A Unified Approach to Singularly Perturbed Quasilinear Schrödinger Equations

AbstractIn this paper we present a unified approach to investigate existence and concentration of positive solutions for the following class of quasilinear Schrödinger equations,$$-\varepsilon^2\Delta u+V(x)u\mp\varepsilon^{2+\gamma}u\Delta u^2=h(u),\ \ x\in \mathbb{R}^N,$$-ε2Δu+V(x)u∓ε2+γuΔu2=h(u),x∈RN,where$$N\geqslant3, \varepsilon > 0, V(x)$$N⩾3,ε>0,V(x)is a positive external potential,his a real function with subcritical or critical growth. The problem is quite sensitive to the sign changing of the quasilinear term as well as to the presence of the parameter$$\gamma>0$$γ>0. Nevertheless, by means of perturbation type techniques, we establish the existence of a positive solution$$u_{\varepsilon,\gamma}$$uε,γconcentrating, as$$\varepsilon\rightarrow 0$$ε→0, around minima points of the potential

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

Concentrating solutions for a fractional Kirchhoff equation with critical growth

In this paper we consider the following class of fractional Kirchhoff equations with critical growth: \begin{equation*} \left\{ \begin{array}{ll} \left(\varepsilon^{2s}a+\varepsilon^{4s-3}b\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s}u+V(x)u=f(u)+|u|^{2^{*}_{s}-2}u \quad &\mbox{ in } \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \quad u>0 &\mbox{ in } \mathbb{R}^{3}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b>0$ are constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-\Delta)^{s}$ is the fractional Laplacian operator, $V$ is a positive continuous potential and $f$ is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we prove the existence of a family of positive solutions $u_{\varepsilon}$ which concentrates around a local minimum of $V$ as $\varepsilon\rightarrow 0$.Comment: arXiv admin note: text overlap with arXiv:1810.0456

Existence of positive solutions for a class of p-Laplacian type generalized quasilinear Schrödinger equations with critical growth and potential vanishing at infinity

In this paper, we study the existence of positive solutions for the following generalized quasilinear Schrödinger equation − div(g p (u)|∇u| p−2∇u) + g p−1 (u)g (u)|∇u| p + V(x)|u| p−2u = K(x)f(u) + Q(x)g(u)|G(u)| p ∗−2G(u), x ∈ R N, where N ≥ 3, 1 < p ≤ N, p Np N−p , g ∈ C1 (R, R+), V(x) and K(x) are positive continuous functions and G(u) = R u 0 g(t)dt. By using a change of variable, we obtain the existence of positive solutions for this problem by using the Mountain Pass Theorem. Our results generalize some existing results

Blow-up phenomena and asymptotic profiles passing from h1-critical to super-critical quasilinear Schr\uf6dinger equations

We study the asymptotic profile, as h \u2192 0, of positive solutions to where \u3b3 650 is a parameter with relevant physical interpretations, V and K are given potentials and the dimension N is greater than or equal to 5, as we look for finite L2-energy solutions. We investigate the concentrating behavior of solutions when \u3b3&gt;0 and, differently from the case \u3b3=0 where the leading potential is V, the concentration is here localized by the source potential K. Moreover, surprisingly for \u3b3&gt;0 we find a different concentration behavior of solutions in the case p=2NN-2 and when 2NN-24 NN-2. This phenomenon does not occur when \u3b3=0

On the logarithmic Schrodinger equation

In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.Comment: 10 page

Blow up dynamics for smooth equivariant solutions to the energy critical Schr\"odinger map

We consider the energy critical Schr\"odinger map problem with the 2-sphere target for equivariant initial data of homotopy index $k=1$. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blow up solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy