Volume 16, Issue 3

In this paper, we study the following Schrödinger–Maxwell system with critical exponents in ℝ3${\mathbb{R}^{3}}$: { - ε 2 ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u + K ⁢ ( x ) ⁢ ϕ ⁢ u = P ⁢ ( x ) ⁢ g ⁢ ( u ) + Q ⁢ ( x ) ⁢ | u | 4 ⁢ u , - ε 2 ⁢ Δ ⁢ ϕ = K ⁢ ( x ) ⁢ u 2 . $\left\{\begin{aligned} &\displaystyle{-}\varepsilon^{2}\Delta u+V(x)u+K(x)\phi u% =P(x)g(u)+Q(x)|u|^{4}u,\\ &\displaystyle{-}\varepsilon^{2}\Delta\phi=K(x)u^{2}.\end{aligned}\right.$ Under suitable assumptions on the potentials V⁢(x)${V(x)}$, K⁢(x)${K(x)}$, P⁢(x)${P(x)}$ and Q⁢(x)${Q(x)}$, we are able to prove the existence of semiclassical ground state solutions (uε,ϕε)${(u_{\varepsilon},\phi_{\varepsilon})}$ with the maximum points xε${x_{\varepsilon}}$ of uε${u_{\varepsilon}}$ concentrating at a special set M described by these potential functions. Moreover, setting vε⁢(x):=uε⁢(ε⁢x+xε)${v_{\varepsilon}(x):=u_{\varepsilon}(\varepsilon x+x_{\varepsilon})}$ for any sequence xε→x0∈M${x_{\varepsilon}\to x_{0}\in M}$, the solutions (vε,ϕε)${(v_{\varepsilon},\phi_{\varepsilon})}$ converge strongly in H1⁢(ℝ3)${H^{1}(\mathbb{R}^{3})}$ to a solution (u,ϕ)${(u,\phi)}$ of { - Δ ⁢ v + V ⁢ ( x 0 ) ⁢ v + K ⁢ ( x 0 ) ⁢ ϕ ⁢ v = P ⁢ ( x 0 ) ⁢ g ⁢ ( v ) + Q ⁢ ( x 0 ) ⁢ | v | 4 ⁢ v , - Δ ⁢ ϕ = K ⁢ ( x 0 ) ⁢ v 2 . $\left\{\begin{aligned} &\displaystyle{-}\Delta v+V(x_{0})v+K(x_{0})\phi v=P(x_% {0})g(v)+Q(x_{0})|v|^{4}v,\\ &\displaystyle{-}\Delta\phi=K(x_{0})v^{2}.\end{aligned}\right.$

We study two classes of nonhomogeneous elliptic problems with Dirichlet boundary condition and involving a fourth-order differential operator with variable exponent and power-type nonlinearities. The first result of this paper establishes the existence of a nontrivial weak solution in the case of a small perturbation of the right-hand side. The proof combines variational methods, including the Ekeland variational principle and the mountain pass theorem of Ambrosetti and Rabinowitz. Next we consider a very related eigenvalue problem and we prove the existence of nontrivial weak solutions for large values of the parameter. The direct method of the calculus of variations, estimates of the levels of the associated energy functional and basic properties of the Lebesgue and Sobolev spaces with variable exponent have an important role in our arguments.

We consider the Rosenau–Korteweg-de Vries-regularized long wave and Rosenau-regularized long wave equations, which contain nonlinear dispersive effects. We prove that by adding small diffusion to the equations, as the diffusion and dispersion parameters tends to zero, the solutions of the duffusive/dispersive equations converge to the unique entropy solution of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the Lp${L^{p}}$ setting.

In this work, we employ minimization arguments and topological degree theory for mappings of type S+${S_{+}}$ to prove the existence of infinite sequences of ordered solutions for a class of nonlinear equations in which the semilinear term vanishes an infinite number of times. Our results extend earlier ones by Hess [10] and by Loc and Schmitt [15].

