Volume 8, Issue 1

We present sufficient conditions on the existence of solutions, with various specific almost periodicity properties, in the context of nonlinear, generally multivalued, non-autonomous initial value differential equations, d ⁢ u d ⁢ t ⁢ ( t ) ∈ A ⁢ ( t ) ⁢ u ⁢ ( t ) , t ≥ 0 , u ⁢ ( 0 ) = u 0 , \frac{du}{dt}(t)\in A(t)u(t),\quad t\geq 0,\qquad u(0)=u_{0}, and their whole line analogues, d ⁢ u d ⁢ t ⁢ ( t ) ∈ A ⁢ ( t ) ⁢ u ⁢ ( t ) {\frac{du}{dt}(t)\in A(t)u(t)} , t ∈ ℝ {t\in\mathbb{R}} , with a family { A ⁢ ( t ) } t ∈ ℝ {\{A(t)\}_{t\in\mathbb{R}}} of ω-dissipative operators A ⁢ ( t ) ⊂ X × X {A(t)\subset X\times X} in a general Banach space X . According to the classical DeLeeuw–Glicksberg theory, functions of various generalized almost periodic types uniquely decompose in a “dominating” and a “damping” part. The second main object of the study – in the above context – is to determine the corresponding “dominating” part [ A ⁢ ( ⋅ ) ] a ⁢ ( t ) {[A(\,\cdot\,)]_{a}(t)} of the operators A ⁢ ( t ) {A(t)} , and the corresponding “dominating” differential equation, d ⁢ u d ⁢ t ⁢ ( t ) ∈ [ A ⁢ ( ⋅ ) ] a ⁢ ( t ) ⁢ u ⁢ ( t ) , t ∈ ℝ . \frac{du}{dt}(t)\in[A(\,\cdot\,)]_{a}(t)u(t),\quad t\in\mathbb{R}.

Closed form formulas for well-defined complex-valued solutions to a product-type system of difference equations of interest with six parameters are presented. The form of the solutions is described in detail in terms of the parameters and initial values.

In this article, we study the following fractional p -Laplacian equation with critical growth and singular non-linearity: ( - Δ p ) s ⁢ u = λ ⁢ u - q + u α , u > 0   in  ⁢ Ω , u = 0   in  ⁢ ℝ n ∖ Ω , (-\Delta_{p})^{s}u=\lambda u^{-q}+u^{\alpha},\quad u>0\quad\text{in }\Omega,% \qquad u=0\quad\text{in }\mathbb{R}^{n}\setminus\Omega, where Ω is a bounded domain in ℝ n {\mathbb{R}^{n}} with smooth boundary ∂ ⁡ Ω {\partial\Omega} , n > s ⁢ p {n>sp} , s ∈ ( 0 , 1 ) {s\in(0,1)} , λ > 0 {\lambda>0} , 0 < q ≤ 1 {0<q\leq 1} and 1 < p < α + 1 ≤ p s * {1<p<\alpha+1\leq p^{*}_{s}} . We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter λ.

We consider non-autonomous functionals of the form ℱ ⁢ ( u , Ω ) = ∫ Ω f ⁢ ( x , D ⁢ u ⁢ ( x ) ) ⁢ 𝑑 x {\mathcal{F}(u,\hskip-0.569055pt\Omega)\hskip-0.853583pt=\hskip-0.853583pt\int% _{\Omega}f(x,\hskip-0.569055ptDu(x))\hskip-0.569055pt\,dx} , where u : Ω → ℝ N {u\colon\kern-0.711319pt\Omega\hskip-0.569055pt\to\hskip-0.569055pt\mathbb{R}^% {N}} , Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} . We assume that f ⁢ ( x , z ) {f(x,z)} grows at least as | z | p {|z|^{p}} and at most as | z | q {|z|^{q}} . Moreover, f ⁢ ( x , z ) {f(x,z)} is Hölder continuous with respect to x and convex with respect to z . In this setting, we give a sufficient condition on the density f ⁢ ( x , z ) {f(x,z)} that ensures the absence of a Lavrentiev gap.

We study the homogeneous Dirichlet problem for the fully nonlinear equation u t = | Δ ⁢ u | m - 2 ⁢ Δ ⁢ u - d ⁢ | u | σ - 2 ⁢ u + f   in  Q T = Ω × ( 0 , T ) , u_{t}=|\Delta u|^{m-2}\Delta u-d|u|^{\sigma-2}u+f\quad\text{in ${Q_{T}=\Omega% \times(0,T)}$,} with the parameters m > 1 {m>1} , σ > 1 {\sigma>1} and d ≥ 0 {d\geq 0} . At the points where Δ ⁢ u = 0 {\Delta u=0} , the equation degenerates if m > 2 {m>2} , or becomes singular if m ∈ ( 1 , 2 ) {m\in(1,2)} . We derive conditions of existence and uniqueness of strong solutions, and study the asymptotic behavior of strong solutions as t → ∞ {t\to\infty} . Sufficient conditions for exponential or power decay of ∥ ∇ ⁡ u ⁢ ( t ) ∥ 2 , Ω {\|\nabla u(t)\|_{2,\Omega}} are derived. It is proved that for certain ranges of the exponents m and σ, every strong solution vanishes in a finite time.

We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous ( p , q ) {(p,q)} -Laplace equations - Δ p ⁢ u - Δ q ⁢ u = α ⁢ | u | p - 2 ⁢ u + β ⁢ | u | q - 2 ⁢ u {-\Delta_{p}u-\Delta_{q}u=\alpha\lvert u\rvert^{p-2}u+\beta\lvert u\rvert^{q-2% }u} where p ≠ q {p\neq q} . By virtue of the Nehari manifolds, the linking theorem, and descending flow, we explicitly characterize subsets of the ( α , β ) {(\alpha,\beta)} -plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore several relations between eigenvalues and eigenfunctions of the p - and q -Laplacians in one dimension.

We consider the Grushin type operator on ℝ x d × ℝ y k {\mathbb{R}^{d}_{x}\times\mathbb{R}^{k}_{y}} of the form G μ = ∑ i = 1 d ∂ x i 2 + ( ∑ i = 1 d x i 2 ) 2 ⁢ μ ⁢ ∑ j = 1 k ∂ y j 2 G_{\mu}=\sum_{i=1}^{d}\partial_{x_{i}}^{2}+\Bigl{(}\sum_{i=1}^{d}x_{i}^{2}% \Bigr{)}^{2\mu}\sum_{j=1}^{k}\partial_{y_{j}}^{2} and derive weighted Hardy–Sobolev type inequalities and weighted Caffarelli–Kohn–Nirenberg type inequalities related to G μ {G_{\mu}} .

