Convex-concave Nonlinearity Research Articles

Double phase systems with convex–concave nonlinearity on complete manifold

Double phase systems with convex–concave nonlinearity on complete manifold

Global existence, blow-up and dynamical behavior in a nonlocal parabolic problem with variational structure

In this paper, we study an initial boundary value problem for a nonlocal parabolic equation that includes a diffusion term and convex–concave nonlinearities. By establishing the Lq-estimate and analyzing its energy, we prove the existence of global solutions and obtain some blow-up conditions. Given the variational structure of the problem, we utilize the Mountain-pass theorem to demonstrate the existence of nontrivial steady-state solutions. Additionally, we establish the dynamical behavior of global solutions whose trajectory is relatively compact in H01(Ω), and which uniformly converge to some non-zero steady state after a long time, owing to the energy functional satisfying the P.S. condition. Finally, we derive an unstable steady states sequence by using another minimax theorem.

Positive Solutions of Nonuniformly Elliptic Equations with Weighted Convex-Concave Nonlinearity

Positive Solutions of Nonuniformly Elliptic Equations with Weighted Convex-Concave Nonlinearity

Positive Solutions of Nonuniformly Elliptic Equations with Weighted Convex-Concave Nonlinearity

В данной статье мы доказываем результаты о существовании двух различных положительных решений задачи $$ \frac{\partial}{\partial z_i}(a_{ij}(z) \frac{\partial u}{\partial z_j})+v(x)u^{q-1}+ \mu u^{p-1}=0, \qquad z\in \Omega, \quad u|_{\partial\Omega}=0, $$ содержащей выпуклую и вогнутую нелинейности, параметр $\mu=\operatorname{const}$; переменные $z=(x,y) \in \mathbb{R}^n \times \mathbb{R}^{N-n}$. Матрица коэффициентов $A=\{a_{ij}(z)\}_{i,j=1}^N$ удовлетворяет условию неравномерной эллиптичности $$ C_1(\omega(x)|\xi|^2+|\eta|^2)\leqslant A(z) \zeta \cdot \zeta \leqslant C_2(\omega(x)|\xi|^2+|\eta|^2) $$ в ограниченной области $\Omega \subset \mathbb{R}^N$, $\zeta=(\xi,\eta) \in \mathbb{R}^n \times \mathbb{R}^{N-n}$, $\zeta \ne 0$. Для достижения своих целей мы рассматриваем условия на диапазон изменения показателей нелинейности $q \in (2,2N/(N-2))$ и $p\in (1,N/(N-1))$ (или $p\in (1,2)$ и дополнительное условие $v^{-p/(q-p)}\in L_1(\Omega)$) и $\mu \in (0,\Lambda)$ при достаточно малой $\Lambda$; положительные весовые функции $v \in A_\infty$, $\omega \in A_2$ принадлежат соответствующим классам Маккенхаупта в метрике $n$-мерного евклидового пространства, а также выполняется условие баланса типа Джанило-Видена. Библиография: 25 названий.

An Investigation of an Integral Equation Involving Convex–Concave Nonlinearities

We investigate the existence and uniqueness of positive solutions to an integral equation involving convex or concave nonlinearities. A numerical algorithm based on Picard iterations is provided to obtain an approximation of the unique solution. The main tools used in this work are based on partial-ordering methods and fixed-point theory. Our results are supported by examples.

Anisotropic problems with unbalanced growth

Abstract The main purpose of this paper is to study a general class of (p, q)-type eigenvalues problems with lack of compactness. The reaction is a convex-concave nonlinearity described by power-type terms. Our main result establishes a complete description of all situations that can occur. We prove the existence of a critical positive value λ * such that the following properties hold: (i) the problem does not have any entire solution in the case of low perturbations (that is, if 0 < λ < λ *); (ii) there is at least one solution if λ = λ *; and (iii) the problem has at least two entire solutions in the case of high perturbations (that is, if λ > λ *). The proof combines variational methods, analytic tools, and monotonicity arguments.

Existence results for a supercritical Neumann problem with a convex–concave non-linearity

We shall consider the following semi-linear problem with a Neumann boundary condition $$\begin{aligned} -\Delta u + u= a(|x|)|u|^{p-2}u- b(|x|)|u|^{q-2}u, \quad x\in B_1, \end{aligned}$$ where \(B_1\) is the unit ball in \({\mathbb {R}}^N\), \(N\ge 2\), a, b are nonnegative radial functions, and p, q are distinct numbers greater than or equal to 2. We shall assume no growth condition on p and q. Our plan is to use a new variational principle that allows one to deal with problems with supercritical Sobolev non-linearities. Indeed, we first find a critical point of the Euler–Lagrange functional associated with this equation over a suitable closed and convex set. Then we shall use this new variational principle to deduce that the restricted critical point of the Euler–Lagrange functional is an actual critical point.

Bifurcation and blow-up results for equations with p-Laplacian and convex-concave nonlinearity

Bifurcation and blow-up results for equations with p-Laplacian and convex-concave nonlinearity

On the existence of stationary solutions for higher-order p-Kirchhoff problems

In this paper, we establish the existence of two nontrivial weak solutions of possibly degenerate nonlinear eigenvalue problems involving the p-polyharmonic Kirchhoff operator in bounded domains. The p-polyharmonic operators [Formula: see text] were recently introduced in [F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal.74 (2011) 5962–5974] for all orders L and independently, in the same volume of the journal, in [V. F. Lubyshev, Multiple solutions of an even-order nonlinear problem with convex-concave nonlinearity, Nonlinear Anal.74 (2011) 1345–1354] only for L even. In Sec. 3, the results are then extended to non-degeneratep(x)-polyharmonic Kirchhoff operators. The main tool of the paper is a three critical points theorem given in [F. Colasuonno, P. Pucci and Cs. Varga, Multiple solutions for an eigenvalue problem involving p-Laplacian type operators, Nonlinear Anal.75 (2012) 4496–4512]. Several useful properties of the underlying functional solution space [Formula: see text], endowed with the natural norm arising from the variational structure of the problem, are also proved both in the homogeneous case p ≡ Const. and in the non-homogeneous case p = p(x). In the latter some sufficient conditions on the variable exponent p are given to prove the positivity of the infimum λ1of the Rayleigh quotient for the p(x)-polyharmonic operator [Formula: see text].