Let (M,g)${(M,g)}$ be a smooth compact n -dimensional Riemannian manifold (n≥2${n\geq 2}$) with smooth (n-1)${(n-1)}$-dimensional boundary ∂⁡M${\partial M}$. We prove that the stable critical points of the mean curvature of the boundary generates H1⁢(M)${H^{1}(M)}$ solutions for the following singularly perturbed elliptic problem with Neumann boundary conditions: - ε 2 ⁢ Δ g ⁢ u + u = u p - 1 ⁢ in ⁢ M , u > 0 ⁢ in ⁢ M , ∂ ⁡ u ∂ ⁡ ν = 0 ⁢ on ⁢ ∂ ⁡ M , $-\varepsilon^{2}\Delta_{g}u+u=u^{p-1}\text{ in }M,\quad u>0\text{ in }M,\quad% \frac{\partial u}{\partial\nu}=0\text{ on }\partial M,$ when ε is small enough. Here p is subcritical.

This paper is focused on the local interior W1,∞${W^{1,\infty}}$-regularity for weak solutions of degenerate elliptic equations of the form div⁡[𝐚⁢(x,u,∇⁡u)]+b⁢(x,u,∇⁡u)=0${\operatorname{div}[\mathbf{a}(x,u,\nabla u)]+b(x,u,\nabla u)=0}$, which include those of p -Laplacian type. We derive an explicit estimate of the local L∞${L^{\infty}}$-norm for the solution’s gradient in terms of its local Lp${L^{p}}$-norm. Specifically, we prove ∥ ∇ ⁡ u ∥ L ∞ ⁢ ( B R / 2 ⁢ ( x 0 ) ) p ≤ C | B R ⁢ ( x 0 ) | ⁢ ∫ B R ⁢ ( x 0 ) | ∇ ⁡ u ⁢ ( x ) | p ⁢ 𝑑 x . $\lVert\nabla u\rVert_{L^{\infty}(B_{R/2}(x_{0}))}^{p}\leq\frac{C}{\lvert B_{R}% (x_{0})\rvert}\int_{B_{R}(x_{0})}\lvert\nabla u(x)\rvert^{p}dx.$ This estimate paves the way for our work [9] in establishing W1,q${W^{1,q}}$-estimates (for q>p${q>p}$) for weak solutions to a much larger class of quasilinear elliptic equations.

In this paper we consider singular semilinear elliptic equations with a variable exponent whose model problem is - Δ ⁢ u = f ⁢ ( x ) u γ ⁢ ( x )   in ⁢ Ω , u = 0   on ⁢ ∂ ⁡ Ω . $-\Delta u=\frac{f(x)}{u^{\gamma(x)}}\quad\text{in }\Omega,\qquad u=0\quad\text% {on }\partial\Omega.$ Here Ω is an open bounded set of ℝN${\mathbb{R}^{N}}$, γ⁢(x)${\gamma(x)}$ is a positive continuous function and f⁢(x)${f(x)}$ is a positive function that belongs to a certain Lebesgue space. We prove that there exists a solution to this problem in the natural energy space H01⁢(Ω)${H^{1}_{0}(\Omega)}$ when γ⁢(x)≤1${\gamma(x)\leq 1}$ in a strip around the boundary. For another case, we prove that the solution belongs to Hloc1⁢(Ω)${H^{1}_{\mathrm{loc}}(\Omega)}$ and that it is zero on the boundary in a suitable sense.

We prove symmetry and monotonicity properties for positive solutions of the singular semilinear elliptic equation - Δ ⁢ u = g ⁢ ( x ) u γ + h ⁢ ( x ) ⁢ f ⁢ ( u ) $-\Delta u=\frac{g(x)}{u^{\gamma}}+h(x)f(u)$ in bounded smooth domains with zero Dirichlet boundary conditions. The well-known moving plane method is applied.

We consider the equation - div ⁡ ( a ⁢ ( x ) ⁢ ∇ ⁡ u ) = b ⁢ ( x ) ⁢ | u | q - 2 ⁢ u + c ⁢ ( x ) ⁢ | u | p - 2 ⁢ u , u ∈ H 0 1 ⁢ ( Ω ) , $-\operatorname{div}(a(x)\nabla u)=b(x)|u|^{q-2}u+c(x)|u|^{p-2}u,\quad u\in H_{% 0}^{1}(\Omega),$ where Ω⊂ℝN${\Omega\subset\mathbb{R}^{N}}$ is a bounded smooth domain and N≥4${N\geq 4}$. The functions a , b and c satisfy some hypotheses which provide a variational structure for the problem. For 1<q<2<p≤2⁢N/(N-2)${1<q<2<p\leq 2N/(N-2)}$ we obtain the existence of two nonzero solutions if the function b has small Lebesgue norm. The proof is based on minimization arguments and the Mountain Pass Theorem.

In this paper, we investigate the following double critical Hardy–Sobolev–Maz’ya problem: { - Δ ⁢ u = μ ⁢ | u | 2 * ⁢ ( t ) - 2 ⁢ u | y | t + | u | 2 * ⁢ ( s ) - 2 ⁢ u | y | s + a ⁢ ( x ) ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , $\left\{\begin{aligned} &\displaystyle{-}\Delta u=\mu\frac{|u|^{2^{*}(t)-2}u}{|% y|^{t}}+\frac{|u|^{2^{*}(s)-2}u}{|y|^{s}}+a(x)u&&\displaystyle\text{in }\Omega% ,\\ &\displaystyle u=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right.$ where μ≥0${\mu\geq 0}$, a⁢(x)>0${a(x)>0}$, 2*⁢(t)=2⁢(N-t)N-2${2^{*}(t)=\frac{2(N-t)}{N-2}}$, 2*⁢(s)=2⁢(N-s)N-2${2^{*}(s)=\frac{2(N-s)}{N-2}}$, 0≤t<s<2${0\leq t<s<2}$, x=(y,z)∈ℝk×ℝN-k${x=(y,z)\in\mathbb{R}^{k}\times\mathbb{R}^{N-k}}$, 2≤k<N,(0,z*)∈Ω¯$2\leq k<N,(0,z^{*})\in\bar{\Omega}$ and Ω is an open bounded domain in ℝN${\mathbb{R}^{N}}$. By applying an abstract theorem presented in [42], we prove that if N>6+t${N>6+t}$ when μ>0,${\mu>0,}$ and N>6+s${N>6+s}$ when μ=0,${\mu=0,}$ and Ω satisfies some geometric conditions, then the above problem has infinitely many sign-changing solutions. The main tool is to estimate the Morse indices of these nodal solutions.

We consider nonlinear nonlocal boundary value problems associated with fractional operators (including the fractional p -Laplace and the regional fractional p -Laplace operators) and subject to general (fractional-like) boundary conditions on bounded domains with Lipschitz boundary. Under suitable conditions on the nonlinearities of our system, we establish the existence of bounded solutions and provide explicit L∞${L^{\infty}}$-estimates of solutions which are optimal with respect to the inhomogeneous “sources” present in the system. As application, these results are shown to apply to a class of nonlinear nonlocal equations for the Dirichlet fractional p -Laplacian and regional fractional p -Laplace with a dissipative nonlinearity, and to a class of semilinear nonlocal boundary value problems with fractional Wentzell–Robin boundary conditions corresponding to the so-called fractional Wentzell Laplacian.

We prove the presence of chaotic dynamics for the classical two-body Kepler problem with a time-periodic gravitational coefficient oscillating between two fixed values. The set of chaotic solutions we detect is coded by the number of revolutions in each period. The chaotic dynamics is obtained for large period T as well as for small angular momentum μ. In particular, we provide an explicit lower bound on T and explicit upper bound on μ which guarantee the existence of complex dynamics. We get our results by applying a simple and well-known topological method, the stretching along the path technique. Our results are robust with respect to small perturbations of the gravitational coefficient and to the addition of a small friction term.

In this paper we establish the existence of standing wave solutions for quasilinear Schrödinger equations involving subcritical growth at resonance. By using a change of variables, the quasilinear equation is reduced to a semilinear one, whose associated functional is well defined in the usual Sobolev space. The “first” eigenvalue type of a nonhomogeneous operator has been studied. Using this fact and a variant of the monotone operator theorem, we show that the problem at resonance has at least one nontrivial solution.

