Existence of solutions for a Neumann problem with double critical exponents and logarithmic perturbation

This paper is concerned with the nonlinear boundary value problem (1) $\beta u''-u'+f(u)=0$, (2) $u'(0)-au(0)=0,u'(1)=0$, where $f(u)=b(c-u)\exp(-k/(1+u))$ and $\beta,a,b,c,k$ are constants. First a formal singular perturbation procedure is applied to reveal the possibility of multiple solutions of (1) and (2). Then an iteration procedure is introduced which yields sequences converging to the maximal solution from above and the minimal solution from below. A criterion for a unique solution of (1), (2) is given. It is mentioned that for certain values of the parameters multiple solutions have been found numerically. Finally, the stability of solutions of (1), (2) is discussed for certain values of the parameters. A solution $u(x)$ of (1), (2) is said to be stable if the first eigenvalue $\sigma$ of the variational equations $(1)' \beta v''-v'+[\sigma\beta+f'(u)]v=0$ and $(2)' v'(0)-av(0)=0, v'(1)=0$, is positive.

As introduced in Chap. 1, we study the ground state solutions of system ( 1.2) in the entire space \(\mathbb R^N\) with \(N=2, 3\). Precisely, motivated by Sirakov’s previous work, we prove some uniqueness results of positive (ground state) solutions for the special case \(\lambda _1=\lambda _2\). These give partial answers to Sirakov’s conjecture. For the general case \(\lambda _1\ne \lambda _2\), we prove a sharp result on the parameter range for the existence of ground state solutions. The asymptotic behaviors of ground state solutions can be investigated as a corollary. We also prove a nonexistence result about positive solutions. These results answer partially some open questions raised by Ambrosetti, Colorado and Sirakov. Our proof is mainly applying asymptotic analysis together with the classical bifurcation theory.

In this paper we deal with positive radially symmetric solutions for a boundary value problem containing a strongly nonlinear operator. The proof of existence of positive solutions that we give uses the blow-up method as a main ingredient for the search of a-priori bounds of solutions. The blow-up argument is one by contradiction and uses a sort of scaling, reminiscent to the one used in the theory of minimal surfaces, see [B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 1981, 883–901], and therefore the homogeneity of the operators, Laplacian or p-Laplacian, and second members powers or power like functions play a fundamental role in the method. Thus, when the differential operators are no longer homogeneous, and similarly for the second members, applying the blow-up method to obtain a-priori bounds of solutions seems an almost impossible task. In spite of this fact, in [M. García-Huidobro, I. Guerra and R. Manásevich, Existence of positive radial solutions for a weakly coupled system via blow up, Abstr. Appl. Anal. 3 1998, 1–2, 105–131], we were able to overcome this difficulty and obtain a-priori bounds for a certain (simpler) type of problems. We show in this paper that the asymptotically homogeneous functions provide, in the same sense, a nonlinear rescaling, that allows us to generalize the blow-up method to our present situation. After the a-priori bounds are obtained, the existence of a solution follows from Leray–Schauder topological degree theory.

We investigate a class of fractional Schrödinger–Poisson systems with critical exponents and steep potential well. By using the constraint variational method and a quantitative deformation lemma, the existence of positive ground state solutions and sign-changing ground state solutions is obtained. It is shown that the energy of the sign-changing ground state solution is strictly larger than twice the energy of the positive ground state solution. Moreover, we also study the asymptotic behavior of the sign-changing ground state

Nonlocal scalar field equations: Qualitative properties, asymptotic profiles and local uniqueness of solutions

In this paper, we consider the following two coupled nonlinear Schrödinger system:$ \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+|x|^{2}u-\lambda_{1}u = \mu_{1}|u|^{2^{*}-2}u+\beta |v|^{\frac{2^{*}}{2}}|u|^{\frac{2^{*}}{2}-2}u, \quad\text{in }\mathbb{R}^{N}, \\ -\Delta v+|x|^{2}v-\lambda_{2}v = \mu_{2}|v|^{2^{*}-2}v+\beta |u|^{\frac{2^{*}}{2}}|v|^{\frac{2^{*}}{2}-2}v, \quad \quad\text{in }\mathbb{R}^{N}, \\ u(x)\to0, \; v(x)\to0, \quad\text{as }|x|\to+\infty, \end{array} \right. \end{equation*} $where $ N\geq3 $, $ 2^{*} = \frac{2N}{N-2} $ is the critical Sobolev exponent, $ \lambda_1, \lambda_2, \mu_{1}, \mu_{2} $ are constants and $ \beta $ is the coupling parameter. By using the variational method, we construct a positive ground state solution of the above system under some suitable assumptions. Our results indicate that, unlike the previous studies on elliptic systems with energy critical nonlinearities but without the harmonic potential, where the physical dimension $ N = 3 $ is the special dimension for the existence of positive ground state solutions, the dimension $ N = 4 $ plays a special role in the existence of positive ground state solutions for the above coupled nonlinear Schrödinger system with both harmonic potential and energy critical nonlinearities.

