Volume 19, Issue 2

The main purpose of this paper is to prove several sharp singular Trudinger–Moser-type inequalities on domains in ℝN{\mathbb{R}^{N}} with infinite volume on the Sobolev-type spaces DN,q⁢(ℝN){D^{N,q}(\mathbb{R}^{N})}, q≥1{q\geq 1}, the completion of C0∞⁢(ℝN){C_{0}^{\infty}(\mathbb{R}^{N})} under the norm ∥∇⁡u∥N+∥u∥q{\|\nabla u\|_{N}+\|u\|_{q}}. The case q=N{q=N} (i.e., DN,q⁢(ℝN)=W1,N⁢(ℝN){D^{N,q}(\mathbb{R}^{N})=W^{1,N}(\mathbb{R}^{N})}) has been well studied to date. Our goal is to investigate which type of Trudinger–Moser inequality holds under different norms when q changes. We will study these inequalities under two types of constraint: semi-norm type ∥∇⁡u∥N≤1{\|\nabla u\|_{N}\leq 1} and full-norm type ∥∇⁡u∥Na+∥u∥qb≤1{\|\nabla u\|_{N}^{a}+\|u\|_{q}^{b}\leq 1}, a>0{a>0}, b>0{b>0}. We will show that the Trudinger–Moser-type inequalities hold if and only if b≤N{b\leq N}. Moreover, the relationship between these inequalities under these two types of constraints will also be investigated. Furthermore, we will also provide versions of exponential type inequalities with exact growth when b>N{b>N}.

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in ℝN{\mathbb{R}^{N}} (N≥2{N\geq 2}): ${(*)_{m}}$ { - Δ ⁢ u = g ⁢ ( u ) - μ ⁢ u   in  ⁢ ℝ N , ∥ u ∥ L 2 ⁢ ( ℝ N ) 2 = m , u ∈ H 1 ⁢ ( ℝ N ) , \displaystyle\begin{cases}-\Delta u=g(u)-\mu u\quad\text{in }\mathbb{R}^{N},% \cr\lVert u\rVert_{L^{2}(\mathbb{R}^{N})}^{2}=m,\cr u\in H^{1}(\mathbb{R}^{N})% ,\end{cases} where g⁢(ξ)∈C⁢(ℝ,ℝ){g(\xi)\in C(\mathbb{R},\mathbb{R})}, m>0{m>0} is a given constant and μ∈ℝ{\mu\in\mathbb{R}} is a Lagrange multiplier. We introduce a new approach using a Lagrange formulation of problem (*)m{(*)_{m}}. We develop a new deformation argument under a new version of the Palais–Smale condition. For a general class of nonlinearities related to [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I: Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313–345], [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 347–375], [J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in ℝN{\mathbb{R}^{N}}: Mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010), no. 2, 253–276], it enables us to apply minimax argument for L2{L^{2}} constraint problems and we show the existence of infinitely many solutions as well as mountain pass characterization of a minimizing solution of the problem inf ⁡ { ∫ ℝ N 1 2 ⁢ | ∇ ⁡ u | 2 - G ⁢ ( u ) ⁢ d ⁢ x : ∥ u ∥ L 2 ⁢ ( ℝ N ) 2 = m } , G ⁢ ( ξ ) = ∫ 0 ξ g ⁢ ( τ ) ⁢ 𝑑 τ . \inf\Bigg{\{}\int_{\mathbb{R}^{N}}{1\over 2}|{\nabla u}|^{2}-G(u)\,dx:\lVert u% \rVert_{L^{2}(\mathbb{R}^{N})}^{2}=m\Bigg{\}},\quad G(\xi)=\int_{0}^{\xi}g(% \tau)\,d\tau.

Necessary and sufficient conditions for a regular positive entire solution u of a biharmonic equation Δ 2 ⁢ u = u p   in  ⁢ ℝ N , N ≥ 5 , p > N + 4 N - 4 \Delta^{2}u=u^{p}\quad\text{in }\mathbb{R}^{N},\,N\geq 5,\,p>\frac{N+4}{N-4} to be a radially symmetric solution are obtained via the exact asymptotic behavior of u at ∞{\infty} and the moving plane method (MPM). It is known that above equation admits a unique positive radial entire solution u⁢(x)=u⁢(|x|){u(x)=u(|x|)} for any given u⁢(0)>0{u(0)>0}, and the asymptotic behavior of u⁢(|x|){u(|x|)} at ∞{\infty} is also known. We will see that the behavior similar to that of a radial entire solution of above equation at ∞{\infty}, in turn, determines the radial symmetry of a general positive entire solution u⁢(x){u(x)} of the equation. To make the procedure of the MPM work, the precise asymptotic behavior of u at ∞{\infty} is obtained.

