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We are concerned with the study of a class of non-autonomous eigenvalue problems driven by two non-homogeneous differential operators with variable ( p 1 , p 2 ) {(p_{1},p_{2})} -growth. The main result of this paper establishes the existence of a continuous spectrum consisting in an unbounded interval and the nonexistence of eigenvalues in a neighbourhood of the origin. The abstract results of this paper are described by two Rayleigh-type quotients and the proofs rely on variational arguments.

This paper ascertains the exact boundary blow-up rates of the large positive solutions of a class of cooperative logistic systems involving n species in a general domain of ℝ N {\mathbb{R}^{N}} of class 𝒞 2 + ν {\mathcal{C}^{2+\nu}} , 0 < ν < 1 {0 . The problem models a population divided in groups whose individuals compete with those of the same group, while simultaneously they cooperate with the members of the remaining groups. Our main result provides with the exact blow-up rates along the edges of Ω from the values of the blow-up rates of the underlying uncoupled system. Rather astonishingly, these blow-up rates are independent of the strength of the cooperative effects, which play a secondary role in the analysis carried out in this paper. No previous result of this nature is available in the specialized literature for more than n = 2 {n=2} species.

This paper deals with maximum principles depending on the domain and ABP estimates associated to the following Lane–Emden system involving fractional Laplace operators: { ( - Δ ) s ⁢ u = λ ⁢ ρ ⁢ ( x ) ⁢ | v | α - 1 ⁢ v in ⁢ Ω , ( - Δ ) t ⁢ v = μ ⁢ τ ⁢ ( x ) ⁢ | u | β - 1 ⁢ u in ⁢ Ω , u = v = 0 in ⁢ ℝ n ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&\displaystyle=\lambda\rho(x% )\lvert v\rvert^{\alpha-1}v&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle(-\Delta)^{t}v&\displaystyle=\mu\tau(x)\lvert u\rvert^{\beta-1}u&% &\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=v=0&&\displaystyle\phantom{}\text{in }\mathbb{R}% ^{n}\setminus\Omega,\end{aligned}\right. where s , t ∈ ( 0 , 1 ) {s,t\in(0,1)} , α , β > 0 {\alpha,\beta>0} satisfy α ⁢ β = 1 {\alpha\beta=1} , Ω is a smooth bounded domain in ℝ n {\mathbb{R}^{n}} , n ≥ 1 {n\geq 1} , and ρ and τ are continuous functions on Ω ¯ {\overline{\Omega}} and positive in Ω. We establish some maximum principles depending on Ω. In particular, we explicitly characterize the measure of Ω for which the maximum principles corresponding to this problem hold in Ω. For this, we derived an explicit lower estimate of principal eigenvalues in terms of the measure of Ω. Aleksandrov–Bakelman–Pucci (ABP) type estimates for the above systems are also proved. We also show the existence of a viscosity solution for a nonlinear perturbation of the nonhomogeneous counterpart of the above problem with polynomial and exponential growths. As an application of the maximum principles, we measure explicitly how small | Ω | {\lvert\Omega\rvert} has to be to ensure the positivity of the obtained solutions.

In this paper, we study the Cauchy problem of the parabolic Monge–Ampère equation - u t ⁢ det ⁡ D 2 ⁢ u = f ⁢ ( x , t ) -u_{t}\det D^{2}u=f(x,t) and obtain the existence and uniqueness of viscosity solutions with asymptotic behavior by using the Perron method.

In this paper, the existence of smooth positive solutions to a Robin boundary-value problem with non-homogeneous differential operator and reaction given by a nonlinear convection term plus a singular one is established. Proofs chiefly exploit sub-super-solution and truncation techniques, set-valued analysis, recursive methods, nonlinear regularity theory, as well as fixed point arguments. A uniqueness result is also presented.

Our purpose in this paper is to study positive solutions of the Lane–Emden equation - Δ ⁢ u = V ⁢ u p in ⁢ ℝ N ∖ { 0 } , -\Delta u=Vu^{p}\quad\text{in }\mathbb{R}^{N}\setminus\{0\}, perturbed by a nonhomogeneous potential V, with p ∈ ( N N - 2 , p c ) p\in(\frac{N}{N-2},p_{c}) , where p c {p_{c}} is the Joseph–Ludgren exponent. We construct a sequence of fast and slow decaying solutions with appropriated restrictions for V.

