Volume 20, Issue 3

In this paper, we study the existence of ground state solutions for the nonlinear Schrödinger–Bopp–Podolsky system with critical Sobolev exponent { - Δ ⁢ u + V ⁢ ( x ) ⁢ u + q 2 ⁢ ϕ ⁢ u = μ ⁢ | u | p - 1 ⁢ u + | u | 4 ⁢ u in  ⁢ ℝ 3 , - Δ ⁢ ϕ + a 2 ⁢ Δ 2 ⁢ ϕ = 4 ⁢ π ⁢ u 2 in  ⁢ ℝ 3 , \left\{\begin{aligned} \displaystyle{}{-}\Delta u+V(x)u+q^{2}\phi u&% \displaystyle=\mu|u|^{p-1}u+|u|^{4}u&&\displaystyle\phantom{}\mbox{in }\mathbb% {R}^{3},\\ \displaystyle{-}\Delta\phi+a^{2}\Delta^{2}\phi&\displaystyle=4\pi u^{2}&&% \displaystyle\phantom{}\mbox{in }\mathbb{R}^{3},\end{aligned}\right. where μ>0{\mu>0} is a parameter and 2<p<5{2<p<5}. Under certain assumptions on V , we prove the existence of a nontrivial ground state solution, using the method of the Pohozaev–Nehari manifold, the arguments of Brézis–Nirenberg, the monotonicity trick and a global compactness lemma.

We prove the validity of the ε-εβ{\varepsilon-\varepsilon^{\beta}} property in the isoperimetric problem with double density, generalising the known properties for the case of single density. As a consequence, we derive regularity for isoperimetric sets.

The aim of this paper is analyzing the positive solutions of the quasilinear problem - ( u ′ / 1 + ( u ′ ) 2 ) ′ = λ ⁢ a ⁢ ( x ) ⁢ f ⁢ ( u )   in  ⁢ ( 0 , 1 ) , u ′ ⁢ ( 0 ) = 0 , u ′ ⁢ ( 1 ) = 0 , -\bigl{(}u^{\prime}/\sqrt{1+(u^{\prime})^{2}}\big{)}^{\prime}=\lambda a(x)f(u)% \quad\text{in }(0,1),\qquad u^{\prime}(0)=0,\quad u^{\prime}(1)=0, where λ∈ℝ{\lambda\in\mathbb{R}} is a parameter, a∈L∞⁢(0,1){a\in L^{\infty}(0,1)} changes sign once in (0,1){(0,1)} and satisfies ∫01a⁢(x)⁢𝑑x<0{\int_{0}^{1}a(x)\,dx<0}, and f∈𝒞1⁢(ℝ){f\in\mathcal{C}^{1}(\mathbb{R})} is positive and increasing in (0,+∞){(0,+\infty)} with a potential, F⁢(s)=∫0sf⁢(t)⁢𝑑t{F(s)=\int_{0}^{s}f(t)\,dt}, quadratic at zero and linear at +∞{+\infty}. The main result of this paper establishes that this problem possesses a component of positive bounded variation solutions, 𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}}, bifurcating from (λ,0){(\lambda,0)} at some λ0>0{\lambda_{0}>0} and from (λ,∞){(\lambda,\infty)} at some λ∞>0{\lambda_{\infty}>0}. It also establishes that 𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}} consists of regular solutions if and only if ∫ 0 z ( ∫ x z a ( t ) d t ) - 1 / 2 d x = + ∞   or   ∫ z 1 ( ∫ x z a ( t ) d t ) - 1 / 2 d x = + ∞ . \int_{0}^{z}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty\quad\text% {or}\quad\int_{z}^{1}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty. Equivalently, the small positive regular solutions of 𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}} become singular as they are sufficiently large if and only if ( ∫ x z a ( t ) d t ) - 1 / 2 ∈ L 1 ( 0 , z )   and   ( ∫ x z a ( t ) d t ) - 1 / 2 ∈ L 1 ( z , 1 ) . \Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}\in L^{1}(0,z)\quad\text{and}% \quad\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}\in L^{1}(z,1). This is achieved by providing a very sharp description of the asymptotic profile, as λ→λ∞{\lambda\to\lambda_{\infty}}, of the solutions. According to the mutual positions of λ0{\lambda_{0}} and λ∞{\lambda_{\infty}}, as well as the bifurcation direction, the occurrence of multiple solutions can also be detected.

