Number Of Positive Solutions Research Articles

High multiplicity of positive solutions in a superlinear problem of Moore–Nehari type

In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in Moore and Nehari (1959). Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number κ≥1 of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending on the value of a parameter λ and on κ.Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as λ↓−∞, in the regions where the weight vanishes. Our result leads us to conjecture the existence of 2κ+1−1 solutions for sufficiently negative λ. On the other hand, we support such a conjecture with the results of numerical simulations which also shed light on the structure of the global bifurcation diagrams in λ and the profiles of positive solutions.Finally, we give additional numerical results suggesting that high multiplicity also holds true for a much larger class of weights, even arbitrarily close to situations where there is uniqueness of positive solutions.

Multiplicity of solutions for a generalized Kadomtsev-Petviashvili equation with potential in R^2

In this article, we study the generalized Kadomtsev-Petviashvili equation witha potential $$ (-u_{xx}+D_{x}^{-2}u_{yy}+V(\varepsilon x,\varepsilon y)u-f(u))_{x}=0\quad \text{in }\mathbb{R}^2, $$ where \(D_{x}^{-2}h(x,y)=\int_{-\infty }^{x}\int_{-\infty }^{t}h(s,y)\,ds\,dt \), \(f\) is a nonlinearity, \(\varepsilon\) is a small positive parameter, and the potential \(V\) satisfies a local condition. We prove the existence of nontrivial solitary waves for the modified problem by applying penalization techniques. The relationship between the number of positive solutions and the topology of the set where \(V\) attains its minimum is obtained by using Ljusternik-Schnirelmann theory. With the help of Moser's iteration method, we verify that the solutions of the modified problem are indeed solutions of the original roblem for \(\varepsilon>0\) small enough.

Concentrating solutions for singularly perturbed double phase problems with nonlocal reaction

This paper focuses on the study of multiplicity and concentration phenomena of positive solutions for the singularly perturbed double phase problem with nonlocal Choquard reaction{−ϵpΔpu−ϵqΔqu+V(x)(|u|p−2u+|u|q−2u)=ϵμ−N(1|x|μ⁎G(u))g(u),inRN,u∈W1,p(RN)∩W1,q(RN),u>0,inRN, where 1<p<q<N, 0<μ<N, ϵ is a small positive parameter and V is the absorption potential. Combining variational and topological arguments from Nehari manifold analysis and Ljusternik-Schnirelmann category theory, we prove the existence of positive ground state solutions that concentrate around global minimum points of the potential V. In the second part of this paper, we establish the relationship between the number of positive solutions and the topology of the set where V attains its global minimum. The main results included in this paper complement several recent contributions to the study of concentration phenomena.

The Bifurcation Curves of a Category of Dirichlet Boundary Value Problems

We study the Dirichlet boundary value problemu″t+λfut=0,−1&lt;t&lt;1,u−1=u1=0,generally and develop a schema for determining the relationship between the values of its parameters and the number of positive solutions. Then, we focus our attention on the special cases whenfu=σ−uexp−K/1+uandfu=∏i=1mai—u, respectively. We prove first that all positive solutions of the first problem are less than or equal toσ, obtain more specific lower and upper bounds for these solutions, and compute a curve in theσK-plane with accuracy up to10−6, below which the first problem has a unique positive solution and above which it has exactly three positive solutions. For the second problem, we determine its number of positive solutions and find a formula for the value ofλthat separates the regions ofλ, in which the problem has different numbers of solutions. We also computed the graphs for some special cases of the second problem, and the results are consistent with the existing results. Our code in Mathematica is available upon request.

New multiplicity results for a class of nonlocal equation with steep potential well

In this paper, we consider a class of Kirchhoff type equation with Hartree nonlinearity: where parameters, , is the Riesz potential, in , and . Under some suitable assumptions for potential and spatial dimensions N, by using the filtration of Nehari manifold and Ekeland variational principle, we prove that above nonlocal equation admits at least two positive solutions. The effect of the steep potential well and spatial dimensions on the number of positive solutions are completely showed.

Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity and critical exponents

&lt;abstract&gt;&lt;p&gt;In this paper, we study the multiplicity results of positive solutions for a class of Kirchhoff type problems with singularity and critical exponents. Combining with the Nehari method and variational method, we prove the existence of positive ground state solutions. Furthermore, we obtain a relationship between the number of positive solutions and the topology of the global maximum set of $ Q(x) $.&lt;/p&gt;&lt;/abstract&gt;

Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation

In this paper, we consider the bifurcation curves and exact multiplicity of positive solutions of the one-dimensional Minkowski-curvature equation \t\t\t{−(u′1−u′2)′=λf(u),x∈(−L,L),u(−L)=0=u(L),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} - (\\frac{u'}{\\sqrt{1-u^{\\prime \\,2}}} )'=\\lambda f(u), &x\\in (-L,L), \\\\ u(-L)=0=u(L), \\end{cases} $$\\end{document} where λ and L are positive parameters, fin C[0,infty ) cap C^{2}(0,infty ), and f(u)>0 for 0< u< L. We give the precise description of the structure of the bifurcation curves and obtain the exact number of positive solutions of the above problem when f satisfies f''(u)>0 and uf'(u)geq f(u)+frac{1}{2}u^{2}f''(u) for 0< u< L. In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of λ. The arguments are based upon a detailed analysis of the time map.