In this paper, we study the following nonlinear Schrödinger–Newton type system: { - ϵ 2 ⁢ Δ ⁢ u + u - Φ ⁢ ( x ) ⁢ u = Q ⁢ ( x ) ⁢ | u | ⁢ u , x ∈ ℝ 3 , - ϵ 2 ⁢ Δ ⁢ Φ = u 2 , x ∈ ℝ 3 , \left\{\begin{aligned} &\displaystyle{-}\epsilon^{2}\Delta u+u-\Phi(x)u=Q(x)|u% |u,&&\displaystyle x\in\mathbb{R}^{3},\\ &\displaystyle{-}\epsilon^{2}\Delta\Phi=u^{2},&&\displaystyle x\in\mathbb{R}^{% 3},\end{aligned}\right. where ϵ > 0 {\epsilon>0} and Q ⁢ ( x ) {Q(x)} is a positive bounded continuous potential on ℝ 3 {\mathbb{R}^{3}} satisfying some suitable conditions. By applying the finite-dimensional reduction method, we prove that for any positive integer k, the system has a positive solution with k-peaks concentrating near a strict local minimum point x 0 {x_{0}} of Q ⁢ ( x ) {Q(x)} in ℝ 3 {\mathbb{R}^{3}} , provided that ϵ > 0 {\epsilon>0} is sufficiently small.

We consider the following Choquard equation: { - Δ ⁢ u = ( ∫ Ω | u ⁢ ( y ) | 2 μ * | x - y | μ ⁢ 𝑑 y ) ⁢ | u | 2 μ * - 2 ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=\Bigg{(}\int_{% \Omega}\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}\,dy\Bigg{)}|u|^{2^{*}_{\mu}-2}% u&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a smooth bounded domain in ℝ N {\mathbb{R}^{N}} ( N ≥ 3 {N\geq 3} ), 2 μ * = 2 ⁢ N - μ N - 2 {2^{*}_{\mu}=\frac{2N-\mu}{N-2}} . This paper is concerned with the existence of a positive high-energy solution of the above problem in an annular-type domain when the inner hole is sufficiently small.

This paper deals with the general Choquard equation - Δ ⁢ u + V ⁢ ( | x | ) ⁢ u = ( I α * | u | p ) ⁢ | u | p - 2 ⁢ u in ⁢ ℝ N , -\Delta u+V(|x|)u=(I_{\alpha}*|u|^{p})|u|^{p-2}u\quad\text{in }\mathbb{R}^{N}, where V ∈ C ⁢ ( [ 0 , ∞ ) , ℝ + ) {V\in C([0,\infty),\mathbb{R}^{+})} is bounded below by a positive constant, and I α {I_{\alpha}} denotes the Riesz potential of order α ∈ ( 0 , N ) {\alpha\in(0,N)} . In view of the convolution term, the nonlocal property makes the variational functional completely different from the one for local pure power-type nonlinearity. By combining the Brouwer degree and developing some new techniques, a family of radial solutions with a prescribed number of zeros is constructed for p ∈ [ 2 , N + α N - 2 ) {p\in[2,\frac{N+\alpha}{N-2})} , while the degeneracy happens for p ∈ ( N + α N , 2 ) {p\in(\frac{N+\alpha}{N},2)} . This result complements and improves the ones in the literature in the aspect of the range of p.

This paper is concerned with the existence, uniqueness and asymptotic behavior of classical solutions to two classes of models - △ ⁢ u ± λ ⁢ | ∇ ⁡ u | 2 u β = b ⁢ ( x ) ⁢ u - α {-\triangle u\pm\lambda\frac{|\nabla u|^{2}}{u^{\beta}}=b(x)u^{-\alpha}} , u > 0 {u>0} , x ∈ Ω {x\in\Omega} , u | ∂ ⁡ Ω = 0 {u|_{\partial\Omega}=0} , where Ω is a bounded domain with smooth boundary in ℝ N {\mathbb{R}^{N}} , λ > 0 {\lambda>0} , β > 0 {\beta>0} , α > - 1 {\alpha>-1} , and b ∈ C loc ν ⁢ ( Ω ) {b\in C^{\nu}_{\mathrm{loc}}(\Omega)} for some ν ∈ ( 0 , 1 ) {\nu\in(0,1)} , and b is positive in Ω but may be vanishing or singular on ∂ ⁡ Ω {\partial\Omega} . Our approach is largely based on nonlinear transformations and the construction of suitable sub- and super-solutions.