In this paper we consider a Lotka–Volterra prey–predator model with cross-diffusion of fractional type. The main purpose is to discuss the existence and nonexistence of positive steady state solutions of such a model. Here a positive solution corresponds to a coexistence state of the model. Firstly we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system. Secondly we derive some necessary conditions to ensure the existence of positive solutions, which demonstrate that if the intrinsic growth rate of the prey is too small or the death rate (or the birth rate) of the predator is too large, the model does not possess positive solutions. Thirdly we study the sufficient conditions to ensure the existence of positive solutions by using degree theory. Finally we characterize the stable/unstable regions of semi-trivial solutions and coexistence regions in parameter plane.

Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent

In this paper, we study the following quadratically coupled Schrödinger system: $\begin{equation*}\left\{\begin{array}{ll}-\Delta u+\lambda_1u=\mu_1u^2+2\alpha uv+\gamma v^2, & \mbox{in }\Omega,\\-\Delta v+\lambda_2v=\mu_2v^2+2\gamma uv+\alpha u^2, & \mbox{in }\Omega,\\u=v=0, & \mbox{on }\partial\Omega,\end{array}\right.\end{equation*}$ where $\Omega\subset\mathbb{R}^6$ is a smooth bounded domain, $-\lambda (\Omega) < \lambda_1, \lambda_2 < 0, \mu_1, \mu_2, \alpha, \gamma>0$, and $\lambda (\Omega)$ is the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition. The main difficulty to investigate this kind of equations is caused by the fact that all the quadratic nonlinearities, including the coupling terms, are of critical growth. By the methods used in [Zhenyu Guo, Positive ground state solutions of a nonlinearly coupled Schrödinger system with critical exponents in [Zhenyu Guo, Positive ground state solutions of a nonlinearly coupled Schrödinger system with critical exponents in $\mathbb{R}^4$, J. Math. Anal. Appl., 430(2):950-970, 2015], the existence of positive ground state solutions of the system is established with more ingenious hypotheses.

Chapter 6 Positive solutions for Lotka-Volterra systems with cross-diffusion

Existence and asymptotic behavior of positive ground state solutions for coupled nonlinear fractional Kirchhoff-type systems

We study the existence of positive ground state solution for Choquard systems. In the autonomous case, we prove the existence of at least one positive ground state solution by the Pohozaev manifold method and symmetric-decreasing rearrangement arguments. Moreover, we show that each positive ground state solution is radial symmetric. While, in the nonautonomous case, a positive ground state solution is obtained by using a monotonicity trick and a global compactness lemma. We remark that, under our assumptions of the nonlinearity W u {W_{u}} , the search of ground state solutions cannot be reduced to the study of critical points of a functional restricted to a Nehari manifold.

In this paper, we study the positive radial solutions for the Hénon equations with weighted critical exponents on the unit ball in with . We first confirm that with is the critical exponent for the embedding from into and name as the Hénon‐Sobolev critical exponent. Then, following the great ideas of Brezis and Nirenberg (Comm Pure Appl Math. 1983;36:437‐477), we establish the existence and nonexistence of positive radial solutions of the problems with single Hénon‐Sobolev critical exponent and linear or nonlinear but subcritical perturbations. We further study the problems with multiple critical exponents, which may be Hénon‐Sobolev critical exponents, Hardy‐Sobolev critical exponents, or Sobolev critical exponents. The methods and arguments involved with are the mountain pass theorem and the strong maximum principle and the Pohozaev identity.

We consider a nonlinear parametric Neumann problem driven by a nonhomogeneous differential operator and a strictly $$(p-1)$$ -sublinear reaction term. We prove a bifurcation-type result establishing the existence of a critical parameter value $$\lambda _*>0$$ such that for all $$\lambda >\lambda _*$$ the problem has at least two positive solutions, for $$\lambda =\lambda _*$$ it has at least one positive solution and for $$\lambda \in (0,\lambda _*)$$ there are no positive solutions. Also, for $$\lambda \ge \lambda _*$$ we show that the problem has a smallest positive solution $$\bar{u}_{\lambda }$$ and we investigate the continuity and monotonicity properties of the map $$\lambda \rightarrow \bar{u}_{\lambda }$$ .

We study the nonlinear coupled Kirchhoff system with purely Sobolev critical exponent. By using appropriate transformation, we get one equivalent system involving a critical Schrödinger system and an algebraic system. Through solving the critical Schrödinger system with a corresponding algebraic system, under suitable conditions we obtain the existence and classification of positive ground states for the Kirchhoff system in dimensions 3 and 4. Furthermore, for the degenerate case, we give a complete classification of positive ground states for the Kirchhoff system in any dimension. To the best of our knowledge, this paper is the first to give classification results for the ground states of Kirchhoff systems. The results in this paper partially extend and complement the main results established by Lü and Peng [Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differ. Equ. 263 (2017) 8947–8978] considering the linearly coupled Kirchhoff system with subcritical exponent and some partial results established by Chen and Zou [Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012) 515–551; Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differ. Equ. 52 (2015) 423–467], where the authors considered the coupled purely critical Schrödinger system.