A multiplicity result for a (p,\xa0q)-Schr\xf6dinger\u2013Kirchhoff type equation

In this paper, we study a class of (p, q)-Schrödinger–Kirchhoff type equations involving a continuous positive potential satisfying del Pino–Felmer type conditions and a continuous nonlinearity with subcritical growth at infinity. By applying variational methods, penalization techniques and Lusternik–Schnirelman category theory, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values.

Positive solutions for the fractional Schrödinger equations with logarithmic and critical non‐linearities

In this paper, we study a class of fractional Schrödinger equations involving logarithmic and critical non-linearities on an unbounded domain, and show that such an equation with positive or sign-changing weight potentials admits at least one positive ground state solution and the associated energy is positive (or negative). By applying the Nehari manifold method and Ljusternik–Schnirelmann category, we investigate how the weight potential affects the multiplicity of positive solutions, and obtain the relationship between the number of positive solutions and the category of some sets related to the weight potential.

Multiplicity and concentration results for a (p, q)-Laplacian problem in $${\mathbb {R}}^{N}$$

In this paper, we study the multiplicity and concentration of positive solutions for the following (p, q)-Laplacian problem: $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p} u -\Delta _{q} u +V(\varepsilon x) \left( |u|^{p-2}u + |u|^{q-2}u\right) = f(u) &{} \text{ in } {\mathbb {R}}^{N}, \\ u\in W^{1, p}({\mathbb {R}}^{N})\cap W^{1, q}({\mathbb {R}}^{N}), \quad u>0 \text{ in } {\mathbb {R}}^{N}, \end{array} \right. \end{aligned}$$where \(\varepsilon >0\) is a small parameter, \(1< p<q<N\), \(\Delta _{r}u={{\,\mathrm{div}\,}}(|\nabla u|^{r-2}\nabla u)\), with \(r\in \{p, q\}\), is the r-Laplacian operator, \(V:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}\) is a continuous function satisfying the global Rabinowitz condition, and \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function with subcritical growth. Using suitable variational arguments and Ljusternik–Schnirelmann category theory, we investigate the relation between the number of positive solutions and the topology of the set where V attains its minimum for small \(\varepsilon \).

Multiple positive solutions for critical elliptic problem with singularity

This paper investigates the following critical elliptic problem with singularity $$\begin{aligned} \left\{ \begin{array}{lll} -\Delta u=\lambda f(x)|u|^{-\gamma }+Q(x)|u|^{4}u, &{}&{}\quad x\in {\mathbb {R}}^3,\\ u>0,&{}&{}\quad x\in {\mathbb {R}}^3, \end{array}\right. \end{aligned}$$ where $$0<\gamma <1$$ , $$\lambda >0$$ , $$f\in L^{\frac{6}{5+\gamma }}({\mathbb {R}}^3)$$ is a positive function and $$0<Q(x)\in C({\mathbb {R}}^3)$$ . By variational method and Nehari type constraint, we establish a relation between the number of maxima of Q(x) and the number of positive solutions.

Multiplicity of Positive Solutions for a Quasilinear Schrödinger Equation with an Almost Critical Nonlinearity

Abstract In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem: { - Δ ⁢ u - Δ ⁢ ( u 2 ) ⁢ u = | u | p - 2 ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\Delta u-\Delta(u^{2})u&amp;\displaystyle=|u|% ^{p-2}u&amp;&amp;\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&amp;\displaystyle=0&amp;&amp;\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a smooth and bounded domain in ℝ N , N ≥ 3 {\mathbb{R}^{N},N\geq 3} . More specifically we prove that, for p near the critical exponent 22 * = 4 ⁢ N / ( N - 2 ) {22^{*}=4N/(N-2)} , the number of positive solutions is estimated below by topological invariants of the domain Ω: the Ljusternick–Schnirelmann category and the Poincaré polynomial. With respect to the case involving semilinear equations, many difficulties appear here and the classical procedure does not apply immediately. We obtain also en passant some new results concerning the critical case.

Positive solutions for a Kirchhoff-type problem involving multiple competitive potentials and critical Sobolev exponent

In this paper, we study the existence and multiplicity of positive solutions for a Kirchhoff-type problem involving several potentials and critical nonlinearities on unbounded domain. The first main result is that the problem possesses a ground state solution concentrating around a concrete set relates to the potentials. The second one is that with the help of Nehari manifold and Ljusternik–Schnirelmann category, the relationship between the number of positive solutions and the category of the global minima set of a suitable ground energy function is established. Moreover, we present a sufficient condition for the nonexistence of ground state solution.

