This article concerns the existence and multiplicity of solutions for the following class of non-linear elliptic equations with variable exponents

\begin{equation*}\left\{\begin{array}{l}-\Delta u+\lambda u=f(x,u), \quad \text{in} \quad \mathbb{R}^N, \\\,u\in H^{1}(\mathbb{R}^N),\\\end{array}\right.\end{equation*}

where λ > 0, $N\geq 2$ and $f:\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$ is a function of the following types:

Type 1: The subcritical case:

\begin{equation*}f(x,t)=|t|^{p(\varepsilon x)-2}t, \quad \forall (x,t) \in \mathbb{R}^N\times \mathbb{R}, N\geq3, \end{equation*}

Type 2: The critical case:

\begin{equation*}f(x,t)=\mu|t|^{p(\varepsilon x)-2}t+|t|^{2^*-2}t, \quad \forall (x,t) \in \mathbb{R}^N\times \mathbb{R}, N\geq3, \end{equation*}

Type 3: The exponential subcritical growth case:

\begin{equation*} f(x,t)=\mu|t|^{p(\varepsilon x)-2}te^{\alpha|t|^\beta}, \quad \forall (x,t) \in \mathbb{R}^2\times \mathbb{R}, \end{equation*}

where parameterɛ > 0, α > 0, $\beta \in (0,2)$, $2^*=\frac{2N}{N-2}$ if $N \geq 3$ and $2^*=+\infty$ if N = 2. Related to the function $p:\mathbb{R}^{N}\rightarrow \mathbb{R}$, we assume that it is a continuous function with $p_{\max}, p_{\min} \in (2,2^*)$, where $p_{\max}=\displaystyle \max_{x \in \mathbb{R}^N}p(x)$ and $p_{\min}=\displaystyle \min_{x \in \mathbb{R}^N}p(x)$.

We show that for each λ > 0 the number of solutions is associated with the number of global maximum or global minimum points of p when ɛ is small enough. The proof of is based on the variational methods, Ekeland’s variational principle, Trundiger–Moser inequality, and the monotonicity of the ground state energy with respect to p. Our results extend those of Cao and Noussair (Multiplicity of positive and nodal solutions for nonlinear elliptic problem in $\mathbb{R}^{N}$. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 13 (1996), 567–588) and Ji, Wang and Wu (A monotone property of the ground state energy to the scalar field equation and applications. J. Lond. Math. Soc., II. Ser. 100 (2019), 804–824).

We consider the pseudorelativistic Hartree equation

$$ \begin{align*} i\partial_t\psi=(\sqrt{-c^2\Delta +m^2c^4}-mc^2)\psi-(|x|^{-1}*|\psi|^2)\psi\quad \text{with } (t,x)\in\mathbb{R}\times\mathbb{R}^3, \end{align*} $$

which describes the dynamics of pseudorelativistic boson stars in the mean-field limit. We study the travelling waves of the form $\psi (t,x)=e^{it\mu }\varphi _{c}(x-vt)$, where $v\in \mathbb {R}^3$ denotes the travelling velocity. We prove that $\varphi _{c}$ converges strongly to the minimiser $\varphi _{\infty }$ of the limit energy $E_{\infty }(N)$ in $H^1(\mathbb {R}^3)$ as the light speed $c\to \infty $, where $E_{\infty }(N)$ is the corresponding energy for the limit equation

$$ \begin{align*} -\frac{1}{2m}\Delta\varphi_{\infty}+i(v\cdot\nabla)\varphi_{\infty}-({|x|^{-1}}*|\varphi_{\infty}|^2)\varphi_{\infty}=-\lambda\varphi_{\infty}. \end{align*} $$

Since the operator $-\Delta $ is the classical kinetic operator, we call this the nonrelativistic limit. We prove the existence of the minimiser for the limit energy $E_{\infty }(N)$ by using concentration-compactness arguments.

In this short note, we review results on equilibrium shapes of minimizers to the sessile drop problem. More precisely, we study the Winterbottom problem and prove that the Winterbottom shape is indeed optimal. The arguments presented here are based on relaxation and the (anisotropic) isoperimetric inequality.

