An improved result for a class of Klein–Gordon–Maxwell system with quasicritical potential vanishing at infinity

The following kind of Klein–Gordon–Maxwell system is investigated where $\omega >0$ is a parameter, and V is vanishing potential. By using some suitable conditions on K and f, we obtain a Palais–Smale sequence by using Pohožaev equality and prove the ground-state solution for this system by employing variational methods. Our result improves the related one in the literature.     

Quasilinear Schrödinger equations with unbounded or decaying potentials in dimension 2

We establish the existence of nontrivial solutions for the following class of quasilinear Schrödinger equations: where κ is a positive parameter, $V(|x|)$ and $Q(|x|)$ are continuous functions that can be singular at the origin, unbounded or vanishing at infinity, and the nonlinearity $h(s)$ has critical exponential growth motivated by the Trudinger–Moser inequality. To prove our main result, we apply variational methods together with careful $L^{\infty }$-estimates.     

Positive ground state solutions for Schrödinger–Poisson system with critical nonlocal term in ℝ3

This paper is dedicated to studying the following Schrödinger–Poisson system $$\left\{\begin{array}{ll}-\mathrm{\Delta }u+V(x)u-K(x)\varphi |u{|}^{3}u=a(x)f(u),& x\in {ℝ}^3,\\ -\mathrm{\Delta }\varphi =K(x)|u{|}^5,& x\in {ℝ}^3.\end{array}\right.$$ Under some different assumptions on functions V(x), K(x), a(x) and f(u), by using the variational approach, we establish the existence of positive ground state solutions.     

Existence of positive ground state solutions for the coupled Choquard system with potential

In this paper, we study the following coupled Choquard system in ${\mathbb{R}}^N$: where $\alpha \in (0,N)$ and , in which $2_{\ast }^{\alpha }$ denotes $\frac{N+\alpha }{N-2}$ if $N\ge 3$ and $2_{\ast }^{\alpha }:=\infty$ if $N=1,2$. The function $I_{\alpha }$ is a Riesz potential. By using Nehari manifold method, we obtain the existence of a positive ground state solution in the case of bounded potential and periodic potential, respectively. In particular, the nonlinear term includes the well-studied case $p=q$ and $u(x)=v(x)$, and the less-studied case $p\ne q$ and $u(x)\ne v(x)$. Moreover, it seems to be the first existence result for the case $p\ne q$.     

Global well-posedness and finite time blow-up for a class of wave equation involving fractional p-Laplacian with logarithmic nonlinearity

In this paper, we consider the following class of wave equation involving fractional p-Laplacian with logarithmic nonlinearity where $\mathrm{\Omega }\subset {\mathbb{R}}^N(N\ge 1)$ is a bounded domain with Lipschitz boundary, $s\in (0,1)$, , and $p_s^{\ast }=\frac{Np}{N-sp}$ is the critical exponent in the Sobolev inequality. First, via the Galerkin approximations, the existence of local solutions are obtained when . Next, by combining the potential well theory with the Nehari manifold, we establish the existence of global solutions when . Then, via the Pohozaev manifold, the existence of global solutions are obtained when . By virtue of a differential inequality technique, we prove that the local solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we discuss the asymptotic behavior of solutions as time tends to infinity. Here, we point out that the main difficulty is the lack of logarithmic Sobolev inequality concerning fractional p-Laplacian.     

Two solutions for a class of singular Kirchhoff‐type problems with Hardy–Sobolev critical exponent II

In this article, we devote ourselves to investigate the following singular Kirchhoff‐type equation: $$\left\{\begin{array}{l}?\left(a+b{\int }_{\mathrm{\Omega }}|?u{|}^{2}dx\right)\mathrm{\Delta }u=\frac{u^{5?2s}}{|x{|}^s}+\frac{\lambda }{|x{|}^{\beta }{u}^{\gamma }},x\in \mathrm{\Omega },\\ u>0,x\in \mathrm{\Omega },\\ u=0,x\in ?\mathrm{\Omega },\end{array}\right.$$ where $\mathrm{\Omega }?{?}^3$ is a bounded domain with smooth boundary ?Ω,0∈Ω,a≥0,b,λ>0,0<γ,s<1, and $0\le \beta <\frac{5+\gamma }{2}.$ By using the variational and perturbation methods, we obtain the existence of two positive solutions, which generalizes and improves the recent results in the literature.     

