In this paper, we introduce a direct method of moving spheres for the fractional Laplacian (−△)α/2 with 0<α<2, in which a key ingredient is the narrow region maximum principle. As immediate ...applications, we classify non-negative solutions for semilinear equations involving the fractional Laplacian in Rn; we prove a non-existence result for the prescribing Qα curvature equation on Sn; then by combining the direct method of moving planes and moving spheres, we establish a Liouville type theorem on a half Euclidean space. We expect to see more applications of this method to many other nonlinear equations involving non-local operators.
The main goal of this paper is the study of an elliptic obstacle problem with a double phase phenomena and a multivalued reaction term which also depends on the gradient of the solution. Such term is ...called multivalued convection term. Under quite general assumptions on the data, we prove that the set of weak solutions to our problem is nonempty, bounded and closed. Our proof is based on a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping.
We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak–Orlicz space avoiding growth restrictions. Namely, we consider−divA(x,∇u)=f∈L1(Ω), on a Lipschitz ...bounded domain in RN. The growth of the monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N-function M. The approach does not require any particular type of growth condition of M or its conjugate M⁎ (neither Δ2, nor ∇2). The condition we impose is log-Hölder continuity of M, which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.
Existence and uniqueness of a specific self-similar solution is established for the following reaction-diffusion equation with Hardy singular potential∂tu=Δum+|x|−2up,(x,t)∈RN×(0,∞), in the range of ...exponents 1≤p<m and dimension N≥3. The self-similar solution is unbounded at x=0 and has a logarithmic vertical asymptote, but it remains bounded at any x≠0 and t∈(0,∞) and it is a weak solution in L1 sense, which moreover satisfies u(t)∈Lp(RN) for any t>0 and p∈1,∞). As an application of this self-similar solution, it is shown that there exists at least a weak solution to the Cauchy problem associated to the previous equation for any bounded, nonnegative and compactly supported initial condition u0, contrasting with previous results in literature for the critical limit p=m.
This paper presents the existence of global solutions to the discrete Safronov-Dubvoskiǐ coagulation equations for a large class of coagulation kernels satisfying Λi,j=θiθj+κi,j with ...κi,j≤Aθiθj,∀i,j≥1 where the sequence (θi)i≥1 grows linearly or superlinearly with respect to i. Moreover, the failure of mass-conservation of the solution is also addressed which confirms the occurrence of the gelation phenomenon.
This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator LK and ...involving a critical nonlinearity. In particular, we consider the problem −M(||u||2)LKu=λf(x,u)+|u|2s∗−2uin Ω,u=0in Rn∖Ω, where Ω⊂Rn is a bounded domain, 2s∗ is the critical exponent of the fractional Sobolev space Hs(Rn), the function f is a subcritical term and λ is a positive parameter. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function M could be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature.
We prove existence and uniqueness of renormalized solutions to general nonlinear parabolic equation in Musielak–Orlicz space avoiding growth restrictions. Namely, we consider∂tu−divA(x,∇u)=f∈L1(ΩT), ...on a Lipschitz bounded domain in RN. The growth of the weakly monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N-function M. The approach does not require any particular type of growth condition of M or its conjugate M⁎ (neither Δ2, nor ∇2). The condition we impose on M is continuity of log-Hölder-type, which results in good approximation properties of the space. However, the requirement of regularity can be skipped in the case of reflexive spaces. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments. Uniqueness results from the comparison principle.
This paper establishes the existence of entire large positive radial solutions for a class of nonlinear equations and systems. The method employed is based on a successive approximation technique. ...Our results improve some existing works recently published.
We consider a second order PDEs system of Parabolic–Elliptic type with chemotactic terms. The system describes the evolution of a biological species “u” moving towards a higher concentration of a ...chemical stimuli “v” in a bounded and open domain of RN. In the system considered, the chemotaxis sensitivity depends on the gradient of v, i.e., the chemotaxis term has the following expression−div(χu|∇v|p−2∇v), where χ is a positive constant and p satisfiesp∈(1,∞), if N=1 and p∈(1,NN−1), if N≥2. We obtain uniform bounds in L∞(Ω) and the existence of global in time solutions. For the one-dimensional case we prove the existence of infinitely many non-constant steady-states for p∈(1,2) for any χ positive and a given positive mass.
In this article, by using the lower and upper solution method, we prove the existence of iterative solutions for a class of fractional initial value problem with non-monotone term $$\displaylines{ ...D_{0+}^\alpha u(t)=f(t, u(t)), \quad t \in (0, h), \cr t^{1-\alpha}u(t)\big|_{t=0} = u_0 \neq 0, }$$ where $0<h<+\infty$, $f\in C(0, h\times \mathbb{R}, \mathbb{R})$, $D_{0+}^\alpha u (t) $ is the standard Riemann-Liouville fractional derivative, $0<\alpha< 1$. A new condition on the nonlinear term is given to guarantee the equivalence between the solution of the IVP and the fixed-point of the corresponding operator. Moreover, we show the existence of maximal and minimal solutions.
PDF