Multiplicity and concentration results for a class of critical fractional Schrödinger–Poisson systems via penalization method

We deal with the multiplicity and concentration of positive solutions for the following fractional Schrödinger–Poisson-type system with critical growth: [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text], [Formula: see text], [Formula: see text], with [Formula: see text], is the fractional Laplacian operator, [Formula: see text] is a continuous positive potential and [Formula: see text] is a superlinear continuous function with subcritical growth. Using penalization techniques and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum value.

Existence of positive solutions for a Schrödinger-Poisson system with critical growth

ABSTRACT In this paper, we are concerned with the existence, multiplicity and concentration of positive solutions for the semilinear Schrödinger-Poisson system where (small) and λ are real parameters, a,b and c are continuous functions, and f is a continuous superlinear and subcritical nonlinearity. First, we prove that there are two families of semiclassical positive solutions for small. Then we show that these solutions are concentrating on some sets which is generated by the minimal points of a and maximal points of b and c. In addition, we investigate the relation between the number of positive solutions and the topology of the set of the global minima(or maxima) of the potentials by minimax theorems and the Ljusternik-Schnirelmann theory. Finally, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.

Multiplicity of Positive Solutions for Schrödinger–Poisson Systems with a Critical Nonlinearity in $$\mathbb {R}^3$$ R 3

This paper is dedicated to studying the multiplicity of positive solutions for the following Schrodinger–Poisson problem $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+u+\phi u=\lambda Q(x)|u|^{q-2}u+ K(x)|u|^4u, \quad &{}\hbox {in} \ \mathbb {R}^3,\\ -\Delta \phi =u^2, \quad &{}\hbox {in} \ \mathbb {R}^3,\\ \end{array}\right. \end{aligned}$$ where $$4<q<6 $$ or $$q=2$$ , $$\lambda >0$$ is a parameter, K(x) and Q(x) satisfy some mild assumptions. With minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set where K(x) attains its global maximum for small $$\lambda $$ .

Ground state and multiple solutions for the fractional Schrödinger–Poisson system with critical Sobolev exponent

In this paper, we study the following doubly singularly perturbed fractional Schrödinger–Poissonsystem with critical Sobolev exponent ε2α(−Δ)αu+V(x)u+ϕu=|u|2α∗−2u+f(u)inRN,εθ(−Δ)s2ϕ=γsu2inRN,where α∈(12,1), N∈(2α,4α), s∈(N−2α,N), θ∈(0,s), f is a subcritical nonlinearity, ε is a small parameter, the positive potential V satisfies a local condition. By combining penalization techniques with Ljusternik–Schnirelmann theory, the number of positive solutions is estimated below by the topology of the set where the potential V attains its minimum.

Concentration phenomena for a fractional Schrödinger‐Kirchhoff type equation

In this paper, we deal with the multiplicity and concentration of positive solutions for the following fractional Schrödinger‐Kirchhoff type equation urn:x-wiley:mma:media:mma4633:mma4633-math-0001 where ε&gt;0 is a small parameter, is the fractional Laplacian, M is a Kirchhoff function, V is a continuous positive potential, and f is a superlinear continuous function with subcritical growth. By using penalization techniques and Ljusternik‐Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum.

Multiplicity of positive solutions for fractional elliptic systems involving sign-changing weight

In this paper, we study the multiplicity results of positive solutions for a fractional elliptic system involving both concave-convex and critical growth terms. With the help of Morse theory and the Ljusternik-Schnirelmann category, we investigate how the coefficient h(x) of the critical nonlinearity affects the number of positive solutions of that problem and we get some results as regards the relationship between the number of positive solutions and the topology of the global maximum set of h.

Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth

In this paper, we study the multiplicity and concentration of solutions for the following critical fractional Schrodinger–Poisson system: \begin{eqnarray*} \left\{ \begin{array}{ll} \epsilon^{2s}(-\triangle)^{s} {u}+ V(x)u+\phi u =f(u)+|u|^{2^*_{s}-2}u &\mbox{in}\,\,\R^3, \\[2.5mm] \epsilon^{2t}(-\triangle)^{t}{\phi}=u^2 &\mbox{in}\,\, \R^3, \end{array} \right. \end{eqnarray*} where ϵ > 0 is a small parameter, (− △ ) α denotes the fractional Laplacian of order α = s,t ∈ (0,1), where 2 α ∗ 6/3−2α is the fractional critical exponent in Dimension 3; V ∈ C 1 (ℝ 3 ,ℝ + ) and f is subcritical. We first prove that for ϵ > 0 sufficiently small, the system has a positive ground state solution. With minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set where V attains its minimum for small ϵ . Moreover, each positive solution u ϵ converges to the least energy solution of the associated limit problem and concentrates around a global minimum point of V .