On the existence of positive solutions of singular nonlinear elliptic equations with Dirichlet boundary conditions

This paper is concerned with the existence of positive solutions of the singular nonlinear elliptic equation with a Dirichlet boundary condition

     

On a Critical Fourth Order PDE with Navier Boundary Condition

We consider a fourth order nonlinear PDE involving the critical Sobolev exponent on a bounded domain of Rⁿ, n ≥ 5 with Navier condition on the boundary. We study the lack of compactness of the problem and we provide an existence theorem through a new index formula.     

Existence of solutions for a damped nonlinear impulsive problem with Dirichlet boundary conditions

In this paper, we consider the existence of solutions for second‐order nonlinear damped impulsive differential equations with Dirichlet boundary condition. By critical point theory, we obtain some existence theorems of solutions for the nonlinear problem. We extend and improve some recent results. Copyright © 2013 John Wiley & Sons, Ltd.     

On a nonlinear elliptic problem with critical potential in R~2

Consider the existence of nontrivial solutions of homogeneous Dirichlet problem for a nonlinear elliptic equation with the critical potential in R2. By establishing a weighted inequality with the best constant, determine the critical potential in R2, and study the eigenvalues of Laplace equation with the critical potential. By the Pohozaev identity of a solution with a singular point and the Cauchy-Kovalevskaya theorem, obtain the nonexis tence result of solutions with singular points to the nonlinear elliptic equation. Moreover, for the same problem, the existence results of multiple solutions are proved by the mountain pass theorem.     

On a nonhomogeneous,nonlinear Dirichlet eigenvalue problem

We consider a nonlinear eigenvalue problem for the Dirichlet $(p,q)$-Laplacian with a sign-changing Carath$\acute{\mathrm{e}}$odory reaction. Using variational tools, truncation and comparison techniques, and critical groups, we prove an existence and multiplicity result which is global in the parameter $\lambda >0$ (bifurcation-type theorem). Our work here complements the recent one by Papageorgiou–Qin–Rădulescu, Bull. Sci. Math. 172 (2021).     

The thermostat problem with a nonlocal nonlinear boundary condition

The thermostat controller for an air-conditioning system isusually placed in a position at some distance from the unitand this can lead to large swings in temperature. This paperaddresses this question by studying a paradigm—a one-dimensionalheat conduction equation with and without heat loss, and wherethe flux of heat extracted or input by the unit is consideredto be a function of the temperature at the other end. The essential results are that the system can be unstable andthat this is exacerbated both by a more powerful air-conditioningunit and by more efficient insulation.     

On nonlinear boundary value problem corresponding to N-dimensional inverse spectral problem

We establish a relationship between an inverse optimization spectral problem for the N-dimensional Schrödinger equation $?\mathrm{\Delta }?+q(x)?=\lambda ?$ and a solution of the nonlinear boundary value problem $?\mathrm{\Delta }u+q(x)u=\lambda u?u^{\gamma ?1},u>0,u{|}_{?\mathrm{\Omega }}=0$. Using this relationship, we find an exact solution for the inverse optimization spectral problem, investigate its stability and obtain new results on the existence and uniqueness of the solution for the nonlinear boundary value problem.     

Multiple solutions for elliptic equations at resonance

We study an asymptotically linear elliptic equation at resonance, with an odd nonlinearity. By a penalization technique and suitable min-max theorems (which give Morse index estimates), we prove the existence of pairs of non trivial solutions, where N is, roughly speaking, the difference between the Morse indexes at zero and at infinity. Received December 1999     

On a Nirenberg-type problem involving the square root of the Laplacian

Through an Euler–Hopf-type formula, we establish existence result to a Nirenberg-type problem involving the square root of the Laplacian in sphere S²$S^2$.     

Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition

We study non-negative solutions of the porous medium equationwith a source and a nonlinear flux boundary condition, u_t =(u^m)_xx + u^p in (0, ), x (0, T); – (u^m)_x (0, t) = u^q (0,t) for t (0, T); u (x, 0) = u₀ (x) in (0, ), where m > 1,p, q > 0 are parameters. For every fixed m we prove thatthere are two critical curves in the (p, q-plane: (i) the criticalexistence curve, separating the region where every solutionis global from the region where there exist blowing-up solutions,and (ii) the Fujita curve, separating a region of parametersin which all solutions blow up from a region where both globalin time solutions and blowing-up solutions exist. In the caseof blow up we find the blow-up rates, the blow-up sets and theblow-up profiles, showing that there is a phenomenon of asymptoticsimplification. If 2q < p + m the asymptotics are governedby the source term. On the other hand, if 2q > p + m theevolution close to blow up is ruled by the boundary flux. If2q = p + m both terms are of the same order.     

A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function

In this paper, we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system (Eλ,μ) involving nonlinear boundary condition and sign-changing weight function. With the help of the Nehari manifold, we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters (λ,μ) belongs to a certain subset of R².     

On a perturbed non-local boundary value problem: a multiplicity result

In the present paper, we deal with a non-local boundary value problem with an exponential non-linearity. The existence of one, two or three solutions is investigated under the presence of a suitable perturbation. Our approach is variational and combines results from critical point theory.     

A generalized Minkowski problem with Dirichlet boundary condition

We prove the existence of hypersurfaces with prescribed boundary whose Weingarten curvature equals a given function that depends on the normal of the hypersurface.

     

Projective Dirichlet boundary condition with applications to a geometric problem

Given a domain Ω ? R~n, let λ 0 be an eigenvalue of the elliptic operator L :=Σ!(i,j)~n =1?/?xi(a~(ij0 ?/?xj) on Ω for Dirichlet condition. For a function f ∈ L~2(Ω), it is known that the linear resonance equation Lu + λu = f in Ω with Dirichlet boundary condition is not always solvable.We give a new boundary condition P_λ(u|? Ω) = g, called to be pro jective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ||u||2,2 ≤ C(||f ||_2 +|| g||_(2,2)) under suitable regularity assumptions on ?Ω and L, where C is a constant depends only on n, Ω, and L. More a priori estimates,such as W~(2,p)-estimates and the C~(2,α)-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean(Cartan) torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.