Qualitative Properties of Anisotropic Elliptic Schrödinger Equations

AbstractWe consider a class of nonlinear stationary Schrödinger-type equations and we are concerned with sufficient properties that guarantee the existence of multiple solutions in a suitable Sobolev space with variable exponents. We first establish that in the case of small perturbations, the problem admits at least two weak solutions. Next, in the case of convexconcave nonlinearities, we obtain conditions for the existence of infinitely many solutions with high (resp., small) energies. The arguments combine variational techniques with a careful analysis of the energy levels.

Exact multiplicity of positive solutions for concave–convex and convex–concave nonlinearities

This note gives an unified treatment of the exact multiplicity results for both S-shaped and reversed S-shaped bifurcation for positive solutions of the two-point problemu″+λf(u)=0,for −1<x<1,u(−1)=u(1)=0, depending on a positive parameter λ, for both concave–convex and convex–concave nonlinearities f(u).

О существовании знакопеременного решения эллиптических уравнений с выпукло-вогнутыми нелинейностями

In a bounded connected domainR N , N > 1, with a smooth boundary, we consider the Dirichlet boundary value problem for elliptic equation with a convex-concave nonlinearity ( −�u = �juj q−2 u + juj −2 u, x 2 uj@ = 0, where 1 < q < 2 < < 2 � . As a main result, we prove the existence of a nodal solution to this equation on the nonlocal interval � 2 (−1 ,� � ), where � � is determined by the

A theorem on S-shaped bifurcation curve for a positone problem with convex–concave nonlinearity and its applications to the perturbed Gelfand problem

We study the bifurcation curve and exact multiplicity of positive solutions of the positone problem { u ″ ( x ) + λ f ( u ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where λ > 0 is a bifurcation parameter, f ∈ C 2 [ 0 , ∞ ) satisfies f ( 0 ) > 0 and f ( u ) > 0 for u > 0 , and f is convex–concave on ( 0 , ∞ ) . Under a mild condition, we prove that the bifurcation curve is S-shaped on the ( λ , ‖ u ‖ ∞ ) -plane. We give an application to the perturbed Gelfand problem { u ″ ( x ) + λ exp ( a u a + u ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where a > 0 is the activation energy parameter. We prove that, if a ⩾ a ⁎ ≈ 4.166 , the bifurcation curve is S-shaped on the ( λ , ‖ u ‖ ∞ ) -plane. Our results improve those in [S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Anal. 22 (1994) 1475–1485] and [P. Korman, Y. Li, On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc. 127 (1999) 1011–1020].

Multiple solutions of an even-order nonlinear problem with convex–concave nonlinearity

We study multiple solutions of an even-order nonlinear partial differential equation with a Dirichlet boundary condition. A related class of nonlinear systems is investigated.

Multiple positive solutions of an elliptic equation with a convex-concave nonlinearity containing a sign-changing term

We study the existence of multiple positive solutions to a nonlinear Dirichlet problem for the p-Laplacian (in a bounded domain in ℝ N ) with a concave nonlinearity and with a nonlinear perturbation involving a function of the spatial variable whose sign can change the character of concavity. Under two different sets of conditions imposed on the perturbation, we prove the existence of two and three positive solutions, respectively.

Existence of bounded solutions for fourth order Dirichlet problems with convex-concave nonlinearity

AbstractWe consider the Dirichlet boundary value problem for higher order O.D.E. with nonlinearity being the sum of a derivative of a convex and of a concave function in case when no growth condition is imposed on the concave part.

EXISTENCE OF BOUNDED SOLUTIONS FOR FOURTH ORDER DIRICHLET PROBLEMS WITH CONVEX-CONCAVE NONLINEARITY

Article EXISTENCE OF BOUNDED SOLUTIONS FOR FOURTH ORDER DIRICHLET PROBLEMS WITH CONVEX-CONCAVE NONLINEARITY was published on January 1, 2009 in the journal Demonstratio Mathematica (volume 42, issue 1).

Multiple periodic solutions for p-Laplacian problems with convex-concave nonlinearities

We prove the existence of at least three periodic solutions of a nonlinear differential equation involving $p$-Laplacian and convex-concave nonlinearities.

Positive solutions of elliptic and parabolic equations with convex-concave nonlinearities

We consider, respectively, the Dirichlet problem and the initial-boundary value problem of elliptic and parabolic equations with two power nonlinearities. We find that these problems are closely related to the so-called quenching problem. We obtain the existence and nonexistence of positive solutions to these problems on bounded and unbounded domains, by using the results of quenching problem and sub-super solution method.

On nonlocal existence results for elliptic equations with convex–concave nonlinearities

In this paper we study the family of nonlinear elliptic Dirichlet boundary value problems with p-Laplacian and with concave–convex nonlinearity which depend on real parameter λ . We introduce nonlocal intervals ( λ i , λ i + 1 ) such that the characteristic points λ i , λ i + 1 (a priori bifurcation values) expressed in terms of exact variational principles. In these intervals, new results on the existence of positive solutions, multiple positive solutions and existence of multiple disjoint sets with infinitely many solutions are proved.