Steep Potential Well Research Articles

Solutions to the coupled Schrödinger systems with steep potential well and critical exponent

Solutions to the coupled Schrödinger systems with steep potential well and critical exponent

Multiplicity and concentration of nontrivial solutions for Kirchhoff–Schrödinger–Poisson system with steep potential well

Multiplicity and concentration of nontrivial solutions for Kirchhoff–Schrödinger–Poisson system with steep potential well

Concentration of ground state solutions for critical generalized quasilinear equation with steep potential well

In this work, we are looking at the generalized quasilinear Schrödinger equations with critical growth and a steep potential well: where and is an even function and satisfies some suitable conditions (such as ). By using a change of variable and variational methods, for big enough , the existence of a ground state solution is obtained and its concentration behavior is also discussed.

Ground states for the superlinear Schrödinger equation involving parametric potential

We consider the existence of ground states for the following Schrödinger equation −Δu+Vλ(x)u=f(x,u),x∈RN, where f satisfies general superlinear (but subcritical) assumptions, Vλ(x)=λV(x)≥0 and the set {V<V0}≔{x∈RN:V(x)<V0} is nonempty and has finite measure for some V0>0. In particular, if V−1(0) has nonempty interior, the parametric potential Vλ(x) is the classical steep potential well. We point out that this point is the key to verifying that the (PS)c sequence satisfies the local (PS)c condition in existing works. In present paper, we removed this condition by establishing a more fine local (PS)c condition. This work partially solves an open problem proposed by Bartsch and Wang (1995).

Multi-bump solutions of Schrödinger–Bopp–Podolsky system with steep potential well

In this paper, we study the existence of multi-bump solutions for the following Schrödinger–Bopp–Podolsky system with steep potential well: { − Δ u + ( λ V ( x ) + V 0 ( x ) ) u + K ( x ) ϕ u = | u | p − 2 u , x ∈ R 3 , − Δ ϕ + a 2 Δ 2 ϕ = K ( x ) u 2 , x ∈ R 3 , where p ∈ ( 4 , 6 ) , a &gt; 0 and λ is a parameter. We require that V ( x ) ≥ 0 and has a bounded potential well Ω = V − 1 ( 0 ) . Combining this with other suitable assumptions on Ω , V 0 and K , when λ is large enough, we obtain the existence of multi-bump-type solutions u λ by using variational methods.

Existence and asymptotic behavior for ground state sign-changing solutions of fractional Schrödinger-Poisson system with steep potential well

&lt;abstract&gt;&lt;p&gt;In this paper, we investigate the existence of ground state sign-changing solutions for the following fractional Schrödinger-Poisson system&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{equation} \begin{cases} (-\Delta)^s u+V_{\lambda} (x)u+\mu\phi u = f(u), &amp;amp; \; \mathrm{in}\; \; \mathbb{R}^3, \\ (-\Delta)^t \phi = u^2, &amp;amp; \; \mathrm{in}\; \; \mathbb{R}^3, \end{cases} \nonumber \end{equation} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where $ \mu &amp;gt; 0, s\in(\frac{3}{4}, 1), t\in(0, 1) $ and $ V_{\lambda}(x) $ = $ \lambda V(x)+1 $ with $ \lambda &amp;gt; 0 $. Under suitable conditions on $ f $ and $ V $, by using the constraint variational method and quantitative deformation lemma, if $ \lambda &amp;gt; 0 $ is large enough, we prove that the above problem has one least energy sign-changing solution. Moreover, for any $ \mu &amp;gt; 0 $, the least energy of the sign-changing solution is strictly more than twice of the energy of the ground state solution. In addition, we discuss the asymptotic behavior of ground state sign-changing solutions as $ \lambda\rightarrow \infty $ and $ \mu\rightarrow0 $.&lt;/p&gt;&lt;/abstract&gt;

Least energy sign-changing solutions for fractional critical Kirchhoff–Schrödinger–Poisson with steep potential well

Least energy sign-changing solutions for fractional critical Kirchhoff–Schrödinger–Poisson with steep potential well

Existence and asymptotic behavior of solutions for the Schrödinger–Born–Infeld system with steep potential well