In this paper, we investigate the existence of nontrivial solutions to the following fractional p -Laplacian system with homogeneous nonlinearities of critical Sobolev growth: { ( - Δ p ) s ⁢ u = Q u ⁢ ( u , v ) + H u ⁢ ( u , v ) in  ⁢ Ω , ( - Δ p ) s ⁢ v = Q v ⁢ ( u , v ) + H v ⁢ ( u , v ) in  ⁢ Ω , u = v = 0 in  ⁢ ℝ N ∖ Ω , u , v ≥ 0 , u , v ≠ 0 in  ⁢ Ω , \left\{\begin{aligned} \displaystyle{}(-\Delta_{p})^{s}u&\displaystyle=Q_{u}(u% ,v)+H_{u}(u,v)&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle(-\Delta_{p})^{s}v&\displaystyle=Q_{v}(u,v)+H_{v}(u,v)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u=v&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}% ^{N}\setminus\Omega,\\ \displaystyle u,v&\displaystyle\geq 0,\quad u,v\neq 0&&\displaystyle\phantom{}% \text{in }\Omega,\end{aligned}\right. where (-Δp)s{(-\Delta_{p})^{s}} denotes the fractional p -Laplacian operator, p>1{p>1}, s∈(0,1){s\in(0,1)}, p⁢s<N{ps<N}, ps*=N⁢pN-p⁢s{p_{s}^{*}=\frac{Np}{N-ps}} is the critical Sobolev exponent, Ω is a bounded domain in ℝN{\mathbb{R}^{N}} with Lipschitz boundary, and Q and H are homogeneous functions of degrees p and q with p<q≤ps∗{p<q\leq p^{\ast}_{s}} and Qu{Q_{u}} and Qv{Q_{v}} are the partial derivatives with respect to u and v , respectively. To establish our existence result, we need to prove a concentration-compactness principle associated with the fractional p -Laplacian system for the fractional order Sobolev spaces in bounded domains which is significantly more difficult to prove than in the case of single fractional p -Laplacian equation and is of its independent interest (see Lemma 5.1). Our existence results can be regarded as an extension and improvement of those corresponding ones both for the nonlinear system of classical p -Laplacian operators (i.e., s=1{s=1}) and for the single fractional p -Laplacian operator in the literature. Even a special case of our main results on systems of fractional Laplacian (-Δ)s{(-\Delta)^{s}} (i.e., p=2{p=2} and 0<s<1{0<s<1}) has not been studied in the literature before.

We prove the existence of extremals for fractional Moser–Trudinger inequalities in an interval and on the whole real line. In both cases we use blow-up analysis for the corresponding Euler–Lagrange equation, which requires new sharp estimates obtained via commutator techniques.

We consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain Ω⊂ℝd{\Omega\subset\mathbb{R}^{d}} and show that the first eigenfunction v satisfies v⁢(x)≥δ>0{v(x)\geq\delta>0} for all x∈Ω¯{x\in\overline{\Omega}}, even if the boundary ∂⁡Ω{\partial\Omega} is only Lipschitz continuous. Under such weak regularity assumptions the Hopf–Oleĭnik boundary lemma is not available; instead we use a new approach based on an abstract positivity improving condition for semigroups that map Lp⁢(Ω){L_{p}(\Omega)} into C⁢(Ω¯){C(\overline{\Omega})}. The same tool also yields corresponding results for Dirichlet or mixed boundary conditions. Finally, we show that our results can be used to derive strong minimum and maximum principles for parabolic and elliptic equations.