The aim of this paper is to study the following problem: { ( - Δ ) p , β s ⁢ u = f ⁢ ( x , u ) in  ⁢ Ω , u = 0 in  ⁢ ℝ N ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta)^{s}_{p,\beta}u&\displaystyle=f(x% ,u)&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% N}\setminus\Omega,\end{aligned}\right. where Ω is a smooth bounded domain of ℝ N {\mathbb{R}^{N}} containing the origin, ( - Δ ) p , β s ⁢ u ⁢ ( x ) := PV ⁢ ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p - 2 ⁢ ( u ⁢ ( x ) - u ⁢ ( y ) ) | x - y | N + p ⁢ s ⁢ d ⁢ y | x | β ⁢ | y | β (-\Delta)^{s}_{p,\beta}u(x):=\mathrm{PV}\int_{\mathbb{R}^{N}}\frac{\lvert u(x)% -u(y)\rvert^{p-2}(u(x)-u(y))}{\lvert x-y\rvert^{N+ps}}\frac{dy}{\lvert x\rvert% ^{\beta}\lvert y\rvert^{\beta}} with 0 ≤ β < N - p ⁢ s 2 {0\leq\beta<\frac{N-ps}{2}} , 1 < p < N {1<p<N} , s ∈ ( 0 , 1 ) {s\in(0,1)} , and p ⁢ s < N {ps<N} . The main purpose of this work is to prove the existence of a weak solution under some hypotheses on f . In particular, we will consider two cases: (i) f ⁢ ( x , σ ) = f ⁢ ( x ) {f(x,\sigma)=f(x)} ; in this case we prove the existence of a weak solution, that is, in a suitable weighted fractional Sobolev space for all f ∈ L 1 ⁢ ( Ω ) {f\in L^{1}(\Omega)} . In addition, if f ⪈ 0 {f\gneq 0} , we show that the problem above has a unique entropy positive solution. (ii) f ⁢ ( x , σ ) = λ ⁢ σ q + g ⁢ ( x ) {f(x,\sigma)=\lambda\sigma^{q}+g(x)} , σ ≥ 0 {\sigma\geq 0} ; in this case, according to the values of λ and q , we get the largest class of data g for which the problem above has a positive solution.

We investigate the problem { - Δ ⁢ u = | u | p - 2 ⁢ u in  Ω , ∂ ⁡ u ∂ ⁡ 𝐧 = λ ⁢ b ⁢ ( x ) ⁢ | u | q - 2 ⁢ u on  ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=\lvert u\rvert^{p-% 2}u&&\displaystyle\phantom{}\text{in ${\Omega}$},\\ \displaystyle\frac{\partial u}{\partial\mathbf{n}}&\displaystyle=\lambda b(x)% \lvert u\rvert^{q-2}u&&\displaystyle\phantom{}\text{on ${\partial\Omega}$},% \end{aligned}\right. where Ω is a bounded and smooth domain of ℝ N {\mathbb{R}^{N}} ( N ≥ 2 {N\geq 2} ), 1 < q < 2 < p {1<q<2<p} , λ > 0 {\lambda>0} , and b ∈ C 1 + α ⁢ ( ∂ ⁡ Ω ) {b\in C^{1+\alpha}(\partial\Omega)} for some α ∈ ( 0 , 1 ) {\alpha\in(0,1)} . We show that ∫ ∂ ⁡ Ω b < 0 {\int_{\partial\Omega}b<0} is a necessary and sufficient condition for the existence of nontrivial non-negative solutions of this problem. Under the additional condition b + ≢ 0 {b^{+}\not\equiv 0} we show that for λ > 0 {\lambda>0} sufficiently small this problem has two nontrivial non-negative solutions which converge to zero in C ⁢ ( Ω ¯ ) {C(\overline{\Omega})} as λ → 0 {\lambda\to 0} . When p < 2 * {p<2^{*}} we also provide the asymptotic profiles of these solutions.

In this paper we study the nonexistence of solutions, which are stable or stable outside a compact set, possibly unbounded and sign-changing, of some nonlinear elliptic equations with mixed boundary value conditions. The main methods used are the integral estimates and the monotonicity formula.

As an essential extension of the well known case β ∈ ( 1 2 , 1 ] {\beta\kern-1.0pt\in\kern-1.0pt({\frac{1}{2}},1]} to the hyper-dissipative case β ∈ ( 1 , ∞ ) {\beta\kern-1.0pt\in\kern-1.0pt(1,\infty)} , this paper establishes both well-posedness and ill-posedness (not only norm inflation but also indifferentiability of the solution map) for the mild solutions of the incompressible Navier–Stokes system with dissipation ( - Δ ) 1 2 < β < ∞ {(-\Delta)^{{\frac{1}{2}}<\beta<\infty}} through the generalized Carleson measure spaces of initial data that unify many diverse spaces, including the Q space ( Q - s = - α ) n {(Q_{-s=-\alpha})^{n}} , the BMO-Sobolev space ( ( - Δ ) - s 2 ⁢ BMO ) n {((-\Delta)^{-{\frac{s}{2}}}\mathrm{BMO})^{n}} , the Lip-Sobolev space ( ( - Δ ) - s 2 ⁢ Lip ⁢ α ) n {((-\Delta)^{-{\frac{s}{2}}}\mathrm{Lip}\alpha)^{n}} , and the Besov space ( B ˙ ∞ , ∞ s ) n {(\dot{B}^{s}_{\infty,\infty})^{n}} .

We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over ℝ d {\mathbb{R}^{d}} and in L p {L^{p}} -spaces with respect to tight evolution systems of measures. Here, the linear part of the equation is a nonautonomous second-order elliptic operator with unbounded coefficients defined in I × ℝ d {I\times\mathbb{R}^{d}} , ( I being a right-halfline). To the above Cauchy problem we associate a nonlinear evolution operator, which we study in detail, proving some summability improving properties. We also study the stability of the null solution to the Cauchy problem.

