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We consider the existence of ground state solutions for a class of zero-mass Chern-Simons-Schr\"{o}dinger systems \[ \left\{ \begin{array}{ll} \displaystyle -\Delta u +A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u)-a(x)|u|^{p-2}u, \newline \displaystyle \partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2,~\partial_1A_1+\partial_2A_2=0, \newline \displaystyle \partial_1A_0=A_2|u|^2,~ \partial_2A_0=-A_1|u|^2, \end{array} \right. \] where $a:\mathbb R^2\to\mathbb R^+$ is an external potential, $p\in(1,2)$ and $f\in \mathcal{C}(\mathbb R)$ denotes a nonlinearity that fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. By introducing an improvement of the version of Trudinger-Moser inequality, we are able to investigate the existence of positive ground state solutions for the given system using variational method., Comment: 20 pages

We investigate the existence and concentration of normalized solutions for a $p$-Laplacian problem with logarithmic nonlinearity of type \[ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^p\Delta_p u+V(x)|u|^{p-2}u=\lambda |u|^{p-2}u+|u|^{p-2}u\log|u|^p ~\text{in}~\mathbb R^N,\newline \displaystyle \int_{\mathbb R^N}|u|^pdx=a^p\varepsilon^N, \end{array} \right. \] where $a,\varepsilon> 0$, $\lambda\in\mathbb R$ is known as the Lagrange multiplier, $\Delta_p\cdot =\text{div} (|\nabla \cdot|^{p-2}\nabla \cdot)$ denotes the usual $p$-Laplacian operator with $2\leq p < N$ and $V \in \mathcal{C}^0(\mathbb R^N)$ is the potential which satisfies some suitable assumptions. We prove that the number of positive solutions depends on the profile of $V$ and each solution concentrates around its corresponding global minimum point of $V$ in the semiclassical limit when $\varepsilon\to0^+$ using variational method. Moreover, we also get the existence of normalized solutions for some logarithmic $p$-Laplacian equations involving mass-supercritical nonlinearities., Comment: 35 pages

We study the fractional Schr\"{o}dinger equations coupled with a neutral scalar field $$ (-\Delta)^s u+V(x)u=K(x)\phi u +g(x)|u|^{q-2}u, \quad x\in \mathbb{R}^3,\qquad (I-\Delta)^t \phi=K(x)u^2, \quad x\in \mathbb{R}^3, $$ where $(-\Delta)^s$ and $(I-\Delta)^t$ denote the fractional Laplacian and Bessel operators with $\frac{3}{4}

We study a class of planar Schr\"{o}dinger-Poisson systems $$ -\Delta u+\lambda V(x)u+\phi u=f(u) , \quad x\in{\mathbb R}^2,\qquad \Delta \phi=u^2, \quad x\in{\mathbb R}^2, $$ where $\lambda>0$ is a parameter, $V\in C({\mathbb R}^2,{\mathbb R}^+)$ has a potential well $\Omega \triangleq\text{int}\, V^{-1}(0)$ and the nonlinearity $f$ fulfills the supercritical exponential growth at infinity in the Trudinger-Moser sense. By exploiting the mountain-pass theorem and elliptic regular theory, we establish the existence and concentrating behavior of ground state solutions for sufficiently large $\lambda$., Comment: 33 pages

We are concerned with the existence of normalized solutions for a class of generalized Chern-Simons-Schr\"{o}dinger type problems with supercritical exponential growth $$ -\Delta u +\lambda u+A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u),\quad \partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2,\quad \partial_1A_1+\partial_2A_2=0,\quad \partial_1A_0=A_2|u|^2,\quad \partial_2A_0=-A_1|u|^2,\quad \int_{\mathbb{R}^2}|u|^2dx=a^2, $$ where $a\neq0$, $\lambda\in \mathbb{R}$ is known as the Lagrange multiplier and $f\in C^1(\mathbb{R})$ denotes the nonlinearity that fulfills the supercritical exponential growth in the Trudinger-Moser sense at infinity. Under suitable assumptions, combining the constrained minimization approach together with the homotopy stable family and elliptic regularity theory, we obtain that the problem has at least a ground state solution., Comment: 39 pages

We study a class of gauged nonlinear Schr\"{o}dinger equations in the plane. We obtain existence of two nontrivial solutions via the Mountain-Pass theorem and Ekeland's variational principle. Moreover, we prove existence of infinitely many solutions for odd nonlinearities., Comment: 22 pages, minor revision, accepted in DCDS-A

The present study is concerned with the following fractional Schr\"{o}dinger-Poisson system with steep potential well: $$ \left\{% \begin{array}{ll} (-\Delta)^s u+ \la V(x)u+K(x)\phi u= f(u), & x\in\R^3, (-\Delta)^t \phi=K(x)u^2, & x\in\R^3, \end{array}% \right. $$ where $s,t\in(0,1)$ with $4s+2t>3$, and $\la>0$ is a parameter. Under certain assumptions on $V(x)$, $K(x)$ and $f(u)$ behaving like $|u|^{q-2}u$ with $2

The present study is concerned with the following Schr\"{o}dinger-Poisson system involving critical nonlocal term $$ \left\{ \begin{array}{ll} -\Delta u+u-K(x)\phi |u|^3u=\lambda f(x)|u|^{q-2}u, & x\in\mathbb{R}^3, -\Delta \phi=K(x)|u|^5, & x\in\mathbb{R}^3,\\ \end{array} \right. $$ where $10$ is a parameter. Under suitable assumptions on $K(x)$ and $f(x)$, there exists $\lambda_0=\lambda_0(q,S,f,K)>0$ such that for any $\lambda\in(0,\lambda_0)$, the above Schr\"{o}dinger-Poisson system possesses at least two positive solutions by standard variational method, where a positive least energy solution will also be obtained., Comment: arXiv admin note: substantial text overlap with arXiv:1702.03785

The present study is concerned with the following Schr\"{o}dinger-Poisson system involving critical nonlocal term with general nonlinearity: $$ \left\{ \begin{array}{ll} -\Delta u+V(x)u- \phi |u|^3u= f(u), & x\in\mathbb{R}^3, -\Delta \phi= |u|^5, & x\in\mathbb{R}^3,\\ \end{array} \right. $$ Under certain assumptions on non-constant $V(x)$, the existence of a positive least energy solution is obtained by using some new analytical skills and Poho\v{z}aev type manifold. In particular, the Ambrosetti-Rabinowitz type condition or monotonicity assumption on the nonlinearity is not necessary., Comment: arXiv admin note: text overlap with arXiv:1702.03792

In this paper, we study the following nonlinear Kirchhoff problem involving critical growth: $$ \left\{% \begin{array}{ll} -(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u=|u|^4u+\lambda|u|^{q-2}u, u=0\ \ \text{on}\ \ \partial\Omega, \end{array}% \right. $$ where $10$ are parameters and $\Omega$ is a bounded domain in $\R^3$. We prove that there exists $\lambda_1=\lambda_1(q,\Omega)>0$ such that for any $\lambda\in(0,\lambda_1)$ and $a,\ b>0$, the above Kirchhoff problem possesses at least two positive solutions and one of them is a positive ground state solution. We also establish the convergence property of the ground state solution as the parameter $b\searrow 0$. More generally, we obtain the same results about the following Kirchhoff problem: $$ \left\{% \begin{array}{ll} -(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx)\Delta u+u=Q(x)|u|^4u+{\lambda}f(x)|u|^{q-2}u, u\in H^1(\mathbb{R}^3), \end{array}% \right. $$ for any $a,\ b>0$ and $\lambda\in \big(0,\lambda_0(q,Q,f)\big)$ under certain conditions of $f(x)$ and $Q(x)$. Finally, we investigate the depending relationship between $\lambda_0$ and $b$ to show that for any (large) $\lambda>0$, there exists a $b_0(\lambda)>0$ such that the above results hold when $b>b_0(\lambda)$ and $a>0$., Comment: 25 Pages