This paper first studies the multiplicity of normalized solutions to the non-autonomous Schrödinger equation with mixed nonlinearities

\begin{equation*}\begin{cases}-\Delta u=\lambda u+h(\epsilon x)|u|^{q-2}u+\eta |u|^{p-2}u,\quad x\in \mathbb{R}^N, \\\int_{\mathbb{R}^N}|u|^2\,\textrm{d}x=a^2,\end{cases}\end{equation*}

$a, \epsilon, \eta \gt 0$

$\lambda\in \mathbb{R}$

$p=2N/(N-2)$

In this paper, we prove some weighted sharp inequalities of Trudinger–Moser type. The weights considered here have a logarithmic growth. These inequalities are completely new and are established in some new Sobolev spaces where the norm is a mixture of the norm of the gradient in two different Lebesgue spaces. This fact allowed us to prove a very interesting result of sharpness for the case of doubly exponential growth at infinity. Some improvements of these inequalities for the weakly convergent sequences are also proved using a version of the Concentration-Compactness principle of P.L. Lions. Taking profit of these inequalities, we treat in the last part of this work some elliptic quasilinear equation involving the weighted $(N,q)-$Laplacian operator where $1 < q < N$ and a nonlinearities enjoying a new type of exponential growth condition at infinity.

We consider the non-linear Schrödinger equation(Pμ)

\begin{equation*}\begin{array}{lc}-\Delta u + V(x) u = \mu f(u) + |u|^{2^*-2}u, &\end{array}\end{equation*}

$\mathbb{R}^N$

$N\geq3$

$f(s)/s$

The problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems have conservation laws of arbitrarily high order that must be found with the aid of computer algebra. Even low-order conservation laws of complex systems can be hard to find and invert. This paper describes a new, efficient approach to the inversion problem. Two main tools are developed: partial Euler operators and partial scalings. These lead to a line integral formula for the inversion of a total derivative and a procedure for inverting a given total divergence concisely.

We establish multiplicity results for the following class of quasilinear problems P

\begin{equation*} \left\{ \begin{array}{@{}l} -\Delta_{\Phi}u=f(x,u) \quad \mbox{in} \quad \Omega, \\ u=0 \quad \mbox{on} \quad \partial \Omega, \end{array} \right. \end{equation*}

$\Delta _{\Phi }u=\text {div}(\varphi (x,|\nabla u|)\nabla u)$

$\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$

$\Omega \subset \mathbb {R}^{N}$

$\Omega _N$

$\Omega _p$

$\overline {\Omega _N}\cap \overline {\Omega _p}=\emptyset$

$(P)$

$-\Delta _{\Phi }$

$-\Delta _N$

$\Omega _N$

$-\Delta _p$

$\Omega _p$

$f:\Omega \times \mathbb {R}\to \mathbb {R}$

$e^{\alpha |t|^{\frac {N}{N-1}}}$

$\Omega _N$

$|t|^{p^{*}-2}t$

$\Omega _p$

$|t|$

$\alpha >0$

$p^{*}=\frac {Np}{N-p}$

$1< p< N$

$f$

$f$

$\Omega$

$(P)$

In this paper, we consider the problem $-\Delta u =-u^{-\beta }\chi _{\{u>0\}} + f(u)$ in $\Omega$ with $u=0$ on $\partial \Omega$, where $0<\beta <1$ and $\Omega$ is a smooth bounded domain in $\mathbb {R}^{N}$, $N\geq 2$. We are able to solve this problem provided $f$ has subcritical growth and satisfy certain hypothesis. We also consider this problem with $f(s)=\lambda s+s^{\frac {N+2}{N-2}}$ and $N\geq 3$. In this case, we are able to obtain a solution for large values of $\lambda$. We replace the singular function $u^{-\beta }$ by a function $g_\epsilon (u)$ which pointwisely converges to $u^{-\beta }$ as $\epsilon \rightarrow 0$. The corresponding energy functional to the perturbed equation $-\Delta u + g_\epsilon (u) = f(u)$ has a critical point $u_\epsilon$ in $H_0^{1}(\Omega )$, which converges to a non-trivial non-negative solution of the original problem as $\epsilon \rightarrow 0$.

In this work we study the existence of nontrivial solution for the following class of semilinear degenerate elliptic equations

\[ -\Delta_{\gamma} u + a(z)u = f(u) \quad \mbox{in}\ \mathbb{R}^{N}, \]

$\Delta _{\gamma }$

$z:=(x,y)\in \mathbb {R}^{m}\times \mathbb {R}^{k}$

$m+k=N\geqslant 3$

$f$

$a$

$a=1$

We prove the existence of a solution for a class of activator–inhibitor system of type $- \Delta u +u = f(u) -v$ , $-\Delta v+ v=u$ in $\mathbb{R}^{N}$ . The function f is a general nonlinearity which can grow polynomially in dimension $N\geq 3$ or exponentiallly if $N=2$ . We are able to treat f when it has critical growth corresponding to the Sobolev space we work with. We transform the system into an equation with a nonlocal term. We find a critical point of the corresponding energy functional defined in the space of functions with norm endowed by a scalar product that takes into account such nonlocal term. For that matter, and due to the lack of compactness, we deal with weak convergent minimizing sequences and sequences of Lagrange multipliers of an action minima problem.