In this paper, we study the following water wave model with a nonlocal viscous term: u t + u x + β ⁢ u x ⁢ x ⁢ x + ν π ⁢ ∂ ∂ ⁡ t ⁢ ∫ 0 t u ⁢ ( s ) t - s ⁢ 𝑑 s + u ⁢ u x = ν ⁢ u x ⁢ x , u_{t}+u_{x}+\beta u_{xxx}+\frac{\sqrt{\nu}}{\sqrt{\pi}}\frac{\partial}{% \partial t}\int_{0}^{t}\frac{u(s)}{\sqrt{t-s}}\,ds+uu_{x}=\nu u_{xx}, where 1 π ⁢ ∂ ∂ ⁡ t ⁢ ∫ 0 t u ⁢ ( s ) t - s ⁢ 𝑑 s {\frac{1}{\sqrt{\pi}}\frac{\partial}{\partial t}\int_{0}^{t}\frac{u(s)}{\sqrt{% t-s}}\,ds} is the Riemann–Liouville half-order derivative. We prove the well-posedness of this model using diffusive realization of the half-order derivative, and we discuss the asymptotic convergence of the solution. Also, we compare our mathematical results with those given in [5] and [14] for similar equations.

In this paper, we study the following Kirchhoff type equation: - ( 1 + b ∫ ℝ N | ∇ u | 2 d x ) Δ u + u = a ( x ) f ( u )   in  ℝ N , u ∈ H 1 ( ℝ N ) , -\bigg{(}1+b\int_{\mathbb{R}^{N}}\lvert\nabla u|^{2}\,dx\biggr{)}\Delta u+u=a(% x)f(u)\quad\text{in }\mathbb{R}^{N},\qquad u\in H^{1}(\mathbb{R}^{N}), where N ≥ 3 {N\geq 3} , b > 0 {b>0} and f ⁢ ( s ) {f(s)} is asymptotically linear at infinity, that is, f ⁢ ( s ) ∼ O ⁢ ( s ) {f(s)\sim O(s)} as s → + ∞ {s\rightarrow+\infty} . By using variational methods, we obtain the existence of a mountain pass type solution and a ground state solution under appropriate assumptions on a ⁢ ( x ) {a(x)} .

Our aim in this paper is to study higher-order (in space) Allen–Cahn and Cahn–Hilliard models. In particular, we obtain well-posedness results, as well as the existence of the global attractor. We also give, for the Allen–Cahn models, numerical simulations which illustrate the effects of the higher-order terms and the anisotropy.

Existence, uniqueness and continuous dependence results together with maximum principles represent key tools in the analysis of lattice reaction-diffusion equations. In this paper, we study these questions in full generality by considering nonautonomous reaction functions, possibly nonsymmetric diffusion and continuous, discrete or mixed time. First, we prove the local existence and global uniqueness of bounded solutions, as well as the continuous dependence of solutions on the underlying time structure and on initial conditions. Next, we obtain the weak maximum principle which enables us to get the global existence of solutions. Finally, we provide the strong maximum principle which exhibits an interesting dependence on the time structure. Our results are illustrated by the autonomous Fisher and Nagumo lattice equations and a nonautonomous logistic population model with a variable carrying capacity.

In this article, we establish the existence of bound state solutions for a class of quasilinear Schrödinger equations whose nonlinear term is asymptotically linear in ℝ N {\mathbb{R}^{N}} . After changing the variables, the quasilinear equation becomes a semilinear equation, whose respective associated functional is well defined in H 1 ⁢ ( ℝ N ) {H^{1}(\mathbb{R}^{N})} . The proofs are based on the Pohozaev manifold and a linking theorem.

In this paper, we study a class of quasilinear elliptic equations involving the Sobolev critical exponent - ε p ⁢ Δ p ⁢ u - ε p ⁢ Δ p ⁢ ( u 2 ) ⁢ u + V ⁢ ( x ) ⁢ | u | p - 2 ⁢ u = h ⁢ ( u ) + | u | 2 ⁢ p * - 2 ⁢ u   in  ⁢ ℝ N , -\varepsilon^{p}\Delta_{p}u-\varepsilon^{p}\Delta_{p}(u^{2})u+V(x)\lvert u% \rvert^{p-2}u=h(u)+\lvert u\rvert^{2p^{*}-2}u\quad\text{in }\mathbb{R}^{N}, where Δ p ⁢ u = div ⁡ ( | ∇ ⁡ u | p - 2 ⁢ ∇ ⁡ u ) {\Delta_{p}u=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplace operator, p * = N ⁢ p N - p {p^{*}=\frac{Np}{N-p}} ( N ≥ 3 {N\geq 3} , N > p ≥ 2 {N>p\geq 2} ) is the usual Sobolev critical exponent, the potential V ⁢ ( x ) {V(x)} is a continuous function, and the nonlinearity h ⁢ ( u ) {h(u)} is a nonnegative function of C 1 {C^{1}} class. Under some suitable assumptions on V and h , we establish the existence, multiplicity and concentration behavior of solutions by using combing variational methods and the theory of the Ljusternik–Schnirelman category.

Under certain assumptions, we prove the existence of homoclinic solutions for almost periodic second order Hamiltonian systems in the strongly indefinite case. The proof relies on a careful analysis of the energy functional restricted to the generalized Nehari manifold, and the existence and fine properties of special Palais–Smale sequences.

In this paper, we present a new method for converting boundary value problems of impulsive fractional differential equations to integral equations. Applications of this method are given to solve some types of anti-periodic boundary value problems for impulsive fractional differential equations. Firstly by using iterative method, we prove existence and uniqueness of solutions of Cauchy problems of differential equations involving Caputo fractional derivative, Riemann–Liouville and Hadamard fractional derivatives with order q ∈ ( 0 , 1 ) {q\in(0,1)} , see Theorem 2, Theorem 4, Theorem 6 and Theorem 8. Then we obtain exact expression of piecewise continuous solutions of these fractional differential equations see Theorem 1, Theorem 2, Theorem 3 and Theorem 4. Finally, four classes of integral type anti-periodic boundary value problems of singular fractional differential equations with impulse effects are proposed. Sufficient conditions are given for the existence of solutions of these problems. See Theorems 4.1–4.4. We allow the nonlinearity p ⁢ ( t ) ⁢ f ⁢ ( t , x ) {p(t)f(t,x)} in fractional differential equations to be singular at t = 0 , 1 {t=0,1} and be involved a super-linear and sub-linear term. The analysis relies on Schaefer’s fixed point theorem. In order to avoid misleading readers, we correct the results in [28] and [65]. We establish sufficient conditions for the existence of solutions of an anti-periodic boundary value problem for impulsive fractional differential equation. The results in [68] are complemented. The results in [81] are corrected. See Lemma 5.1, Lemma 5.7, Lemma 5.10 and Lemma 5.13.