Existence and asymptotic behavior of solutions for the Schrödinger–Born–Infeld system with steep potential well

Positive solutions for fractional Kirchhoff–Schrödinger–Poisson system with steep potential well

In this paper, we deal with the following fractional Kirchhoff–Schrödinger–Poisson system: [Formula: see text] where [Formula: see text] and [Formula: see text] is a constant, [Formula: see text] are positive parameters, [Formula: see text] represents a potential well with the bottom [Formula: see text]. By applying the truncation technique and the parameter-dependent compactness lemma, we first prove the existence of positive solutions for b small, [Formula: see text] large and [Formula: see text] small in the case [Formula: see text]. Moreover, we investigate the decay rate of positive solutions as [Formula: see text] as well as their asymptotic behavior as [Formula: see text] and [Formula: see text], respectively.

Existence result for the critical Klein-Gordon-Maxwell system involving steep potential well

&lt;abstract&gt;&lt;p&gt;The Klein-Gordon-Maxwell system has received great attention in the community of mathematical physics. Under a special superlinear condition on the nonlinear term, the existence of solution for the critical Klein-Gordon-Maxwell system with a steep potential well has been solved. In this paper, under two general superlinear conditions, we obtain the existence of ground state solution for the critical Klein-Gordon-Maxwell system with a steep potential well. The general superlinear conditions bring challenge in proving the boundedness of Cerami sequence, which is a key step in the proof of the existence. To solve this, we construct a Pohožaev identity and adopt some analytical techniques. Our results extend the previous results in the literature.&lt;/p&gt;&lt;/abstract&gt;

Existence and asymptotic behavior of nontrivial solution for Klein–Gordon–Maxwell system with steep potential well

In this paper, we consider the following nonlinear Klein–Gordon–Maxwell system with a steep potential well { − Δ u + ( λ a ( x ) + 1 ) u − μ ( 2 ω + ϕ ) ϕ u = f ( x , u ) , in R 3 , Δ ϕ = μ ( ω + ϕ ) u 2 , in R 3 , where ω &gt; 0 is a constant, μ and λ are positive parameters, f ∈ C ( R 3 × R , R ) and the nonlinearity f satisfies the Ambrosetti–Rabinowitz condition. We use parameter-dependent compactness lemma to prove the existence of nontrivial solution for μ small and λ large enough, then explore the asymptotic behavior as μ → 0 and λ → ∞ . Moreover, we also use truncation technique to study the existence and asymptotic behavior of positive solution of Klein–Gordon–Maxwell system when f ( u ) := | u | q − 2 u where 2 &lt; q &lt; 4 .

Existence and concentration of solutions for a Kirchhoff-type problem with sublinear perturbation and steep potential well

&lt;abstract&gt;&lt;p&gt;In this paper, we consider the following nonlinear Kirchhoff-type problem with sublinear perturbation and steep potential well&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{eqnarray*} \left \{\begin{array}{ll} -\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\Big)\Delta u+\lambda V(x)u = f(x,u)+g(x)|u|^{q-2}u\ \ \mbox{in}\ \mathbb{R}^3,\\ \\ u\in H^1(\mathbb{R}^3), \\ \end{array} \right. \label{1} \end{eqnarray*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where $ a $ and $ b $ are positive constants, $ \lambda &amp;gt; 0 $ is a parameter, $ 1 &amp;lt; q &amp;lt; 2 $, the potential $ V\in C(\mathbb{R}^3, \mathbb{R}) $ and $ V^{-1}(0) $ has a nonempty interior. The functions $ f $ and $ g $ are assumed to obey a certain set of conditions. The existence of two nontrivial solutions are obtained by using variational methods. Furthermore, the concentration behavior of solutions as $ \lambda\rightarrow \infty $ is also explored.&lt;/p&gt;&lt;/abstract&gt;

Existence and concentration of ground state solutions for a class of subcritical, critical or supercritical problems with steep potential well

Existence and concentration of ground state solutions for a class of subcritical, critical or supercritical problems with steep potential well

Multiplicity of solutions for a class of upper critical Choquard equation with steep potential well

Multiplicity of solutions for a class of upper critical Choquard equation with steep potential well

Existence of ground state solutions for critical quasilinear Schrödinger equations with steep potential well

Abstract We study the existence of solutions for the quasilinear Schrödinger equation with the critical exponent and steep potential well. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals satisfy the geometric conditions of the Mountain Pass Theorem for suitable assumptions. The existence of a ground state solution is obtained, and its concentration behavior is also considered.

