Search

The existence of multiple pairs of smooth positive solutions for a Carrier problem, driven by a p (x) -Laplacian operator, is studied. The approach adopted combines sub-super solutions, truncation, and variational techniques. In particular, after an explicit computation of a sub-solution, obtained combining a monotonicity type hypothesis on the reaction term and the Giacomoni–Takáč's version of the celebrated Díaz–Saá's inequality, we derive a multiplicity of solution by investigating an associated one-dimensional fixed point problem. The nonlocal term involved may be a sign-changing function and permit us to obtain the existence of multiple pairs of positive solutions, one for each "positive bump" of the nonlocal term. A new result, also for a constant exponent, is established and an illustrative example is proposed. [ABSTRACT FROM AUTHOR]

This paper is divided in two parts. In the first part, we prove coercivity results and minimization of the Euler energy functional. In the second part, we focus on the existence and multiplicity of a positive solution of fractional Dirichlet problem involving the γ(ξ)-Laplacian equation with non-negative weight functions in Hα,β;χ γ(ξ) (Λ,R) using some variational techniques and Nehari manifold. [ABSTRACT FROM AUTHOR]

The main aim of the present work is to review and study a variational method in existence and multiplicity of positive solutions for quasilinear elliptic systems with critical Hardy-Sobolev and sign-changing function exponents. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem ..., where ε>0 is a small parameter, a,b>0 are constants, 4

In this article we study a nonlocal equation involving singular and critical Hardy-Sobolev non-linearities, (–Δp)su – μ |u|p–2u/ |x|sp = λu–α + |u|p*s(t)–2u/|x|t, in Ω, u > 0, in Ω, u = 0, in RN \ Ω, where Ω ⊂ RN is a bounded domain with Lipschitz boundary and (–Δp)s is the fractional p-Laplacian operator. We combine some variational techniques with a perturbation method to show the existence of multiple solutions. [ABSTRACT FROM AUTHOR]

In this work, we study the fractional Laplacian equation with singular nonlinearity: (− Δ) s u = λ a (x) | u | q − 2 u + 1 − α 2 − α − β c (x) | u | − α | v | 1 − β in Ω , (− Δ) s v = μ b (x) | v | q − 2 v + 1 − β 2 − α − β c (x) | u | 1 − α | v | − β in  Ω , u = v = 0 in ℝ N ∖ Ω , where Ω is a bounded domain in ℝ n with smooth boundary ∂ Ω , N > 2 s , s ∈ (0 , 1) , 0 < α < 1 , 0 < β < 1 , 1 < q < 2 < 2 s ∗ , 2 s ∗ = 2 N N − 2 s is the fractional Sobolev exponent, λ , μ are two parameters, a , b , c ∈ C (Ω ¯) are nonnegative weight functions, and (− Δ) s is the fractional Laplace operator. We use the Nehari manifold approach and some variational techniques in order to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter λ and μ. [ABSTRACT FROM AUTHOR]

The purpose of this work is to study the following singular problem: ( P λ) − div (a (x , ∇ u)) = u − α + λ u q in Ω ; u > 0 , in Ω ; u = 0 , in ℝ n ∖ Ω ; where Ω ⊂ ℝ N , N ≥ 2 be a bounded smooth domain, λ is a positive parameter, p ≥ 2 such that N ≥ p and 0 < α ≤ 1 , p − 1 < q ≤ p ∗ − 1 , where p ∗ = N p N − p. We employ the Nehari manifold approach and the fibering maps in order to show the existence of T q , α such that for all λ ∈ (0 , T q , α) , problem (P λ) has at least two solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the following critical nonlocal problem... where a>0, b≥0, 20 is a parameter, Ω is a smooth bounded domain in R4 and Q(x)∈C(Ω¯) is a nonnegative function. By virtue of variational methods and delicate estimates, we prove that problem admits k positive solutions for λ>0 sufficiently small, provided that the maximum of Q(x) is achieved at k interior points in Ω. [ABSTRACT FROM AUTHOR]

In this article, we consider the multiplicity of solutions for nonhomogeneous Schrödinger-Poisson systems under the Berestycki-Lions type conditions. With the aid of Ekeland's variational principle, the mountain pass theorem and a Pohožaev type identity, we prove that the system has at least two positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following two-component system of Schrödinger equations with Hartree nonlinearity:where is a small parameter, , , and are constants. Under some suitable assumptions on the potentials and the coupled function , we prove the existence and multiplicity of positive solutions for the above system by using energy estimates, the Nehari manifold technique and the Lusternik-Schnirelmann theory. Furthermore, the existence and nonexistence of the least energy positive solutions are also explored. [ABSTRACT FROM AUTHOR]

This paper is dedicated to studying the multiplicity of positive solutions for the following Schrödinger–Poisson problem - Δ u + u + ϕ u = λ Q (x) | u | q - 2 u + K (x) | u | 4 u , in R 3 , - Δ ϕ = u 2 , in R 3 , where 4 < q < 6 or q = 2 , λ > 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 λ . [ABSTRACT FROM AUTHOR]