We consider the equation - Δ ⁢ u = a ⁢ u - b ⁢ ( x ) ⁢ u 2 - c ⁢ h ⁢ ( x )   in  ⁢ Ω , u = 0   on  ⁢ ∂ ⁡ Ω , -\Delta u=au-b(x)u^{2}-ch(x)\quad\text{in }\Omega,\qquad u=0\quad\text{on }% \partial\Omega, where Ω is a smooth bounded domain in ℝ N {\mathbb{R}^{N}} , b ⁢ ( x ) {b(x)} and h ⁢ ( x ) {h(x)} are nonnegative functions, and there exists Ω 0 ⊂ ⊂ Ω {\Omega_{0}\subset\subset\Omega} such that { x : b ⁢ ( x ) = 0 } = Ω ¯ 0 {\{x:b(x)=0\}=\overline{\Omega}_{0}} . We investigate the existence of positive solutions of this equation for c large under the strong growth rate assumption a ≥ λ 1 ⁢ ( Ω 0 ) {a\geq\lambda_{1}(\Omega_{0})} , where λ 1 ⁢ ( Ω 0 ) {\lambda_{1}(\Omega_{0})} is the first eigenvalue of the - Δ {-\Delta} in Ω 0 {\Omega_{0}} with Dirichlet boundary condition. We show that if h ≡ 0 {h\equiv 0} in Ω ∖ Ω ¯ 0 {\Omega\setminus\overline{\Omega}_{0}} , then our equation has a unique positive solution for all c large, provided that a is in a right neighborhood of λ 1 ⁢ ( Ω 0 ) {\lambda_{1}(\Omega_{0})} . For this purpose, we prove and utilize some new results on the positive solution set of this equation in the weak growth rate case.

In this manuscript, we study the relation between viscosity and weak solutions for non-homogeneous p -Laplace equations with lower-order term depending on x , u and ∇ ⁡ u {\nabla u} . More precisely, we prove that any locally bounded viscosity solution constitutes a weak solution, extending results presented in Juutinen, Lindqvist and Manfredi [9], and Julin and Juutinen [6]. Moreover, we provide a converse statement in the full case under extra assumptions on the data.

This paper deals with the existence of periodic solutions of fractional differential equations with periodic impulses. The first part of the paper is devoted to the uniqueness, existence and asymptotic stability results for periodic solutions of impulsive fractional differential equations with varying lower limits for standard nonlinear cases as well as for cases of weak nonlinearities, equidistant and periodically shifted impulses. We also apply our result to an impulsive fractional Lorenz system. The second part extends the study to periodic impulsive fractional differential equations with fixed lower limit. We show that in general, there are no solutions with long periodic boundary value conditions for the case of bounded nonlinearities.

We derive the uniqueness of weak solutions to the Shigesada–Kawasaki–Teramoto (SKT) systems using the adjoint problem argument. Combining with [10], we then derive the well-posedness for the SKT systems in space dimension d ≤ 4 {d\leq 4} .

Given the supremal functional E ∞ ⁢ ( u , Ω ′ ) = ess ⁢ sup Ω ′ ⁡ H ⁢ ( ⋅ , D ⁢ u ) {E_{\infty}(u,\Omega^{\prime})=\operatornamewithlimits{ess\,sup}_{\Omega^{% \prime}}H(\,\cdot\,,\mathrm{D}u)} , defined on W loc 1 , ∞ ⁢ ( Ω , ℝ N ) {W^{1,\infty}_{\mathrm{loc}}(\Omega,\mathbb{R}^{N})} , with Ω ′ ⋐ Ω ⊆ ℝ n {\Omega^{\prime}\Subset\Omega\subseteq\mathbb{R}^{n}} , we identify a class of vectorial rank-one absolute minimisers by proving a statement slightly stronger than the next claim: vectorial solutions of the Hamilton–Jacobi equation H ⁢ ( ⋅ , D ⁢ u ) = c {H(\,\cdot\,,\mathrm{D}u)=c} are rank-one absolute minimisers if they are C 1 {C^{1}} . Our minimality notion is a generalisation of the classical L ∞ {L^{\infty}} variational principle of Aronsson to the vector case, and emerged in earlier work of the author. The assumptions are minimal, requiring only continuity and rank-one convexity of the level sets.

Motivated by the work [9], in this paper we investigate the infinite boundary value problem associated with the semilinear PDE L ⁢ u = f ⁢ ( u ) + h ⁢ ( x ) {Lu=f(u)+h(x)} on bounded smooth domains Ω ⊆ ℝ n {\Omega\subseteq\mathbb{R}^{n}} , where L is a non-divergence structure uniformly elliptic operator with singular lower-order terms. In the equation, f is a continuous non-decreasing function that satisfies the Keller–Osserman condition, while h is a continuous function in Ω that may change sign, and which may be unbounded on Ω. Our purpose is two-fold. First we study some sufficient conditions on f and h that would ensure existence of boundary blow-up solutions of the above equation, in which we allow the lower-order coefficients to be singular on the boundary. The second objective is to provide sufficient conditions on f and h for the uniqueness of boundary blow-up solutions. However, to obtain uniqueness, we need the lower-order coefficients of L to be bounded in Ω, but we still allow h to be unbounded on Ω.

We study the elliptic sinh-Gordon equation posed in a semi-strip by applying the so-called Fokas method, a generalization of the inverse scattering transform for boundary value problems. Based on the spectral analysis for the Lax pair formulation, we show that the spectral functions can be characterized from the boundary values. We express the solution of the equation in terms of the unique solution of the matrix Riemann–Hilbert problem whose jump matrices are defined by the spectral functions. Moreover, we derive the global algebraic relation that involves the boundary values. In addition, it can be verified that the solution of the elliptic sinh-Gordon equation posed in the semi-strip exists if the spectral functions defined by the boundary values satisfy this global relation.

In this paper, we study the existence of viscosity solutions to the Gelfand problem for the 1-homogeneous p -Laplacian in a bounded domain Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} , that is, we deal with - 1 p - 1 ⁢ | ∇ ⁡ u | 2 - p ⁢ div ⁡ ( | ∇ ⁡ u | p - 2 ⁢ ∇ ⁡ u ) = λ ⁢ e u -\frac{1}{p-1}\lvert\nabla u\rvert^{2-p}\operatorname{div}(\lvert\nabla u% \rvert^{p-2}\nabla u)=\lambda e^{u} in Ω with u = 0 {u=0} on ∂ ⁡ Ω {\partial\Omega} . For this problem we show that, for p ∈ [ 2 , ∞ ] {p\in[2,\infty]} , there exists a positive critical value λ * = λ * ⁢ ( Ω , N , p ) {\lambda^{*}=\lambda^{*}(\Omega,N,p)} such that the following holds: • If λ < λ * {\lambda<\lambda^{*}} , the problem admits a minimal positive solution w λ {w_{\lambda}} . • If λ > λ * {\lambda>\lambda^{*}} , the problem admits no solution. Moreover, the branch of minimal solutions { w λ } {\{w_{\lambda}\}} is increasing with λ. In addition, using degree theory, for fixed p we show that there exists an unbounded continuum of solutions that emanates from the trivial solution u = 0 {u=0} with λ = 0 {\lambda=0} , and for a small fixed λ we also obtain a continuum of solutions with p ∈ [ 2 , ∞ ] {p\in[2,\infty]} .

