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In this paper, we prove existence and regularity of positive solutions for singular quasilinear elliptic systems involving gradient terms. Our approach is based on comparison properties, a priori estimates and Schauder's fixed point…
The present paper is concerned a class of quasi-linear elliptic degenerate equations. The degenerate operator comes from the analysis of manifolds with corner singularity. Variational methods are applied to verify the existence of infinity…
We deal with nonlinear weighted biharmonic problem in the unit ball of $\mathbb{R}^{4}$. The weight is of logarithm type. The nonlinearity is critical in view of Adam's inequalities in the weighted Sobolev space $W^{2,2}_{0}(B,w)$. We prove…
The aim of this paper is to prove the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint: \begin{align*} \left\{ \begin{array}{ll}…
Equations of motion corresponding to the H\'{e}non - Heiles system are considered. A method enabling one to find all elliptic solutions of an autonomous ordinary differential equation or a system of autonomous ordinary differential…
We establish the existence of positive solutions for a nonlinear elliptic Dirichlet problem in dimension $N$ involving the $N$-Laplacian. The nonlinearity considered depends on the gradient of the unknown function and an exponential term.…
We deal with the existence of infinitely many solutions for a class of elliptic problems with non-symmetric nonlinearities. Our result, which is motivated by a well known conjecture formulated by A. Bahri and P.L. Lions, suggests a new…
We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a W^{1,1}_0 solution which is distributional or entropic, according to the growth assumptions on a lower order term in…
We study a semilinear elliptic equation with a pure power nonlinearity with exponent $p>1$, and provide sufficient conditions for the existence of positive solutions. These conditions involve expected exit times from the domain, $D$, where…
We consider the existence and multiplicity of solutions for a class of quasi-linear Schr\"{o}dinger equations which include the modified nonlinear Schr\"{o}dinger equations. A new perturbation approach is used to treat the sub-cubic…
We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation…
A $p$-Laplacian elliptic problem in the presence of both strongly singular and $(p-1)$-superlinear nonlinearities is considered. We employ bifurcation theory, approximation techniques and sub-supersolution method to establish the existence…
Elliptic equation $(y')^2=a_0+a_2y^2+a_4y^4$ is the foundation of the elliptic function expansion method of finding exact solutions to nonlinear differential equation. In some references, some new form solutions to the elliptic equation…
We develop a unified framework for a broad class of nonlocal elliptic problems, encompassing a wide spectrum of nonlocal terms, including the classical Kirchhoff and Carrier-type equations as particular cases, and nonlinearities having…
In this paper, we consider the solvability of a class of nonlinear fourth order integro-differential equations with Navier boundary condition. We first deal with a corresponding linear problem and establish a maximum principle. Using the…
We provide new results on the existence, non-existence and multiplicity of non-negative radial solutions for semilinear elliptic systems with Neumann boundary conditions on an annulus. Our approach is topological and relies on the classical…
We consider nonlinear elliptic equations which contains global coupling as a nonlinear term. We classify the existence of all possible positive solutions to this problem.
New problem is studied that is to find nonlinear differential equations with special solutions expressed via the Weierstrass function. Method is discussed to construct nonlinear ordinary differential equations with exact solutions. Main…
In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic…
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.