Existence and concentration of solutions for a class of biharmonic equations

Some superlinear fourth order elliptic equations are considered. Ground states are proved to exist and to concentrate at a point in the limit. The proof relies on variational methods, where the existence and concentration of nontrivial solutions are related to a suitable truncated equation.Comment: 18 page

Existence and concentration of ground state solutions for a critical nonlocal Schr\"odinger equation in R2\R^2

We study the following singularly perturbed nonlocal Schr\"{o}dinger equation -\vr^2\Delta u +V(x)u =\vr^{\mu-2}\Big[\frac{1}{|x|^{\mu}}\ast F(u)\Big]f(u) \quad \mbox{in} \quad \R^2, where $V(x)$ is a continuous real function on $\R^2$, $F(s)$ is the primitive of $f(s)$, $0<\mu<2$ and \vr is a positive parameter. Assuming that the nonlinearity $f(s)$ has critical exponential growth in the sense of Trudinger-Moser, we establish the existence and concentration of solutions by variational methods.Comment: 3

Asymptotic behavior of multiple solutions for quasilinear Schrödinger equations

This paper establishes the multiplicity of solutions for a class of quasilinear Schrödinger elliptic equations: −∆u + V(x)u − 2 ∆(u 2 )u = f(x, u), x ∈ R 3 where V(x) : R3 → R is a given potential and γ > 0. Furthermore, by the variational argument and L ∞-estimates, we are able to obtain the precise asymptotic behavior of these solutions as γ → 0

Multiplicity of solutions for Dirichlet boundary conditions of second-order quasilinear equations with impulsive effects

This paper deals with the multiplicity of solutions for Dirichlet boundary conditions of second-order quasilinear equations with impulsive effects. By using critical point theory, a new result is obtained. An example is given to illustrate the main result

Existence, nonexistence and multiplicity of positive solutions for singular quasilinear problems

In the present paper we deal with a quasilinear problem involving a singular term and a parametric superlinear perturbation. We are interested in the existence, nonexistence and multiplicity of positive solutions as the parameter λ > 0 varies. In our first result, the superlinear perturbation has an arbitrary growth and we obtain the existence of a solution for the problem by using the sub-supersolution method. For the second result, the superlinear perturbation has subcritical growth and we employ the Mountain Pass Theorem to show the existence of a second solution

Existence of solutions to a class of quasilinear Schrödinger systems involving the fractional pp-Laplacian

The purpose of this paper is to investigate the existence of solutions to the following quasilinear Schr\"{o}dinger type system driven by the fractional $p$-Laplacian \begin{align*} (-\Delta)^{s}_pu+a(x)|u|^{p-2}u&=H_u(x,u,v)\quad \mbox{in } \mathbb{R}^N,\\ (-\Delta)^s_qv+b(x)|v|^{q-2}v&=H_v(x,u,v)\quad \mbox{in } \mathbb{R}^N, \end{align*} where $1<q\leq p$, $sp<N$, $(-\Delta )_m^s$ is the fractional $m$-Laplacian, the coefficients $a, b$ are two continuous and positive functions, and $H_u,H_v$ denote the partial derivatives of $H$ with respect to the second variable and the third variable. By using the mountain pass theorem, we obtain the existence of nontrivial and nonnegative solutions for the above system. The main feature of this paper is that the nonlinearities do not necessarily satisfy the Ambrosetti-Rabinowitz condition