Positive solutions for a class of quasilinear problems with critical growth in ℝN

2015 ◽  
Vol 145 (2) ◽  
pp. 411-444
Author(s):  
Jun Wang ◽  
Tianqing An ◽  
Fubao Zhang
Keyword(s):  
Positive Solutions ◽  
Ground State ◽  
Sufficient Conditions ◽  
Ground State Solution ◽  
Laplacian Operator ◽  
Minimax Theorems ◽  
Global Minima ◽  
Concentration Phenomena ◽  
Subcritical Nonlinearity ◽  
Quasilinear Problems

In this paper, we study the existence, multiplicity and concentration of positive solutions for a class of quasilinear problemswhere —Δp is the p-Laplacian operator for is a small parameter, f(u) is a superlinear and subcritical nonlinearity that is continuous in u. Using a variational method, we first prove that for sufficiently small ε > 0 the system has a positive ground state solution uε with some concentration phenomena as ε → 0. Then, by the minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set of the global minima of the potentials. Finally, we obtain some sufficient conditions for the non-existence of ground state solutions.

Existence of semiclassical ground-state solutions for semilinear elliptic systems

2012 ◽  
Vol 142 (4) ◽  
pp. 867-895 ◽  
Author(s):  
Jun Wang ◽  
Junxiang Xu ◽  
Fubao Zhang
Keyword(s):  
Ground State ◽  
Sufficient Conditions ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Limit Problem ◽  
State Solution ◽  
Ground State Solutions ◽  
Semilinear Elliptic ◽  
Small Positive Parameter ◽  
Image Position

This paper is concerned with the following semilinear elliptic equations of the formwhere ε is a small positive parameter, and where f and g denote superlinear and subcritical nonlinearity. Suppose that b(x) has at least one maximum. We prove that the system has a ground-state solution (ψε, φε) for all sufficiently small ε > 0. Moreover, we show that (ψε, φε) converges to the ground-state solution of the associated limit problem and concentrates to a maxima point of b(x) in certain sense, as ε → 0. Furthermore, we obtain sufficient conditions for nonexistence of ground-state solutions.

Existence and Concentration of Positive Ground State Solutions for Schrödinger-Poisson Systems

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Jun Wang ◽  
Lixin Tian ◽  
Junxiang Xu ◽  
Fubao Zhang
Keyword(s):  
Small Parameter ◽  
Ground State ◽  
Ground State Solution ◽  
Real Parameter ◽  
State Solution ◽  
Poisson System ◽  
Ground State Solutions ◽  
Subcritical Nonlinearity

AbstractIn this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger-Poisson systemwhere ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that b(x) has a maximum. We prove that the system has a positive ground state solution

New Super-quadratic Conditions on Ground State Solutions for Superlinear Schrödinger Equation

2014 ◽  
Vol 14 (2) ◽  
Author(s):  
X. H. Tang
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Ground State Solution ◽  
Nontrivial Solutions ◽  
Subcritical Nonlinearity ◽  
A New Technique ◽  
Cerami Sequences ◽  
Semilinear Schrödinger Equation ◽  
Funct Anal

AbstractConsider the semilinear Schrödinger equationwhere f is a superlinear, subcritical nonlinearity. We mainly study the case where both V and f are periodic in x and 0 belongs to a spectral gap of −Δ + V. Based on the work of Szulkin and Weth [J Funct Anal 257: 3802-3822, 2009], we develop a new technique to show the boundedness of Cerami sequences and derive a new super-quadratic condition that there exists a θfor the existence a “ground state solution” which minimizes the corresponding energy among all nontrivial solutions. Our result unifies and improves some known ones and the recent ones of Szulkin and Weth [J Funct Anal 257: 3802-3822, 2009] and Liu [Calc. Var. 45: 1-9, 2012].

scholarly journals On the resonant Lane–Emden problem for the p-Laplacian

2014 ◽  
Vol 16 (04) ◽  
pp. 1350033 ◽  
Author(s):  
Grey Ercole
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Upper Bound ◽  
Ground State ◽  
Laplacian Operator ◽  
First Eigenvalue ◽  
Explicit Estimate ◽  
Best Constant ◽  
The First Eigenvalue ◽  
First Eigenfunction

We study the positive solutions of the Lane–Emden problem -Δpu = λp|u|q-2u in Ω, u = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded and smooth domain, N ≥ 2, λp is the first eigenvalue of the p-Laplacian operator Δp, p > 1, and q is close to p. We prove that any family of positive solutions of this problem converges in [Formula: see text] to the function θpep when q → p, where ep is the positive and L∞-normalized first eigenfunction of the p-Laplacian and [Formula: see text]. A consequence of this result is that the best constant of the immersion [Formula: see text] is differentiable at q = p. Previous results on the asymptotic behavior (as q → p) of the positive solutions of the nonresonant Lane–Emden problem (i.e. with λp replaced by a positive λ ≠ λp) are also generalized to the space [Formula: see text] and to arbitrary families of these solutions. Moreover, if uλ,q denotes a solution of the nonresonant problem for an arbitrarily fixed λ > 0, we show how to obtain the first eigenpair of the p-Laplacian as the limit in [Formula: see text], when q → p, of a suitable scaling of the pair (λ, uλ,q). For computational purposes the advantage of this approach is that λ does not need to be close to λp. Finally, an explicit estimate involving L∞- and L1-norms of uλ,q is also derived using set level techniques. It is applied to any ground state family {vq} in order to produce an explicit upper bound for ‖vq‖∞ which is valid for q ∈ [1, p + ϵ] where [Formula: see text].