Properties of given and detected unbounded solutions to a class of chemotaxis models

This paper deals with unbounded solutions to a class of chemotaxis systems. In particular, for a rather general attraction–repulsion model, with nonlinear productions, diffusion, sensitivities, and logistic term, we detect Lebesgue spaces where given unbounded solutions also blow up in the corresponding norms of those spaces; subsequently, estimates for the blow-up time are established. Finally, for a simplified version of the model, some blow-up criteria are proved. More precisely, we analyze a zero-flux chemotaxis system essentially described as (⋄) The problem is formulated in a bounded and smooth domain Ω of ${\mathbb{R}}^n$, with $n\ge 1$, for some $m_1,m_2,m_3\in \mathbb{R}$, $\chi ,\xi ,\alpha ,\beta ,\lambda ,\mu >0$, $k>1$, and with $T_{max}\in (0,\infty ]$. A sufficiently regular initial data $u_0\ge 0$ is also fixed. Under specific relations involving the above parameters, one of these always requiring some largeness conditions on $m_2+\alpha$,

• (i) we prove that any given solution to ($\diamond$), blowing up at some finite time $T_{max}$ becomes also unbounded in $L^{\mathfrak{p}}(\mathrm{\Omega })$-norm, for all $\mathfrak{p}>\frac{n}{2}(m_2-m_1+\alpha )$;
• (ii) we give lower bounds T (depending on ${\int }_{\mathrm{\Omega }}{u}_0^{\overline{p}}$) of $T_{max}$ for the aforementioned solutions in some $L^{\overline{p}}(\mathrm{\Omega })$-norm, being $\overline{p}=\overline{p}(n,m_1,m_2,m_3,\alpha ,\beta )\ge \mathfrak{p}$;
• (iii) whenever $m_2=m_3$, we establish sufficient conditions on the parameters ensuring that for some u₀ solutions to ($\diamond$) effectively are unbounded at some finite time.

Within the context of blow-up phenomena connected to problem ($\diamond$), this research partially improves the analysis in Wang et al. (J Math Anal Appl. 2023;518(1):126679) and, moreover, contributes to enrich the level of knowledge on the topic.     

Standing waves for a class of fractional p‐Laplacian equations with a general critical nonlinearity

In this paper, we concern with the following fractional p‐Laplacian equation with critical Sobolev exponent $$\left\{\begin{array}{l}{\epsilon }^{ps}{\left\{?\mathrm{\Delta }\right.}_p^{s}u+V(x){\left|u\right|}^{p?2}u=\lambda f(x){\left|u\right|}^{q?2}u+{\left|u\right|}^{p_s^{?}?2}u\text{in}{?}^N,\\ u\in W^{s,p}\left\{{?}^N\right.,u>0,\end{array}\right.$$ where ε > 0 is a small parameter,  λ > 0 , N is a positive integer, and N > ps with s ∈ (0, 1) fixed, $1<q\le p,p_s^{?}:=Np/\left(N?ps\right)$. Since the nonlinearity $h(x,u):=\lambda f(x){\left|u\right|}^{q?2}u+{\left|u\right|}^{p_s^{?}?2}u$ does not satisfy the following Ambrosetti‐Rabinowitz condition: $$0<\mu H(x,u):=\mu {\int }_0^{u}h(x,t)dt\le h(x,u)u,x\in {?}^N,0\ne u\in ?,$$ with μ > p , it is difficult to obtain the boundedness of Palais‐Smale sequence, which is important to prove the existence of positive solutions. In order to overcome the above difficulty, we introduce a penalization method of fractional p‐Laplacian type.     

A critical Kirchhoff‐type problem driven by a p (·)‐fractional Laplace operator with variable s (·) ‐order

The paper deals with the following Kirchhoff‐type problem $$\left\{\begin{array}{lll}& M\left({?}_{{?}^{2N}}\frac{1}{p(x,y)}\frac{|v(x)?v(y){|}^{p(x,y)}}{|x?y{|}^{N+p(x,y)s(x,y)}}dxdy\right){(?\mathrm{\Delta })}_{p(·)}^{s(·)}v(x)=\mu g(x,v)+|v{|}^{r(x)?2}v& \text{in}\mathrm{\Omega },\\ & v=0& \text{in}{?}^N\backslash \mathrm{\Omega },\end{array}\right.$$ where M models a Kirchhoff coefficient, ${(?\mathrm{\Delta })}_{p(·)}^{s(·)}$ is a variable s(·) ‐order p(·) ‐fractional Laplace operator, with $s(·):{?}^{2N}\to (0,1)$ and $p(·):{?}^{2N}\to (1,\infty )$. Here, $\mathrm{\Omega }?{?}^N$ is a bounded smooth domain with N > p(x, y)s(x, y) for any $(x,y)\in \overline{\mathrm{\Omega }}\times \overline{\mathrm{\Omega }}$, μ is a positive parameter, g is a continuous and subcritical function, while variable exponent r(x) could be close to the critical exponent $p_s^{?}(x)=N\overline{p}(x)/(N?\overline{s}(x)\overline{p}(x))$, given with $\overline{p}(x)=p(x,x)$ and $\overline{s}(x)=s(x,x)$ for $x\in \overline{\mathrm{\Omega }}$. We prove the existence and asymptotic behavior of at least one non‐trivial solution. For this, we exploit a suitable tricky step analysis of the critical mountain pass level, combined with a Brézis and Lieb‐type lemma for fractional Sobolev spaces with variable order and variable exponent.     