We study the semiclassical limit to a singularly perturbed nonlinear Klein–Gordon–Maxwell–Proca system, with Neumann boundary conditions, on a Riemannian manifold 𝔐 {\mathfrak{M}} with boundary. We exhibit examples of manifolds, of arbitrary dimension, on which these systems have a solution which concentrates at a closed submanifold of the boundary of 𝔐 {\mathfrak{M}} , forming a positive layer, as the singular perturbation parameter goes to zero. Our results allow supercritical nonlinearities and apply, in particular, to bounded domains in ℝ N {\mathbb{R}^{N}} . Similar results are obtained for the more classical electrostatic Klein–Gordon–Maxwell system with appropriate boundary conditions.

We prove the existence and multiplicity of subharmonic solutions for Hamiltonian systems obtained as perturbations of N planar uncoupled systems which, e.g., model some type of asymmetric oscillators. The nonlinearities are assumed to satisfy Landesman–Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is carried out by the use of a generalized version of the Poincaré–Birkhoff Theorem. Different situations, including Lotka–Volterra systems, or systems with singularities, are also illustrated.

Multiplicity of solutions is proved for an elliptic system with an indefinite Robin boundary condition under an assumption that links the linearisation at 0 and the eigenvalues of the associated linear scalar operator. Our result is based on a precise calculation of the topological degree of a suitable fixed point operator over large and small balls.

We consider the elliptic equation - Δ ⁢ u + u = 0 {-\Delta u+u=0} in a bounded, smooth domain Ω in ℝ 2 {\mathbb{R}^{2}} subject to the nonlinear Neumann boundary condition ∂ ⁡ u ∂ ⁡ ν = λ ⁢ u ⁢ e u 2 {\frac{\partial u}{\partial\nu}=\lambda ue^{u^{2}}} , where ν denotes the outer normal vector of ∂ ⁡ Ω {\partial\Omega} . Here λ > 0 {\lambda>0} is a small parameter. For any λ small we construct positive solutions concentrating, as λ → 0 {\lambda\to 0} , around points of the boundary of Ω.

In this paper, we consider the following critical nonlocal problem: { M ⁢ ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) ⁢ ( - Δ ) s ⁢ u = λ u γ + u 2 s * - 1 in  ⁢ Ω , u > 0 in  ⁢ Ω , u = 0 in  ⁢ ℝ N ∖ Ω , \left\{\begin{aligned} &\displaystyle M\bigg{(}\iint_{\mathbb{R}^{2N}}\frac{% \lvert u(x)-u(y)\rvert^{2}}{\lvert x-y\rvert^{N+2s}}\,dx\,dy\biggr{)}(-\Delta)% ^{s}u=\frac{\lambda}{u^{\gamma}}+u^{2^{*}_{s}-1}&&\displaystyle\phantom{}\text% {in }\Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% N}\setminus\Omega,\end{aligned}\right. where Ω is an open bounded subset of ℝ N {\mathbb{R}^{N}} with continuous boundary, dimension N > 2 ⁢ s {N>2s} with parameter s ∈ ( 0 , 1 ) {s\in(0,1)} , 2 s * = 2 ⁢ N / ( N - 2 ⁢ s ) {2^{*}_{s}=2N/(N-2s)} is the fractional critical Sobolev exponent, λ > 0 {\lambda>0} is a real parameter, γ ∈ ( 0 , 1 ) {\gamma\in(0,1)} and M models a Kirchhoff-type coefficient, while ( - Δ ) s {(-\Delta)^{s}} is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is, when the Kirchhoff function M is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.

We study the non-existence and existence of infinitely many solutions to the semilinear degenerate elliptic system { - Δ λ ⁢ u = | v | p - 1 ⁢ v in  ⁢ Ω , - Δ λ ⁢ v = | u | q - 1 ⁢ u in  ⁢ Ω , u = v = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} &\displaystyle{-}\Delta_{\lambda}u=\lvert v\rvert^{p-1}% v&&\displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle{-}\Delta_{\lambda}v=\lvert u\rvert^{q-1}u&&\displaystyle% \phantom{}\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\phantom{}\text{on }\partial\Omega,\end{% aligned}\right. in a bounded domain Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} with smooth boundary ∂ ⁡ Ω {\partial\Omega} . Here p , q > 1 {p,q>1} , and Δ λ {\Delta_{\lambda}} is the strongly degenerate operator of the form Δ λ ⁢ u = ∑ j = 1 N ∂ ∂ ⁡ x j ⁢ ( λ j 2 ⁢ ( x ) ⁢ ∂ ⁡ u ∂ ⁡ x j ) , \Delta_{\lambda}u=\sum^{N}_{j=1}\frac{\partial}{\partial x_{j}}\Bigl{(}\lambda% _{j}^{2}(x)\frac{\partial u}{\partial x_{j}}\Bigr{)}, where λ ⁢ ( x ) = ( λ 1 ⁢ ( x ) , … , λ N ⁢ ( x ) ) {\lambda(x)=(\lambda_{1}(x),\dots,\lambda_{N}(x))} satisfies certain conditions.

The aim of this paper is to consider the asymptotic behavior of boundary value problems in n -dimensional domains with periodically placed particles, with a general microscopic boundary condition on the particles and a p -Laplace diffusion operator on the interior, in the case in which the particles are of critical size. We consider the cases in which 1 < p < n {1<p<n} , n ≥ 3 {n\geq 3} . In fact, in contrast to previous results in the literature, we formulate the microscopic boundary condition in terms of a Robin type condition, involving a general maximal monotone graph, which also includes the case of microscopic Dirichlet boundary conditions. In this way we unify the treatment of apparently different formulations, which before were considered separately. We characterize the so called “strange term” in the homogenized problem for the case in which the particles are balls of critical size. Moreover, by studying an application in Chemical Engineering, we show that the critically sized particles lead to a more effective homogeneous reaction than noncritically sized particles.