In this paper, existence and multiplicity of positive solutions of the elliptic system − Δ u + V 1 (x) u = μ 1 (x) u 3 + β (x) u v 2 in  Ω 𝜀 , − Δ v + V 2 (x) v = β (x) u 2 v + μ 2 (x) v 3 in  Ω 𝜀 , u = v = 0 on  ∂ Ω 𝜀 is proved, where Ω 𝜀 is an exterior domain in ℝ N such that ℝ N \ Ω 𝜀 is far away from the origin and contains a sufficiently large ball, N = 1 , 2 , 3 , and the coefficients V j , μ j , β are continuous functions on ℝ N which tend to positive constants at infinity. We do not assume μ 1 , μ 2 , β to be positive functions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider double phase problems with unbalanced and balanced growth and logistic reaction of the superdiffusive type. We prove existence and multiplicity results and discuss how the two cases differ. [ABSTRACT FROM AUTHOR]

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 the Nehari manifold and the Ljusternik-Schnirelmann category, we prove that the problem admits at least $\operatorname{cat}(\Omega)+1$ distinct positive solutions. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the existence and multiplicity of positive solutions of the following equation where . We will prove that the modified term is helpful to find multiple positive solutions in the case of the nonlinearity with sign changing coefficient . Surprisingly, we do not need a sign condition , which has been proven to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with sign changing nonlinearity (Alama and Tarantello (1993) [1]). [Copyright &y& Elsevier]

Abstract: For an open, bounded set with smooth boundary , , , and , we prove the existence of such that the singular b.v.p. has two positive solutions for every . Given with and a bounded elliptic matrix, we also study the Dirichlet b.v.p. associated to the related singular equation [Copyright &y& Elsevier]

In this article, we study a class of indefinite semilinear elliptic problems in ℝN. By using the fibering maps and studying some properties of the Nehari manifold, we obtain the existence and multiplicity of positive solutions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the multiplicity results of positive solutions for a semi-linear elliptic system involving both concave–convex and critical growth terms. With the help of the Nehari manifold and the Lusternik–Schnirelmann category, we investigate how the coefficient of the critical nonlinearity affects the number of positive solutions of that problem and get a relationship between the number of positive solutions and the topology of the global maximum set of . [Copyright &y& Elsevier]

Abstract: This paper is concerned with a singular elliptic system, which involves the Caffarelli–Kohn–Nirenberg inequality and critical Sobolev–Hardy exponents. The existence and multiplicity results of positive solutions are obtained by variational methods. [Copyright &y& Elsevier]

Abstract: In this paper, we study the multiplicity of positive solutions for the nonhomogeneous elliptic problem: in . We will show how the shape of the graph of affects the number of positive solutions. [Copyright &y& Elsevier]

In this paper, we consider a class of quasilinear elliptic systems with weights and the nonlinearity involving the critical Hardy–Sobolev exponent and one sign-changing function. The existence and multiplicity results of positive solutions are obtained by variational methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the existence and multiplicity of positive solutions for the Dirichlet problem where ,,, is a bounded domain with smooth boundary , and , are nonnegative continuous functions on . [Copyright &y& Elsevier]

Abstract: In this paper, we study the multiplicity of positive solutions for the following elliptic equation: where ( if if ), is an upper half space in and the functions and are satisfying some suitable conditions. Using the decomposition of the Nehari manifold, we prove that the equation has at least two positive solutions provided is sufficiently small. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with the existence and multiplicity of positive solutions for the multi-singular semilinear elliptic equations where is a smooth bounded domain such that the points , , , are different, , , , and . The results depend crucially on the parameters , and for . [Copyright &y& Elsevier]

Abstract: In this paper, we study a class of semilinear elliptic equations in . By means of the Lusternik–Schnirelman category and Bahri and Li’s minimax argument, multiple positive solutions are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we study the following singular, critical elliptic problem : where is a bounded domain with smooth boundary , and a smooth function with subcritical asymptotic behaviour at . We show the existence of multiple positive solutions of the problem by combining variational and sub super-solution methods. [Copyright &y& Elsevier]

Abstract: In this paper, existence and multiplicity results to the following Dirichlet problem are established, where , is bounded smooth domain and . Here , is a positive real parameter, the function , among other conditions, can possibly change sign in , and the function satisfies suitable conditions. The study is based on the comparison of energy levels on Nehari manifold. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

We study the multiplicity of positive solutions for the following Kirchhoff type equations: − ε 2 M ε 2 − N ∇ u L 2 2 Δ u + u = Q x u p − 2 u in R N , u ∈ H 1 (R N) , where N ≥ 4 , ε > 0 is a small parameter, M t = a t + b a , b > 0 and 2 < p < 2 ∗ = 2 N N − 2. We assume that the weight function Q ∈ C (R N , R +) has k maximum points in R N. By using a novel constraint approach as well as the barycenter map, we show that the above problem admits k positive solutions when N = 4 while 2 k positive solutions when N ≥ 5 for ε , a sufficiently small. [ABSTRACT FROM AUTHOR]

We establish the existence, nonexistence, and multiplicity of positive solutions to semilinear elliptic systems with integral boundary conditions when positive multiparameters vary on the boundary. We prove the results by using sub- and supersolution argument and fixed point index theory. [ABSTRACT FROM AUTHOR]

In this paper, we present an existence criterion for multiple positive solutions of nonlinear neutral delay partial difference equations. Such equations can be regarded as a discrete analog of neutral delay partial differential equations. Our main result relies on fixed point index theory. An example is constructed to show the applicability of the obtained result. [ABSTRACT FROM AUTHOR]