Efficient conditions guaranteeing the existence and multiplicity of T -periodic solutions to the second order differential equation u′′=h⁢(t)⁢g⁢(u){u^{\prime\prime}=h(t)g(u)} are established. Here, g:(A,B)→(0,+∞){g\colon(A,B)\to(0,+\infty)} is a positive function with two singularities, and h∈L⁢(ℝ/T⁢ℤ){h\in L(\mathbb{R}/T\mathbb{Z})} is a general sign-changing function. The obtained results have a form of relation between multiplicities of zeros of the weight function h and orders of singularities of the nonlinear term. Our results have applications in a physical model, where from the equation u′′=h⁢(t)sin2⁡u{u^{\prime\prime}=\frac{h(t)}{\sin^{2}u}} one can study the existence and multiplicity of periodic motions of a charged particle in an oscillating magnetic field on the sphere. The approach is based on the classical properties of the Leray–Schauder degree.

In this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem: { ( - Δ ) s ⁢ u = λ ⁢ f ⁢ ( u ) u q , u > 0 ⁢  in  ⁢ Ω , u = 0 in  ⁢ ℝ n ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&\displaystyle=\lambda\frac{% f(u)}{u^{q}},&&\displaystyle u>0\text{ in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% n}\setminus\Omega,\end{aligned}\right. where (-Δ)s{(-\Delta)^{s}} denotes the fractional Laplace operator for s∈(0,1){s\in(0,1)}, n>2⁢s{n>2s}, q∈(0,1){q\in(0,1)}, λ>0{\lambda>0} and Ω is a smooth bounded domain in ℝn{\mathbb{R}^{n}}. Here f:[0,∞)→[0,∞){f:[0,\infty)\to[0,\infty)} is a continuous nondecreasing map satisfying lim u → ∞ ⁡ f ⁢ ( u ) u q + 1 = 0 . \lim_{u\to\infty}\frac{f(u)}{u^{q+1}}=0. We show that under certain additional assumptions on f , the above problem possesses at least three distinct solutions for a certain range of λ. We use the method of sub-supersolutions and a critical point theorem by Amann [H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 1976, 4, 620–709] to prove our results. Moreover, we prove a new existence result for a suitable infinite semipositone nonlocal problem which played a crucial role to obtain our main result and is of independent interest.

In this paper, we are concerned with quasilinear equations of Kirchhoff type, and prove the existence of ground state signed solutions and sign-changing solutions by using the Nehari method.

We consider a higher-order evolution differential inequality in an exterior domain of ℝN{\mathbb{R}^{N}}, N≥3{N\geq 3}, with Dirichlet and Neumann boundary conditions. Using a unified approach, we obtain the critical exponents in the sense of Fujita for the considered problems. Moreover, the behavior of the solutions with respect to the initial data is discussed.

We establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and global bifurcation analysis from the zero solution in a nonregular setting, since the nonlinearities considered are not differentiable at zero, so that the standard bifurcation theory does not apply. To overcome this difficulty, we combine a regularization scheme with a priori bounds, and Whyburn’s topological method. Furthermore, via a continuity argument we prove a positivity property for subcontinua of nonnegative solutions. These results are based on a positivity theorem for the associated concave problem proved by us, and extend previous results established in the powerlike case.

In this paper, we study, using critical point theory for strongly indefinite functionals, the Neumann problem associated to some prescribed mean curvature problems in a FLRW spacetime with one spatial dimension. We assume that the warping function is even and positive and the prescribed mean curvature function is odd and sublinear. Then, we show that our problem has infinitely many solutions. The keypoint is that our problem has a Hamiltonian formulation. The main tool is an abstract result of Clark type for strongly indefinite functionals.

We prove that whenever a sequence of bounded operators (Am)m∈ℤ{(A_{m})_{m\in\mathbb{Z}}} acting on a Banach space X admits an exponential dichotomy and a sequence of differentiable maps fm:X→X{f_{m}\colon X\to X}, m∈ℤ{m\in\mathbb{Z}}, has bounded and Hölder derivatives, the nonautonomous dynamics given by xm+1=Am⁢xm+fm⁢(xm){x_{m+1}=A_{m}x_{m}+f_{m}(x_{m})}, m∈ℤ{m\in\mathbb{Z}}, has various shadowing properties. Hence, we extend recent results of Bernardes Jr. et al. in several directions. As a nontrivial application of our results, we give a new proof of the nonautonomous Grobman–Hartman theorem.