We study the equation ( - Δ ) s ⁢ u + V ⁢ ( x ) ⁢ u = ( I α * | u | p ) ⁢ | u | p - 2 ⁢ u + λ ⁢ ( I β * | u | q ) ⁢ | u | q - 2 ⁢ u   in  ⁢ ℝ N , (-\Delta)^{s}u+V(x)u=(I_{\alpha}*\lvert u\rvert^{p})\lvert u\rvert^{p-2}u+% \lambda(I_{\beta}*\lvert u\rvert^{q})\lvert u\rvert^{q-2}u\quad\text{in }{% \mathbb{R}}^{N}, where I γ ⁢ ( x ) = | x | - γ {I_{\gamma}(x)=\lvert x\rvert^{-\gamma}} for any γ ∈ ( 0 , N ) {\gamma\in(0,N)} , p , q > 0 {p,q>0} , α , β ∈ ( 0 , N ) {\alpha,\beta\in(0,N)} , N ≥ 3 {N\geq 3} , and λ ∈ ℝ {\lambda\in{\mathbb{R}}} . First, the existence of groundstate solutions by using a minimization method on the associated Nehari manifold is obtained. Next, the existence of least energy sign-changing solutions is investigated by considering the Nehari nodal set.

We construct an open set Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} on which an eigenvalue problem for the p -Laplacian has no isolated first eigenvalue and the spectrum is not discrete. The same example shows that the usual Lusternik–Schnirelmann minimax construction does not exhaust the whole spectrum of this eigenvalue problem.

We extend Chen, Wei and Yan’s constructions of families of solutions with unbounded energies [5] to the case of cubic nonlinear Schrödinger equations in the optimal dimension four.

In this paper we explore Liouville-type theorems to solutions of PDEs involving the ϕ-Laplace operator in the setting of Orlicz–Sobolev spaces. Our results extend Liouville-type theorems that have been obtained recently.

We prove in this paper some results on the complex and fractional powers of the Stokes operator with slip frictionless boundary conditions involving the stress tensor. This is fundamental and plays an important role in the associated parabolic problem and will be used to prove maximal L p L^{p} - L q L^{q} regularity results for the non-homogeneous Stokes problem.

We establish the higher fractional differentiability of the solutions to nonlinear elliptic equations in divergence form, i.e., div ⁡ 𝒜 ⁢ ( x , D ⁢ u ) = div ⁡ F , {\operatorname{div}\mathcal{A}(x,Du)=\operatorname{div}F,} when 𝒜 {\mathcal{A}} is a p -harmonic type operator, and under the assumption that x ↦ 𝒜 ⁢ ( x , ξ ) {x\mapsto\mathcal{A}(x,\xi\/)} belongs to the critical Besov–Lipschitz space B n / α , q α {B^{\alpha}_{{n/\alpha},q}} . We prove that some fractional differentiability assumptions on F transfer to Du with no losses in the natural exponent of integrability. When div ⁡ F = 0 {\operatorname{div}F=0} , we show that an analogous extra differentiability property for Du holds true under a Triebel–Lizorkin assumption on the partial map x ↦ 𝒜 ⁢ ( x , ξ ) {x\mapsto\mathcal{A}(x,\xi\/)} .

We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler–Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi’s theorem on conjugate points and Noether’s theorem. We close with an application to low regularity Riemannian geometry.

In this article, we deal with the stochastic perturbation of degenerate parabolic partial differential equations (PDEs). The particular emphasis is on analyzing the effects of a multiplicative Lévy noise on such problems and on establishing a well-posedness theory by developing a suitable weak entropy solution framework. The proof of the existence of a solution is based on the vanishing viscosity technique. The uniqueness of the solution is settled by interpreting Kruzhkov’s doubling technique in the presence of a noise.

We prove interior Hölder estimates for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation u t = | ∇ ⁡ u | κ ⁢ div ⁡ ( | ∇ ⁡ u | p - 2 ⁢ ∇ ⁡ u ) , u_{t}=\lvert\nabla u\rvert^{\kappa}\operatorname{div}(\lvert\nabla u\rvert^{p-% 2}\nabla u), where p ∈ ( 1 , ∞ ) {p\in(1,\infty)} and κ ∈ ( 1 - p , ∞ ) {\kappa\in(1-p,\infty)} . This includes the from L ∞ {L^{\infty}} to C 1 , α {C^{1,\alpha}} regularity for parabolic p -Laplacian equations in both divergence form with κ = 0 {\kappa=0} , and non-divergence form with κ = 2 - p {\kappa=2-p} .

In this paper, we are concerned with the critical and subcritical Trudinger–Moser-type inequalities for functions in a fractional Sobolev space H 1 / 2 , 2 {H^{1/2,2}} on the whole real line. We prove the relation between two inequalities and discuss the attainability of the suprema.

Many existence and nonexistence results are known for nonnegative radial solutions to the equation - △ ⁢ u + A | x | α ⁢ u = f ⁢ ( u )   in  ⁢ ℝ N , N ≥ 3 , A , α > 0 , u ∈ D 1 , 2 ⁢ ( ℝ N ) ∩ L 2 ⁢ ( ℝ N , | x | - α ⁢ d ⁢ x ) , -\triangle u+\frac{A}{|x|^{\alpha}}u=f(u)\quad\text{in }\mathbb{R}^{N},\,N\geq 3% ,\,A,\alpha>0,\,u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},|x|^{-% \alpha}\,dx), with the nonlinearities satisfying | f ( u ) | ≤ ( const . ) u p - 1 {|f(u)|\leq(\mathrm{const.})u^{p-1}} for some p > 2 {p>2} . The existence of nonradial solutions, by contrast, is known only for N ≥ 4 {N\geq 4} , α = 2 {\alpha=2} , f ⁢ ( u ) = u ( N + 2 ) / ( N - 2 ) {f(u)=u^{(N+2)/(N-2)}} and A large enough. Here we show that the above equation has multiple nonradial solutions as A → + ∞ {A\rightarrow+\infty} for N ≥ 4 {N\geq 4} , 2 / ( N - 1 ) < α < 2 ⁢ N - 2 {2/(N-1)<\alpha<2N-2} and α ≠ 2 {\alpha\neq 2} , with the nonlinearities satisfying suitable assumptions. Our argument essentially relies on the compact embeddings between some suitable functional spaces of symmetric functions, which yields the existence of nonnegative solutions of mountain-pass type, and the separation of the corresponding mountain-pass levels from the energy levels associated to radial solutions.

We consider a second-order equation with a linear “elastic” part and a nonlinear damping term depending on a power of the norm of the velocity. We investigate the asymptotic behavior of solutions, after rescaling them suitably in order to take into account the decay rate and bound their energy away from zero. We find a rather unexpected dichotomy phenomenon. Solutions with finitely many Fourier components are asymptotic to solutions of the linearized equation without damping and exhibit some sort of equipartition of the total energy among the components. Solutions with infinitely many Fourier components tend to zero weakly but not strongly. We show also that the limit of the energy of the solutions depends only on the number of their Fourier components. The proof of our results is inspired by the analysis of a simplified model, which we devise through an averaging procedure, and whose solutions exhibit the same asymptotic properties as the solutions to the original equation.

