Ground state solution for the Schrödinger equation with Hardy potential and critical Sobolev exponent

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ < μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

Ground State Solution for an Autonomous Nonlinear Schrödinger System

In this paper, we study the following autonomous nonlinear Schrödinger system (discussed in the paper), where λ , μ , and ν are positive parameters; 2 ∗ = 2 N / N − 2 is the critical Sobolev exponent; and f satisfies general subcritical growth conditions. With the help of the Pohožaev manifold, a ground state solution is obtained.

A singular perturbed problem with critical Sobolev exponent

This paper deals with the following nonlinear elliptic problem \begin{equation}\label{eq0.1} -\varepsilon^2\Delta u+\omega V(x)u=u^{p}+u^{2^{*}-1},\quad u> 0\quad\text{in}\ \R^N, \end{equation} where $\omega\in\R^{+}$, $N\geq 3$, $p\in (1,2^{*}-1)$ with $2^{*}={2N}/({N-2})$, $\varepsilon> 0$ is a small parameter and $V(x)$ is a given function. Under suitable assumptions, we prove that problem (\ref{eq0.1}) has multi-peak solutions by the Lyapunov-Schmidt reduction method for sufficiently small $\varepsilon$, which concentrate at local minimum points of potential function $V(x)$. Moreover, we show the local uniqueness of positive multi-peak solutions by using the local Pohozaev identity.

Compact Sobolev-Slobodeckij embeddings and positive solutions to fractional Laplacian equations

In this work, we study the existence of a positive solution to an elliptic equation involving the fractional Laplacian (−Δ) s in ℝ n , for n ≥ 2, such as (0.1) ( − Δ ) s u + E ( x ) u + V ( x ) u q − 1 = K ( x ) f ( u ) + u 2 s ⋆ − 1 . $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=K(x) f(u)+u^{2_{s}^{\star}-1}.$$ Here, s ∈ (0, 1), q ∈ 2 , 2 s ⋆ $q \in\left[2,2_{s}^{\star}\right)$ with 2 s ⋆ := 2 n n − 2 s $2_{s}^{\star}:=\frac{2 n}{n-2 s}$ being the fractional critical Sobolev exponent, E(x), K(x), V(x) > 0 : ℝ n → ℝ are measurable functions which satisfy joint “vanishing at infinity” conditions in a measure-theoretic sense, and f (u) is a continuous function on ℝ of quasi-critical, super-q-linear growth with f (u) ≥ 0 if u ≥ 0. Besides, we study the existence of multiple positive solutions to an elliptic equation in ℝ n such as (0.2) ( − Δ ) s u + E ( x ) u + V ( x ) u q − 1 = λ K ( x ) u r − 1 , $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=\lambda K(x) u^{r-1},$$ where 2 < r < q < ∞(both possibly (super-)critical), E(x), K(x), V(x) > 0 : ℝ n → ℝ are measurable functions satisfying joint integrability conditions, and λ > 0 is a parameter. To study (0.1)-(0.2), we first describe a family of general fractional Sobolev-Slobodeckij spaces Ms ;q,p (ℝ n ) as well as their associated compact embedding results.

Large Energy Bubble Solutions for Schrödinger Equation with Supercritical Growth

We consider the following nonlinear Schrödinger equation involving supercritical growth: { - Δ ⁢ u + V ⁢ ( y ) ⁢ u = Q ⁢ ( y ) ⁢ u 2 * - 1 + ε in ⁢ ℝ N , u > 0 , u ∈ H 1 ⁢ ( ℝ N ) , \left\{\begin{aligned} &\displaystyle{-}\Delta u+V(y)u=Q(y)u^{2^{*}-1+% \varepsilon}&&\displaystyle\phantom{}\text{in }\mathbb{R}^{N},\\ &\displaystyle u>0,\quad u\in H^{1}(\mathbb{R}^{N}),\end{aligned}\right.{} where 2 * = 2 ⁢ N N - 2 {2^{*}=\frac{2N}{N-2}} is the critical Sobolev exponent, N ≥ 5 {N\geq 5} , and V ⁢ ( y ) {V(y)} and Q ⁢ ( y ) {Q(y)} are bounded nonnegative functions in ℝ N {\mathbb{R}^{N}} . By using the finite reduction argument and local Pohozaev-type identities, under some suitable assumptions on the functions V and Q, we prove that for ε > 0 {\varepsilon>0} is small enough, problem ( * ) {(*)} has large number of bubble solutions whose functional energy is in the order ε - N - 4 ( N - 2 ) 2 . {\varepsilon^{-\frac{N-4}{(N-2)^{2}}}.}

The Brezis-Nirenberg problem for fractional systems with Hardy potentials

In this work we study the existence of positive solutions to the following fractional elliptic systems with Hardy-type singular potentials, and coupled by critical homogeneous nonlinearities \begin{equation*} \begin{cases} (-\Delta)^{s}u-\mu_{1}\frac{u}{|x|^{2s}}=|u|^{2^{\ast}_{s}-2}u+\frac{\eta\alpha}{2^{\ast}_{s}}|u|^{\alpha-2} |v|^{\beta}u+\frac{1}{2}Q_{u}(u,v) \ \ in \ \Omega, \\[2mm] (-\Delta)^{s}v-\mu_{2}\frac{v}{|x|^{2s}}=|v|^{2^{\ast}_{s}-2}v+\frac{\eta\beta}{2^{\ast}_{s}}|u|^{\alpha} |v|^{\beta-2}v+\frac{1}{2}Q_{v}(u,v) \ \ in \ \Omega, \\[2mm] \ \ u, \ v>0 \ \ \ \ \ in \ \ \Omega, \\[2mm] \ u=v=0 \ \ \ \ in \ \ \mathbb{R}^{N}\backslash\Omega, \end{cases} \end{equation*} where $(-\Delta)^{s}$ denotes the fractional Laplace operator, $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain such that $0\in\Omega$, $\mu_{1}, \mu_{2}\in [0,\Lambda_{N,s})$, $\Lambda_{N,s}=2^{2s}\frac{\Gamma^{2}(\frac{N+2s}{4})}{\Gamma^{2}(\frac{N-2s}{4})}$ is the best constant of the fractional Hardy inequality and $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent. In order to prove the main result, we establish some refined estimates on the extremal functions of the fractional Hardy-Sobolev type inequalities and we get the existence of positive solutions to the systems through variational methods.

Energy asymptotics in the three-dimensional Brezis–Nirenberg problem

For a bounded open set $$\Omega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 we consider the minimization problem $$\begin{aligned} S(a+\epsilon V) = \inf _{0\not \equiv u\in H^1_0(\Omega )} \frac{\int _\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int _\Omega u^6\,dx)^{1/3}} \end{aligned}$$ S ( a + ϵ V ) = inf 0 ≢ u ∈ H 0 1 ( Ω ) ∫ Ω ( | ∇ u | 2 + ( a + ϵ V ) | u | 2 ) d x ( ∫ Ω u 6 d x ) 1 / 3 involving the critical Sobolev exponent. The function a is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on a and V we compute the asymptotics of $$S(a+\epsilon V)-S$$ S ( a + ϵ V ) - S as $$\epsilon \rightarrow 0+$$ ϵ → 0 + , where S is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to a and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have $$S(a+\epsilon V)<S$$ S ( a + ϵ V ) < S for all sufficiently small $$\epsilon >0$$ ϵ > 0 .