In this paper, we study the following gauged nonlinear Schrödinger equation \begin{align*} -\Delta u + \lambda V(|x|)u & + \mu\Big(\int_{|x|}^{ + \infty} \frac{h(s)}{s} u^2(s)\text{d}s + \frac{h^2(|x|)}{|x|^2}\Big)u \\ & = a(|x|)|u|^{q-2}u + b(|x|)|u|^{p-2}u~~~~~~~ \text{in}~~\mathbb R^2, \end{align*} where $\lambda > 0$, $\mu > 0$, $h(s) = \frac{1}{2}\int_0^s{u^2(l)l}\text{d}l$ and $a\in L^\infty(\mathbb R^2)$, $b\in L^{\frac{q}{q-p}}(\mathbb R^2, \mathbb R^ + )$, $1 < p < 2$ $ < q < 6$ and the potential well $V \in \mathcal{C}(\mathbb R^2,\mathbb R)$, $V^{-1}(0)$ has nonempty interior. First, the existence of the positive energy solutions is obtained by using the truncation technique. Secondly, we also explore the asymptotic behavior of the positive energy solutions as $\lambda\rightarrow + \infty$ and $\mu\rightarrow 0$. Finally, we give the existence of a negative energy solution via Ekeland variational principle. Our result extend and supplement some recent results in related literatures.

We consider the following singularly perturbed Schrödinger equation involving the $ \frac Ns $-fractional Laplacian operator,$$ \varepsilon^{N}(-\Delta)_{ \frac Ns } ^{s}u+V(x)|u|^{\frac{N}{s}-2}u=f(u) \quad \text{in } \mathbb R^{N},$$where $\varepsilon$ is a positive parameter, $s\in (0,1)$, the potential $V$ is positive and away from zero, and $f$ is a Trudinger-Moser type nonlinearity. By using penalization methods andLusternik-Schnirelmann's theory, we examine existence, multiplicity and concentration of non-trivial non-negative solutions for small values of $\varepsilon$.

This paper is devoted to the study of a quasilinear elliptic inclusion problem driven by a double phase differential operator with variable exponents, an obstacle effect and a multivalued reaction term with gradient dependence. By using an existence result for mixed variational inequalities with multivalued pseudomonotone operators and the theory of nonsmooth analysis, we examine the nonemptiness, boundedness and closedness of the solution set to the problem under consideration. In the second part of the paper, we present some convergence analysis for approximated problems. To be more precise, when the obstacle function is approximated by a suitable sequence, applying a generalized penalty technique, we introduce a family of perturbed problems without constraints associated to our problem and prove that the solution set of the original problem can be approached by the solution sets of the perturbed problems in the sense of Kuratowski.

A generalized Cahn--Hilliard model in a bounded interval of the real line with no-flux boundary conditions is considered. The label ``generalized'' refers to the fact that we consider a concentration dependent mobility, the $p$-Laplace operator with $p > 1$ and a double well potential of the form $F(u)=\frac{1}{2\theta}|1-u^2|^\theta$, with $\theta > 1$; these terms replace, respectively, the constant mobility, the linear Laplace operator and the $C^2$ potential satisfying $F''(\pm1) > 0$, which are typical of the standard Cahn--Hilliard model. After investigating the associated stationary problem and highlighting the differences with the standard results, we focus the attention on the long time dynamics of solutions when $\theta\geq p > 1$. In the critical case $\theta= p > 1$, we prove exponentially slow motion of profiles with a transition layer structure, thus extending the well know results of the standard model, where $\theta=p=2$; conversely, in the supercritical case $\theta > p > 1$, we prove algebraic slow motion of layered profiles.