We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.

We study the existence and concentration of positive solutions for the following class of fractional p-Kirchhoff type problems:

$$ \left\{\begin{array}{@{}ll} \left(\varepsilon^{sp}a+\varepsilon^{2sp-3}b \,[u]_{s, p}^{p}\right)(-\Delta)_{p}^{s}u+V(x)u^{p-1}=f(u) & \text{in}\ \mathbb{R}^{3},\\ \noalign{ u\in W^{s, p}(\mathbb{R}^{3}), \quad u>0 & \text{in}\ \mathbb{R}^{3}, \end{array}\right.$$

$sp \in (\frac {3}{2}, 3)$

$(-\Delta )^{s}_{p}$

Suppose that $G=(V,E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $\unicode[STIX]{x1D6E5}$ be the usual graph Laplacian. Consider the nonlinear Schrödinger equation of the form

$$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u-\unicode[STIX]{x1D6FC}u=f(x,u),\quad u\in W^{1,2}(V),\end{eqnarray}$$

$G$

$f(x,u):V\times \mathbb{R}\rightarrow \mathbb{R}$

$\unicode[STIX]{x1D6FC}$

$G$

$G$

$CD(m,\unicode[STIX]{x1D709})$

$G$

$\unicode[STIX]{x1D6FC}<2\unicode[STIX]{x1D706}_{1}^{2}/m(\unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D709})$

$\unicode[STIX]{x1D706}_{1}$

We consider the minimizing problem for the energy functional with prescribed mass constraint related to the fractional non-linear Schrödinger equation with periodic potentials. Using the concentration-compactness principle, we show a complete classification for the existence and non-existence of minimizers for the problem. In the mass-critical case, under a suitable assumption of the potential, we give a detailed description of blow-up behaviour of minimizers once the mass tends to a critical value.

We prove the existence of global minimisers for a class of attractive–repulsive interaction potentials that are in general not radially symmetric. The global minimisers have compact support. For potentials including degenerate power-law diffusion, the interaction potential can be unbounded from below. Further, a formal calculation indicates that for non-symmetric potentials global minimisers may neither be radial symmetric nor unique.

In this paper, we study the existence of positive solutions to a semilinear nonlocal elliptic problem with the fractional α-Laplacian on Rn, 0 < α < n. We show that the problem has infinitely many positive solutions in $ {C^\tau}({R^n})\bigcap H_{loc}^{\alpha /2}({R^n}) $. Moreover, each of these solutions tends to some positive constant limit at infinity. We can extend our previous result about sub-elliptic problem to the nonlocal problem on Rn. We also show for α ∊ (0, 2) that in some cases, by the use of Hardy’s inequality, there is a nontrivial non-negative $ H_{loc}^{\alpha /2}({R^n}) $ weak solution to the problem

$$ {( - \Delta )^{\alpha /2}}u(x) = K(x){u^p} \quad {\rm{ in}} \ {R^n}, $$

In this paper, we consider the nonlinear Choquard equation

$$-\Delta u + V(x)u = \left( {\int_{{\open R}^N} {\displaystyle{{G(u)} \over { \vert x-y \vert ^\mu }}} \,{\rm d}y} \right)g(u)\quad {\rm in}\;{\open R}^N, $$

$G(u)=\int ^u_0g(s)\,{\rm d}s$

This paper is devoted to the study of fractional Schrödinger-Poisson type equations with magnetic field of the type

$$\varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u + V(x)u + {\rm e}^{-2t}(\vert x \vert^{2t-3} \ast \vert u\vert ^{2})u = f(\vert u \vert^{2})u \quad \hbox{in} \ \open{R}^{3},$$

We study the existence and multiplicity of periodic weak solutions for a non-local equation involving an odd subcritical nonlinearity which is asymptotically linear at infinity. We investigate such problem by applying the pseudo-index theory developed by Bartolo, Benci and Fortunato [11] after transforming the problem to a degenerate elliptic problem in a half-cylinder with a Neumann boundary condition, via a Caffarelli-Silvestre type extension in periodic setting. The periodic nonlocal case, considered here, presents, respect to the cases studied in the literature, some new additional difficulties and a careful analysis of the fractional spaces involved is necessary.