Criterion of Bari basis property for 2 × 2 Dirac-type operators with strictly regular boundary conditions

The paper is concerned with the Bari basis property of a boundary value problem associated in $L^2([0,1];{\mathbb{C}}^2)$ with the following 2 × 2 Dirac-type equation for $y=col(y_1,y_2)$: with a potential matrix $Q\in L^2([0,1];{\mathbb{C}}^{2\times 2})$ and subject to the strictly regular boundary conditions $Uy:=\{U_1,U_2\}y=0$. If $b_2=-b_1=1$, this equation is equivalent to one-dimensional Dirac equation. We show that the normalized system of root vectors ${\{f_n\}}_{n\in \mathbb{Z}}$ of the operator $L_U(Q)$ is a Bari basis in $L^2([0,1];{\mathbb{C}}^2)$ if and only if the unperturbed operator $L_U(0)$ is self-adjoint. We also give explicit conditions for this in terms of coefficients in the boundary conditions. The Bari basis criterion is a consequence of our more general result: Let $Q\in L^p([0,1];{\mathbb{C}}^{2\times 2})$, $p\in [1,2]$, boundary conditions be strictly regular, and let ${\{g_n\}}_{n\in \mathbb{Z}}$ be the sequence biorthogonal to the normalized system of root vectors ${\{f_n\}}_{n\in \mathbb{Z}}$ of the operator $L_U(Q)$. Then, $$\{\parallel f_n-g_n{{\parallel }_2\}}_{n\in \mathbb{Z}}\in {({\ell }^p(\mathbb{Z}))}^{\ast }\iff L_U(0)=L_U{(0)}^{\ast }.$$ These abstract results are applied to noncanonical initial-boundary value problem for a damped string equation.     

Existence of solutions to a Neumann boundary value problem with exponential nonlinearity

This paper is concerned with the solutions to the following sinh-Poisson equation with Hénon term$\{\begin{array}{cc}?\mathrm{\Delta }u+u={\epsilon }^2|x?q_1{|}^{2{\alpha }_1}?|x?q_n{|}^{2{\alpha }_n}({\text{e}}^u?{\text{e}}^{?u}),u>0,\hfill & \text{in}\mathrm{\Omega },\hfill \\ \frac{?u}{?\nu }=0,\hfill & \text{on}?\mathrm{\Omega },\hfill \end{array}$ where $\mathrm{\Omega }?{\mathbb{R}}^2$ is a bounded, smooth domain, $\epsilon >0$, ${\alpha }_1,...,{\alpha }_n\in (0,\infty )?\mathbb{N}$, and $q_1,...,q_n\in \mathrm{\Omega }$ are fixed. Given any two non-negative integers $k,l$ with $k+l?1$, it is shown that, for sufficiently small $\epsilon >0$, there exists a solution $u_{\epsilon }$ for which ${\epsilon }^2|x?q_1{|}^{2{\alpha }_1}?|x?q_n{|}^{2{\alpha }_n}({\text{e}}^u?{\text{e}}^{?u})$ asymptotically (i.e. the limit as $\epsilon \to 0$) develops $k+n$ interior Dirac measures and l boundary Dirac measures. The location of blow-up points is characterized explicitly in terms of Green's function of Neumann problem and the function $k(x)=|x?q_1{|}^{2{\alpha }_1}?|x?q_n{|}^{2{\alpha }_n}$.     

Normalized ground state for the Sobolev critical Schrödinger equation involving Hardy term with combined nonlinearities

In this paper, we study the existence and properties of normalized solutions for the following Sobolev critical Schrödinger equation involving Hardy term: $$-\mathrm{\Delta }u-\frac{\mu }{{|x|}^2}u=\lambda u+{|u|}^{2^{\ast }-2}u+\nu {|u|}^{p-2}u\text{in}{\mathbb{R}}^N,N\ge 3,$$ with prescribed mass $${\int }_{{\mathbb{R}}^N}{u}^2=a^2,$$ where 2* is the Sobolev critical exponent. For a L²-subcritical, L²-critical, or L²-supercritical perturbation ${\nu |u|}^{p-2}u$, we prove several existence results of normalized ground state when $\nu \ge 0$ and non-existence results when $\nu \le 0$. Furthermore, we also consider the asymptotic behavior of the normalized solutions u as $\mu \to 0$ or $\nu \to 0$.     

Single-peak solutions for a subcritical Schrödinger equation with non-power nonlinearity

This paper deals with the following slightly subcritical Schrödinger equation: $$-\mathrm{\Delta }u+V(x)u=f_{\epsilon }(u),u>0\text{in}{\mathbb{R}}^N,$$ where $V(x)$ is a nonnegative smooth function, $f_{\epsilon }(u)=\frac{u^p}{{[\ln (e+u)]}^{\epsilon }}$, $p=\frac{N+2}{N-2}$, $\epsilon >0$, $N\ge 7$. Most of the previous works for the Schrödinger equations were mainly investigated for power-type nonlinearity. In this paper, we will study the case when the nonlinearity $f_{\epsilon }(u)$ is a non-power nonlinearity. We show that, for ε small enough, there exists a family of single-peak solutions concentrating at the positive stable critical point of the potential $V(x)$.