This article ascertains the global structure of the diagram of positive solutions of a very general class of elliptic boundary value problems with spatial heterogeneities and nonlinear mixed boundary conditions, considering as bifurcation-continuation parameter a certain parameter γ that appears in the boundary conditions. In particular, in this work are obtained, in terms of such a parameter γ, the exact decay rate to zero and blow-up rate to infinity of the continuum of positive solutions of the problem, at the bifurcations from the trivial branch and from infinity. The new findings of this work complement, in some sense, those previously obtained for Robin linear boundary conditions by J. García-Melián, J. D. Rossi and J. C. Sabina de Lis in 2007. The main technical tools used to develop the mathematical analysis carried out in this paper are local and global bifurcation, continuation, comparison and monotonicity techniques and blow-up arguments.

In this paper, we obtain the existence and multiplicity of sign-changing solutions of the fractional p-Laplacian problems by applying the method of invariant sets of descending flow and minimax theory. In addition, we prove that the problem admits at least one least energy sign-changing solution by combining the Nehari manifold method with the constrained variational method and Brouwer degree theory. Furthermore, the least energy of sign-changing solutions is shown to exceed twice that of the least energy solutions.

We consider a parameter dependent parabolic logistic population model with diffusion and degenerate logistic term allowing for refuges for the population. The aim of the paper is to remove quite restrictive geometric and smoothness conditions on the refuge used in the existing literature. The key is a new simplified construction of a supersolution that does not make use of any regularity condition of the refuge. At the same time we also simplify other arguments commonly used in the literature.

In this paper, we consider the following parabolic variational inequality containing a multivalued term and a convex functional: Find u ∈ L p ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) {u\in L^{p}(0,T;W^{1,p}_{0}(\Omega))} and f ∈ F ⁢ ( ⋅ , ⋅ , u ) {f\in F(\cdot,\cdot,u)} such that u ⁢ ( ⋅ , 0 ) = u 0 {u(\cdot,0)=u_{0}} and 〈 u t + A ⁢ u , v - u 〉 + Ψ ⁢ ( v ) - Ψ ⁢ ( u ) ≥ ∫ Q f ⁢ ( v - u ) ⁢ 𝑑 x ⁢ 𝑑 t for all ⁢ v ∈ L p ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) , \langle u_{t}+Au,v-u\rangle+\Psi(v)-\Psi(u)\geq\int_{Q}f(v-u)\,dx\,dt\quad% \text{for all }v\in L^{p}(0,T;W^{1,p}_{0}(\Omega)), where A is the principal term; F is a multivalued lower-order term; Ψ ⁢ ( u ) = ∫ 0 T ψ ⁢ ( t , u ) ⁢ 𝑑 t {\Psi(u)=\int_{0}^{T}\psi(t,u)\,dt} is a convex functional. Moreover, we study the existence and other properties of solutions of this inequality assuming certain growth conditions on the lower-order term F.

In this paper, we are concerned with the existence and asymptotic behavior of minimizers of a minimization problem related to some quasilinear elliptic equations. Firstly, we prove that there exist minimizers when the exponent q is the critical one q * = 2 + 4 N {q^{*}=2+\frac{4}{N}} . Then, we prove that all minimizers are compact as q tends to the critical case q * {q^{*}} when a < a q * {a is fixed. Moreover, we find that all the minimizers must blow up as the exponent q tends to the critical case q * {q^{*}} for any fixed a > a q * {a>a_{q^{*}}} .

This paper discusses a high-order Melnikov method for periodically perturbed equations. We introduce a new method to compute M k ⁢ ( t 0 ) {M_{k}(t_{0})} for all k ≥ 0 {k\geq 0} , among which M 0 ⁢ ( t 0 ) {M_{0}(t_{0})} is the traditional Melnikov function, and M 1 ⁢ ( t 0 ) , M 2 ⁢ ( t 0 ) , … {M_{1}(t_{0}),M_{2}(t_{0}),\ldots\,} are its high-order correspondences. We prove that, for all k ≥ 0 {k\geq 0} , M k ⁢ ( t 0 ) {M_{k}(t_{0})} is a sum of certain multiple integrals, the integrand of which we can explicitly compute. In particular, we obtain explicit integral formulas for M 0 ⁢ ( t 0 ) {M_{0}(t_{0})} and M 1 ⁢ ( t 0 ) {M_{1}(t_{0})} . We also study a concrete equation for which the explicit formula of M 1 ⁢ ( t 0 ) {M_{1}(t_{0})} is used to prove the existence of a transversal homoclinic intersection in the case of M 0 ⁢ ( t 0 ) ≡ 0 {M_{0}(t_{0})\equiv 0} .