scholarly journals Positive Solutions to a Fractional-Order Two-Point Boundary Value Problem with p-Laplacian Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Xiangshan Kong ◽  
Haitao Li
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Operator Equation ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Laplacian Operator ◽  
Point Boundary ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions

This paper systematically investigates positive solutions to a kind of two-point boundary value problem (BVP) for nonlinear fractional differential equations with p-Laplacian operator and presents a number of new results. First, the considered BVP is converted to an operator equation by using the property of the Caputo derivative. Second, based on the operator equation and some fixed point theorems, several sufficient conditions are presented for the nonexistence, the uniqueness, and the multiplicity of positive solutions. Finally, several illustrative examples are given to support the obtained new results. The study of illustrative examples shows that the obtained results are effective.

Ground State and Multiple Solutions for Kirchhoff Type Equations With Critical Exponent

2018 ◽  
Vol 61 (2) ◽  
pp. 353-369 ◽  
Author(s):  
Dongdong Qin ◽  
Yubo He ◽  
Xianhua Tang
Keyword(s):  
Variational Methods ◽  
Critical Exponent ◽  
Positive Solutions ◽  
Ground State ◽  
Multiple Solutions ◽  
Type Equation ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Kirchhoff Type ◽  
The Nehari Manifold

AbstractIn this paper, we consider the following critical Kirchhoff type equation:By using variational methods that are constrained to the Nehari manifold, we prove that the above equation has a ground state solution for the case when 3 < q < 5. The relation between the number of maxima of Q and the number of positive solutions for the problem is also investigated.

scholarly journals Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains

2007 ◽  
Vol 2007 ◽  
pp. 1-21
Author(s):  
Tsung-Fang Wu
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Ground State ◽  
Unbounded Domains ◽  
Ground State Solution ◽  
Multiple Positive Solutions ◽  
Dirichlet Problems ◽  
Existence And Multiplicity ◽  
Sobolev Constant ◽  
Multiplicity Of Positive Solutions

We consider the elliptic problem−Δu+u=b(x)|u|p−2u+h(x)inΩ,u∈H01(Ω), where2<p<(2N/(N−2)) (N≥3), 2<p<∞ (N=2), Ωis a smooth unbounded domain inℝN, b(x)∈C(Ω), andh(x)∈H−1(Ω). We use the shape of domainΩto prove that the above elliptic problem has a ground-state solution if the coefficientb(x)satisfiesb(x)→b∞>0as|x|→∞andb(x)≥cfor some suitable constantsc∈(0,b∞), andh(x)≡0. Furthermore, we prove that the above elliptic problem has multiple positive solutions if the coefficientb(x)also satisfies the above conditions,h(x)≥0and0<‖h‖H−1<(p−2)(1/(p−1))(p−1)/(p−2)[bsupSp(Ω)]1/(2−p), whereS(Ω)is the best Sobolev constant of subcritical operator inH01(Ω)andbsup=supx∈Ωb(x).

scholarly journals Existence and concentration of positive solutions for a p-fractional Choquard equation

2021 ◽  
Vol 6 (11) ◽  
pp. 12929-12951
Author(s):  
Xudong Shang ◽  
Keyword(s):  
Small Parameter ◽  
Positive Solutions ◽  
Ground State ◽  
Global Minimum ◽  
Ground State Solution ◽  
Global Maximum ◽  
Choquard Equation ◽  
Number Of Solutions ◽  
Concentration Behavior ◽  
Fractional Choquard Equation

<abstract><p>In this work, we study the existence, multiplicity and concentration behavior of positive solutions for the following problem involving the fractional $ p $-Laplacian</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \varepsilon^{ps}(-\Delta )^{s}_{p}u + V(x)|u|^{p-2}u = \varepsilon^{\mu-N}(\frac{1}{|x|^{\mu}}\ast K|u|^{q})K(x)|u|^{q-2}u \hskip0.2cm\text{in}\hskip0.1cm \mathbb{R}^{N}, \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> <p>where $ 0 &lt; s &lt; 1 &lt; p &lt; \infty $, $ N &gt; ps $, $ 0 &lt; \mu &lt; ps $, $ p &lt; q &lt; \frac{p^{*}_{s}}{2}(2-\frac{\mu}{N}) $, $ (-\Delta)^{s}_{p} $ is the fractional $ p $-Laplacian and $ \varepsilon &gt; 0 $ is a small parameter. Under certain conditions on $ V $ and $ K $, we prove the existence of a positive ground state solution and express the location of concentration in terms of the potential functions $ V $ and $ K $. In particular, we relate the number of solutions with the topology of the set where $ V $ attains its global minimum and $ K $ attains its global maximum.</p></abstract>

scholarly journals Positive Solutions for Three-Point Boundary Value Problem of Fractional Differential Equation with -Laplacian Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Shang-lin Yao ◽  
Guo-hui Wang ◽  
Zhi-ping Li ◽  
Li-jun Yu
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Laplacian Operator ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Fractional Differential

We investigate the existence of multiple positive solutions for three-point boundary value problem of fractional differential equation with -Laplacian operator , where are the standard Riemann-Liouville derivatives with , and the constant is a positive number satisfying ; -Laplacian operator is defined as . By applying monotone iterative technique, some sufficient conditions for the existence of multiple positive solutions are established; moreover iterative schemes for approximating these solutions are also obtained, which start off a known simple linear function. In the end, an example is worked out to illustrate our main results.

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