In the present paper, we study the existence of solutions for some classes of singular systems involving the Δ p ⁢ ( x ) {\Delta_{p(x)}} and Δ q ⁢ ( x ) {\Delta_{q(x)}} Laplacian operators. The approach is based on bifurcation theory and the sub-supersolution method for systems of quasilinear equations involving singular terms.

Ultrafunctions are a particular class of functions defined on a hyperreal field ℝ ∗ ⊃ ℝ {\mathbb{R}^{\ast}\supset\mathbb{R}} . They have been introduced and studied in some previous works [2, 6, 7]. In this paper we introduce a particular space of ultrafunctions which has special properties, especially in term of localization of functions together with their derivatives. An appropriate notion of integral is then introduced which allows to extend in a consistent way the integration by parts formula, the Gauss theorem and the notion of perimeter. This new space we introduce, seems suitable for applications to Partial Differential Equations and Calculus of Variations. This fact will be illustrated by a simple, but meaningful example.

In this paper, we consider a production economy with an unbounded attainable set where the consumers may have non-complete non-transitive preferences. To get the existence of an equilibrium, we provide an asymptotic property on preferences for the attainable consumptions and we use a combination of the nonlinear optimization and fixed point theorems on truncated economies together with an asymptotic argument. We show that this condition holds true if the set of attainable allocations is compact or, when the preferences are representable by utility functions, if the set of attainable individually rational utility levels is compact. This assumption generalizes the CPP condition of [N. Allouch, An equilibrium existence result with short selling, J. Math. Econom. 37 2002, 2, 81–94] and covers the example of [F. H. Page, Jr., M. H. Wooders and P. K. Monteiro, Inconsequential arbitrage, J. Math. Econom. 34 2000, 4, 439–469] when the attainable utility levels set is not compact. So we extend the previous existence results with non-compact attainable sets in two ways by adding a production sector and considering general preferences.

We study the semilinear elliptic equation - Δ u = u α | log u | β   in  B 1 ∖ { 0 } , -\Delta u=u^{\alpha}\lvert\log u|^{\beta}\quad\text{in }B_{1}\setminus\{0\}, where B 1 ⊂ ℝ n {B_{1}\subset{\mathbb{R}}^{n}} , with n ≥ 3 {n\geq 3} , n n - 2 < α < n + 2 n - 2 {\frac{n}{n-2}<\alpha<\frac{n+2}{n-2}} and - ∞ < β < ∞ {-\infty<\beta<\infty} . Our main result establishes that the nonnegative solution u ∈ C 2 ⁢ ( B 1 ∖ { 0 } ) {u\in C^{2}(B_{1}\setminus\{0\})} of the above equation either has a removable singularity at the origin or it behaves like u ⁢ ( x ) = A ⁢ ( 1 + o ⁢ ( 1 ) ) ⁢ | x | - 2 α - 1 ⁢ ( log ⁡ 1 | x | ) - β α - 1   as  ⁢ x → 0 , u(x)=A(1+o(1))|x|^{-\frac{2}{\alpha-1}}\Bigl{(}\log\frac{1}{|x|}\Big{)}^{-% \frac{\beta}{\alpha-1}}\quad\text{as }x\rightarrow 0, with A = [ ( 2 α - 1 ) 1 - β ⁢ ( n - 2 - 2 α - 1 ) ] 1 α - 1 . {A=[(\frac{2}{\alpha-1})^{1-\beta}(n-2-\frac{2}{\alpha-1})]^{\frac{1}{\alpha-1% }}.}

In this paper we establish that the gradient of weak solutions to porous medium-type systems admits the self-improving property of higher integrability.

We present a geometric formula of Poincaré type, which is inspired by a classical work of Sternberg and Zumbrun, and we provide a classification result of stable solutions of linear elliptic problems with nonlinear Robin conditions on Riemannian manifolds with nonnegative Ricci curvature. The result obtained here is a refinement of a result recently established by Bandle, Mastrolia, Monticelli and Punzo.

We study obstacle problems involving p -Laplace-type operators in non-convex polygons. We establish regularity results in terms of weighted Sobolev spaces. As applications, we obtain estimates for the FEM approximation for obstacle problems in pre-fractal Koch Islands.

In this paper, we consider control systems governed by a class of semilinear parabolic equations, which are singular at the boundary and possess singular convection and reaction terms. The systems are shown to be null controllable by establishing Carleman estimates, observability inequalities and energy estimates for solutions to linearized equations.

Consider nonlinear Choquard equations { - Δ ⁢ u + u = ( I α * | u | p ) ⁢ | u | p - 2 ⁢ u in  ⁢ ℝ N , lim x → ∞ ⁡ u ⁢ ( x ) = 0 , \left\{\begin{aligned} \displaystyle-\Delta u+u&\displaystyle=(I_{\alpha}*% \lvert u\rvert^{p})\lvert u\rvert^{p-2}u&&\displaystyle\phantom{}\text{in }% \mathbb{R}^{N},\\ \displaystyle\lim_{x\to\infty}u(x)&\displaystyle=0,\end{aligned}\right. where I α {I_{\alpha}} denotes the Riesz potential and α ∈ ( 0 , N ) {\alpha\in(0,N)} . In this paper, we investigate limit profiles of ground states of nonlinear Choquard equations as α → 0 {\alpha\to 0} or α → N {\alpha\to N} . This leads to the uniqueness and nondegeneracy of ground states when α is sufficiently close to 0 or close to N .

In this paper we formulate a criterion for relative compactness in the space of functions regulated on a bounded and closed interval. We prove that the mentioned criterion is equivalent to a known criterion obtained earlier by D. Fraňkova, but it turns out to be very convenient in applications. Among others, it creates the basis to construct a regular measure of noncompactness in the space of regulated functions. We show the applicability of the constructed measure of noncompactness in proving the existence of solutions of a quadratic Hammerstein integral equation in the space of regulated functions.

This paper deals with the existence of nontrivial solutions for critical Hardy–Schrödinger–Kirchhoff systems driven by the fractional p -Laplacian operator. Existence is derived as an application of the mountain pass theorem and the Ekeland variational principle. The main features and novelty of the paper are the presence of the Hardy terms as well as critical nonlinearities.