Existence and concentration of ground state solutions for an equation with steep potential well and exponential critical growth

In this paper we study the existence and concentration of ground state solution for the following class of elliptic equations{−div(a(|∇u|p)|∇u|p−2∇u)+(1+μV(x))b(|u|p)|u|p−2u=f(u)in RN,u∈W1,p(RN)∩W1,N(RN), where 1<p≤N, N≥2, μ>0 is a parameter, the functions a,b∈C1(R) and the nonlinearity f has an exponential critical growth. To show the existence and concentration results we obtain interesting estimates which do not depend on μ.

Positive solutions for the Schrödinger–Poisson system with steep potential well

In this paper, we consider the following Schrödinger–Poisson system [Formula: see text] where [Formula: see text] are real parameters and [Formula: see text]. Suppose that [Formula: see text] represents a potential well with the bottom [Formula: see text], the system has been widely studied in the case [Formula: see text]. In contrast, no existence result of solutions is available for the case [Formula: see text] due to the presence of the nonlocal term [Formula: see text]. With the aid of the truncation technique and the parameter-dependent compactness lemma, we first prove the existence of positive solutions for [Formula: see text] large and [Formula: see text] small in the case [Formula: see text]. Then we obtain the nonexistence of nontrivial solutions for [Formula: see text] large and [Formula: see text] large in the case [Formula: see text]. Finally, we explore the decay rate of the positive solutions as [Formula: see text] as well as their asymptotic behavior as [Formula: see text] and [Formula: see text].

On Chern-Simons-Schrödinger systems involving steep potential well and concave-convex nonlinearities

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.

Normalized multibump solutions to nonlinear Schrödinger equations with steep potential well

We are concerned with the existence of multibump solutions to the nonlinear Schrödinger equation −Δu+λa(x)u+μu=|u|2σuinRN with an L 2-constraint in the L 2-subcritical case σ ∈ (0, 2/N) and the L 2-supercritical case σ ∈ (2/N, 2*/N), where the usual critical Sobolev exponent is 2* = +∞ if N = 1, 2 and 2* = 2N/(N − 2) if N ⩾ 3. Here will arise as a Lagrange multiplier, and has a bottom int a −1(0) composed of ℓ 0 (ℓ 0 ⩾ 1) connected components , where int a −1(0) is the interior of the zero set of a. When ρ is fixed either large in the L 2-subcritical case or small in the L 2-supercritical case, we construct a ℓ-bump (1 ⩽ ℓ ⩽ ℓ 0) positive normalized solution which is localised at ℓ prescribed components for large λ. The asymptotic profile of the solution is also analysed through taking the limit as λ → +∞, and subsequently as ρ → +∞ in the L 2-subcritical case or ρ → 0+ in the L 2-supercritical case. In particular, we find ℓ-bump normalized solutions to the related Dirichlet problem 0\\quad \ ext{for}\\ i=1,\\dots ,\\ell .\\hfill \\end{aligned}\\right.\\end{equation*}?> −Δv+μv=|v|2σv,v∈H01(∪i=1ℓΩi),∑i=1ℓ∫Ωiv2=ρ,v|Ωi>0fori=1,…,ℓ.

Combined effects of concave and convex nonlinearities for the generalized Chern–Simons–Schrödinger systems with steep potential well and 1 &amp;lt; p &amp;lt; 2 &amp;lt; q &amp;lt; 6

In this paper, we study the Chern–Simons–Schrödinger system with a steep potential well and 1 &amp;lt; p &amp;lt; 2 &amp;lt; q &amp;lt; 6. First, by using the truncation technique, we prove that this system possesses a positive energy solution. Second, the concentration behavior of the positive energy solutions as λ → +∞ and κ → 0 are also considered. Finally, we obtain a negative energy solution via the Ekeland variational principle.