In this paper, we study properties of the lambda constants and the existence of ground states of Perelman’s famous W -functional from a variational formulation. We have two kinds of results. One is about the estimation of the lambda constant of G. Perelman, and the other is about the existence of ground states of his W -functional, both on a complete non-compact Riemannian manifold (M,g){(M,g)}. One consequence of our estimation is that, on an ALE (or asymptotic flat) manifold (M,g){(M,g)}, if the scalar curvature s of (M,g){(M,g)} is non-negative and has quadratical decay at infinity, then M is scalar flat, i.e., s=0{s=0} in M . We also introduce a new constant d⁢(M,g){d(M,g)}. For the existence of the ground states, we use Lions’ concentration-compactness method.

In this paper we establish a new critical point theorem for a class of perturbed differentiable functionals without satisfying the Palais–Smale condition. We prove the existence of at least one critical point to such functionals, provided that the perturbation is sufficiently small. The main abstract result of this paper is applied both to perturbed nonhomogeneous equations in Orlicz–Sobolev spaces and to nonlocal problems in fractional Sobolev spaces.

We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equation u ′′ + c ⁢ u ′ + ( λ ⁢ a + ⁢ ( x ) - μ ⁢ a - ⁢ ( x ) ) ⁢ g ⁢ ( u ) = 0 , u^{\prime\prime}+cu^{\prime}+\bigl{(}\lambda a^{+}(x)-\mu a^{-}(x)\bigr{)}g(u)% =0, where λ,μ>0{\lambda,\mu>0} are parameters, c∈ℝ{c\in\mathbb{R}}, a⁢(x){a(x)} is a locally integrable P -periodic sign-changing weight function, and g:[0,1]→ℝ{g\colon{[0,1]}\to\mathbb{R}} is a continuous function such that g⁢(0)=g⁢(1)=0{g(0)=g(1)=0}, g⁢(u)>0{g(u)>0} for all u∈]0,1[{u\in{]0,1[}}, with superlinear growth at zero. A typical example for g⁢(u){g(u)}, that is of interest in population genetics, is the logistic-type nonlinearity g⁢(u)=u2⁢(1-u){g(u)=u^{2}(1-u)}. Using a topological degree approach, we provide high multiplicity results by exploiting the nodal behavior of a⁢(x){a(x)}. More precisely, when m is the number of intervals of positivity of a⁢(x){a(x)} in a P -periodicity interval, we prove the existence of 3m-1{3^{m}-1} non-constant positive P -periodic solutions, whenever the parameters λ and μ are positive and large enough. Such a result extends to the case of subharmonic solutions. Moreover, by an approximation argument, we show the existence of a family of globally defined solutions with a complex behavior, coded by (possibly non-periodic) bi-infinite sequences of three symbols.

By means of a suitable degree theory, we prove persistence of eigenvalues and eigenvectors for set-valued perturbations of a Fredholm linear operator. As a consequence, we prove existence of a bifurcation point for a non-linear inclusion problem in abstract Banach spaces. Finally, we provide applications to differential inclusions.

We consider periodic solutions of the following problem associated with the fractional Laplacian: (-∂x⁢x)s⁢u⁢(x)+∂u⁡F⁢(x,u⁢(x))=0{(-\partial_{xx})^{s}u(x)+\partial_{u}F(x,u(x))=0} in ℝ{\mathbb{R}}. The smooth function F⁢(x,u){F(x,u)} is periodic about x and is a double-well potential with respect to u with wells at +1{+1} and -1 for any x∈ℝ{x\in\mathbb{R}}. We prove the existence of periodic solutions whose periods are large integer multiples of the period of F about the variable x by using variational methods. An estimate of the energy functional, Hamiltonian identity and Modica-type inequality for periodic solutions are also established.