We consider the mathematical treatment of a system of nonlinear partial differential equations based on a model, proposed in 1972 by J. Newman, in which the coupling between the Lithium concentration, the phase potentials and temperature in the electrodes and the electrolyte of a Lithium battery cell is considered. After introducing some functional spaces well-adapted to our framework, we obtain some rigorous results showing the well-posedness of the system, first for some short time and then, by considering some hypothesis on the nonlinearities, globally in time. As far as we know, this is the first result in the literature proving existence in time of the full Newman model, which follows previous results by the third author in 2016 regarding a simplified case.

We study the Cauchy problem for a system of equations corresponding to a singular limit of radiative hydrodynamics, namely, the 3D radiative compressible Euler system coupled to an electromagnetic field. Assuming smallness hypotheses for the data, we prove that the problem admits a unique global smooth solution and study its asymptotics.

In the present paper, we investigate the existence of solutions for the following inhomogeneous singular equation involving the p ⁢ ( x ) {p(x)} -biharmonic operator: { Δ ⁢ ( | Δ ⁢ u | p ⁢ ( x ) - 2 ⁢ Δ ⁢ u ) = g ⁢ ( x ) ⁢ u - γ ⁢ ( x ) ∓ λ ⁢ f ⁢ ( x , u ) in  ⁢ Ω , Δ ⁢ u = u = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} &\displaystyle\Delta(\lvert\Delta u\rvert^{p(x)-2}% \Delta u)=g(x)u^{-\gamma(x)}\mp\lambda f(x,u)&&\displaystyle\phantom{}\text{in% }\Omega,\\ &\displaystyle\Delta u=u=0&&\displaystyle\phantom{}\text{on }\partial\Omega,% \end{aligned}\right. where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} ( N ≥ 3 {N\geq 3} ) is a bounded domain with C 2 {C^{2}} boundary, λ is a positive parameter, γ : Ω ¯ → ( 0 , 1 ) {\gamma:\overline{\Omega}\rightarrow(0,1)} is a continuous function, p ∈ C ⁢ ( Ω ¯ ) {p\kern-1.0pt\in\kern-1.0ptC(\overline{\Omega})} with 1 < p - := inf x ∈ Ω ⁡ p ⁢ ( x ) ≤ p + := sup x ∈ Ω ⁡ p ⁢ ( x ) < N 2 {1\kern-1.0pt<\kern-1.0ptp^{-}\kern-1.0pt:=\kern-1.0pt\inf_{x\in\Omega}p(x)% \kern-1.0pt\leq\kern-1.0ptp^{+}\kern-1.0pt:=\kern-1.0pt\sup_{x\in\Omega}p(x)% \kern-1.0pt<\kern-1.0pt\frac{N}{2}} , as usual, p * ⁢ ( x ) = N ⁢ p ⁢ ( x ) N - 2 ⁢ p ⁢ ( x ) {p^{*}(x)\kern-1.0pt=\kern-1.0pt\frac{Np(x)}{N-2p(x)}} , g ∈ L p * ⁢ ( x ) p * ⁢ ( x ) + γ ⁢ ( x ) - 1 ⁢ ( Ω ) , g\in L^{\frac{p^{*}(x)}{p^{*}(x)+\gamma(x)-1}}(\Omega), and f ⁢ ( x , u ) {f(x,u)} is assumed to satisfy assumptions (f1)–(f6) in Section 3. In the proofs of our results, we use variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces. In addition, an example to illustrate our result is given.

We are concerned with the existence of ground states and qualitative properties of solutions for a class of nonlocal Schrödinger equations. We consider the case in which the nonlinearity exhibits critical growth in the sense of the Hardy–Littlewood–Sobolev inequality, in the range of the so-called upper-critical exponent. Qualitative behavior and concentration phenomena of solutions are also studied. Our approach turns out to be robust, as we do not require the nonlinearity to enjoy monotonicity nor Ambrosetti–Rabinowitz-type conditions, still using variational methods.

Let ( M , g ) {(M,g)} be a smooth compact Riemannian manifold of dimension n ≥ 5 {n\geq 5} . We are concerned with the following elliptic problem: - Δ g ⁢ u + a ⁢ ( x ) ⁢ u = u n + 2 n - 2 + ε , u > 0 ⁢  in  ⁢ M , -\Delta_{g}u+a(x)u=u^{\frac{n+2}{n-2}+\varepsilon},\quad u>0\text{ in }M, where Δ g = div g ⁢ ( ∇ ) {\Delta_{g}=\mathrm{div}_{g}(\nabla)} is the Laplace–Beltrami operator on M , a ⁢ ( x ) {a(x)} is a C 2 {C^{2}} function on M such that the operator - Δ g + a {-\Delta_{g}+a} is coercive, and ε > 0 {\varepsilon>0} is a small real parameter. Using the Lyapunov–Schmidt reduction procedure, we obtain that the problem under consideration has a k -peaks solution for positive integer k ≥ 2 {k\geq 2} , which blow up and concentrate at one point in M .

In this paper, we consider the analogous of the Hénon equation for the prescribed mean curvature problem in ℝ N {{\mathbb{R}^{N}}} , both in the Euclidean and in the Minkowski spaces. Motivated by the studies of Ni and Serrin [W. M. Ni and J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Att. Convegni Lincei 77 1985, 231–257], we have been interested in finding the relations between the growth of the potential and that of the local nonlinearity in order to prove the nonexistence of a radial ground state. We also present a partial result on the existence of a ground state solution in the Minkowski space.

In this paper, we are concerned with quasilinear equations with indefinite nonlinearity and explore the existence of infinitely many solutions.

Let Ω ⊂ ℝ 2 be a bounded domain with smooth boundary and b ( x ) > 0 a smooth function defined on ∂Ω . We study the following Robin boundary value problem: Δu+up=0in Ω,u>0in Ω,∂u∂ν+λb(x)u=0on ∂Ω, $$\begin{array}{} \displaystyle \left\{ \begin{alignedat}{2} &{\it\Delta} u+u^p=0 &\quad& \text{in }{\it\Omega},\\ &u>0 &\quad& \text{in }{\it\Omega},\\ &\frac{\partial u}{\partial\nu} +\lambda b(x)u=0 &\quad& \text{on } \partial{\it\Omega}, \end{alignedat} \right. \end{array}$$ where ν denotes the exterior unit vector normal to ∂Ω , 0 < λ < +∞ and p > 1 is a large exponent. We construct solutions of this problem which exhibit concentration as p → +∞ and simultaneously as λ → +∞ at points that get close to the boundary, and show that in general the set of solutions of this problem exhibits a richer structure than the problem with Dirichlet boundary condition.