In this paper, we first establish a singular (0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any bounded domain in ℝn${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.1). Next, we prove the critical singular (0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any unbounded domain in ℝn${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.2). Then, we set up a subcritical singular (0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any unbounded domain in ℝn${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.3). Finally, we establish the subcritical nonsingular (β=0${(\beta=0}$) Trudinger–Moser inequality on any unbounded domain in ℝn${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.5). The constants in all these inequalities are sharp. In [9], for the proof of Theorem 1.2 in the nonsingular case β=0${\beta=0}$, the following inequality was used (see [17]): u ∗ ⁢ ( r ) - u ∗ ⁢ ( r 0 ) ≤ 1 n ⁢ w n 1 / n ⁢ ∫ r r 0 U ⁢ ( s ) ⁢ s 1 / n ⁢ d ⁢ s s , $u^{\ast}(r)-u^{\ast}(r_{0})\leq\frac{1}{nw_{n}^{{1/n}}}\int_{r}^{r_{0}}U(s)s^{% {1/n}}\frac{ds}{s},$ where U⁢(x)${U(x)}$ is the radial function built from |∇⁡u|${|\nabla u|}$ on the level set of u , i.e., ∫ | u | > t | ∇ u | d x = ∫ 0 | { | u | > t } | U ( s ) d s . $\int_{|u|>t}\lvert\nabla u|\,dx=\int_{0}^{|\{|u|>t\}|}U(s)\,ds.$ The construction of such U uses the deep Fleming–Rishel co-area formula and the isoperimetric inequality and is highly nontrivial. Moreover, this argument will not work in the singular case 0<β<n${0<\beta<n}$. One of the main novelties of this paper is that we can avoid the use of this deep construction of such a radial function U (see remarks at the end of the introduction). Moreover, our approach adapts the symmetrization-free argument developed in [19, 21, 23], where we derive the global inequalities on unbounded domains from the local inequalities on bounded domains using the level sets of the functions under consideration.

In this paper we consider a nonlinear elliptic problem driven by a nonhomogeneous differential operator with Robin boundary conditions. We produce conditions on the reaction term near 0+${0^{+}}$ and near +∞${+\infty}$, which imply the existence and uniqueness of a positive solution. In the particular case of equations driven by the p -Laplacian, we show that these conditions are also necessary, extending in this way the semilinear Dirichlet work of Brezis and Oswald [5].

In this paper, we show that the minimizing problem Λ s , N , k , α = inf u ∈ H ˙ s ⁢ ( ℝ N ) , u ≢ 0 ⁡ ∫ ℝ N | ( - Δ ) s 2 ⁢ u ⁢ ( x ) | 2 ⁢ 𝑑 x ( ∫ ℝ N | u ⁢ ( x ) | 2 s , α * | y | α ⁢ 𝑑 x ) 2 2 s , α * $\Lambda_{s,N,k,\alpha}=\inf_{u\in\dot{H}^{s}(\mathbb{R}^{N}),u\not\equiv 0}% \frac{\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u(x)|^{2}\,dx}{\bigl{(}\int% _{\mathbb{R}^{N}}\frac{|u(x)|^{2^{*}_{s,\alpha}}}{|y|^{\alpha}}\,dx\bigr{)}^{% \frac{2}{2^{*}_{s,\alpha}}}}$ is achieved by a positive, cylindrically symmetric and strictly decreasing function u⁢(x)${u(x)}$ provided 0<s<N2${0<s<\frac{N}{2}}$, 0<α<2⁢s${0<\alpha<2s}$, where x=(y,z)∈ℝk×ℝN-k${x=(y,z)\in\mathbb{R}^{k}\times\mathbb{R}^{N-k}}$ and 2s,α*=2⁢(N-α)N-2⁢s${2^{*}_{s,\alpha}=\frac{2(N-\alpha)}{N-2s}}$. Decaying laws for the minimizer u are also established.