In this paper, we consider the existence of solutions to spinorial Yamabe type equations on S3 with a prescribed function H. We give a new condition about H under which at least one non-trivial solution exists. It is proved by a method based on Conley index theory applied to a reduced functional defined on a certain finite dimensional non-compact manifold. Our proof is also applicable to similar equations on Sm for all m ≥ 2.

In this paper we study multi-valued parabolic variational inequalities involving quasilinear parabolic operators, and multi-valued integral terms over the underlying parabolic cylinder as well as over parts of the lateral parabolic boundary, where the multi-valued functions involved are assumed to be upper semicontinuous only. Note, since lower semicontinuous multi-valued functions allow always for a Carathéodory selection, this case can be considered as the trivial case, and therefore will be omitted. Our main goal is threefold: First, we provide an analytical frame work and an existence theory for the problems under consideration. Unlike in recent publications on multi-valued parabolic variational inequalities, the closed convex set K representing the constraints is not required to possess a nonempty interior. Second, we prove enclosure and comparison results based on a recently developed sub-supersolution method due to the authors. Third, we consider classes of relevant generalized parabolic variational-hemivariational inequalities that will be shown to be special cases of the multi-valued parabolic variational inequalities under consideration.

Consider in a real Hilbert space H the differential equation (inclusion) (E): p(t)u″(t) + q(t)u′(t) ∈ Au(t) + f (t) for a.a. t ∈ ℝ+ = [0,∞), subject to the condition , where A: D(A) ⊂ H → H is a (possibly set-valued) maximal monotone operator whose range contains 0; p, q ∈ L∞(ℝ+), with ess inf p > 0 and either ess inf q > 0 or ess sup q < 0. Eq. (E) has been investigated under various assumptions by Barbu [1]-[2], Brezis [4], Véron [9]-[10] and others. Recently (see Moroşanu [8]), we proved existence and uniqueness of a strong solution to Eq. (E) subject to u(0) = x ∈ D(A) in the weighted space X = Lb 2(ℝ+; H), where b(t) = a(t)/p(t), a(t) = exp(∫0 t q(s)/p(s) ds), under our weak conditions on p and q (see above). Here we extend this result to the general case , while keeping the other assumptions unchanged, thus solving a long standing problem. In addition, the proof of our recent result, which is the starting point of the present paper, is considerably simplified. Furthermore, some qualitative properties of solutions are pointed out, an application to a minimization problem is given, and some open problems are formulated.

In this paper, the precompactness of minimizing sequences under multiconstraint conditions are discussed. This minimizing problem is related to a coupled nonlinear Schrödinger system which appears in the field of nonlinear optics. As a consequence of the compactness of each minimizing sequence, the orbital stability of the set of all minimizers is obtained.

In this paper we study semilinear variational inequalities driven by an elliptic operator not in divergence form modeled by where Ω is a bounded domain of RN, N ≥ 3, with smooth boundary, A is the elliptic operator, not in divergence form, given by Here aij, ai , i,j = 1,...,N , and a0 satisfy suitable regularity conditions, while 1 < s < 4/(N − 2) and the obstacle ψ is a function sufficiently smooth. Even if this problem is not variational in nature, we will prove the existence of non-trivial non-negative solutions for it, performing a variational approach combined with a penalization technique. This kind of approach seems to be new for problems of this type. We also prove a C1,α-regularity result for the solutions of our problem.

This paper is concerned with the following nonlinear Schrödinger equation: where f(x, t)t-1 < μ* := inf σ(-Δ + V) < f(x, t)t-1 < V(∞) = V(x). The existence of a positive solution and a least energy solution for problem (P) are discussed. Here, we assume neither that f(x, t) ≡ f(t), nor that f(x, t)t-1 is non-decreasing on t ≥ 0. Furthermore, we do not require that f(x, t) have a limit as |x| → +∞.