ We prove the existence of solutions under general hypotheses, investigating in particular…

We consider the nonlinear elliptic equation \begin{equation*} -\Delta u + V(x)u = f(u), \qquad u\in D^{1,2}_0(\Omega), \end{equation*} in an exterior domain $\Omega$ of $\mathbb{R}^N$, where $V$ is a scalar potential that decays to zero at…

We construct a new family of entire solutions for the nonlinear Schr\"odinger equation \begin{align*} \begin{cases} -\Delta u+ V(y ) u = u^p, \quad u>0, \quad \text{in}~ \mathbb{R}^N, \\[2mm] u \in H^1(\mathbb{R}^N), \end{cases}…

In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents (-\Delta)_{p(\cdot)}^{s(\cdot)}u(x)&=\lambda|u(x)|^{\alpha(x)-2}u(x)+…

This paper is concerned with the multiplicity and concentration behavior of nontrivial solutions for the following fractional Kirchhoff equation in presence of a magnetic field: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3}…

We look for ground state solutions to the following nonlinear Schr\"{o}dinger equation $$-\Delta u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where $V=V_{per}+V_{loc}\in L^{\infty}(\mathbb{R}^N)$ is the sum of a periodic…

We study the non-existence and multiplicity of positive solutions of the nonlinear Choquard type equation $$ -\Delta u+ \varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ |u|^{q-2}u, \quad {\rm in} \ \mathbb R^N, \qquad (P_\varepsilon)$$…

We prove the existence of solutions for the following critical Choquard type problem with a variable-order fractional Laplacian and a variable singular exponent \begin{align*} \begin{split} a(-\Delta)^{s(\cdot)}u+b(-\Delta)u&=\lambda…

This article concerns the existence of multi-bump positive solutions for the following logarithmic Schr\"{o}dinger equation $$ \left\{ \begin{array}{lc} -\Delta u+ \lambda V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ u \in…

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear…

In this paper we consider the following class of elliptic problems $$- \Delta_A u + u = a_{\lambda}(x) |u|^{q-2}u+b_{\mu}(x) |u|^{p-2}u ,$$ for $x \in \mathbb{R}^N$, $1<q<2<p<2^*-1= \frac{N+2}{N-2}$, $a_{\lambda}(x)$ is a sign-changing…

This article deals with the study of the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left(\, \int\limits_{\mathbb{R}^N}|\nabla u|^p\right) (-\Delta_p) u + V(x)|u|^{p-2}u = \left(\,…

We are interested in the multiplicity of solutions to the following scalar field equation $$ -\Delta u - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \mbox{in } \mathbb{R}^N \setminus \{0\}. $$ We establish the existence of infinitely many radial…

The present work is concerned with the following version of Choquard Logarithmic equations $ -\Delta_p u -\Delta_N u + a|u|^{p-2}u + b|u|^{N-2}u + \lambda (\ln|\cdot|\ast G(u))g(u) = f(u) \textrm{ in } \mathbb{R}^N $ , where $ a, b, \lambda…

In this paper, we study asymptotic behavior of positive ground state solutions for the nonlinear Choquard equation: \begin{equation}\label{0.1} -\Delta u+\varepsilon u=\big(I_{\alpha}\ast F(u)\big)F'(u),\quad u\in H^1(\mathbb R^N),…

We are interested in the nonlinear, time-harmonic Maxwell equation $$ \nabla \times (\nabla \times \mathbf{E} ) + V(x) \mathbf{E} = h(x, \mathbf{E})\mbox{ in } \mathbb{R}^3 $$ with sign-changing nonlinear term $h$, i.e. we assume that $h$…

We consider the existence of multiple positive solutions to the nonlinear Schr\"odinger systems sets on $H^1(\mathbb{R}^N) \times H^1(\mathbb{R}^N)$, \[ \left\{ \begin{aligned} -\Delta u_1 &= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + \beta…