Multiple positive solutions for a Kirchhoff type problem with a critical nonlinearity

In this paper, we study the multiplicity results of positive solutions for a Kirchhoff type problem with critical growth, with the help of the concentration compactness principle, and we prove that problem admits two positive solutions, and one of the solutions is a positive ground state solution.     

Multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin

In this paper, we investigate the existence of multiple solutions for a class of biharmonic equations where the nonlinearity involves a concave term at the origin. The solutions are obtained from the versions of mountain pass lemma and linking theorem.     

Multiplicity of solutions for a class of Kirchhoff type problems

In this paper we apply the （variant） fountain theorems to study the symmetric nonlinear Kirch- hoff nonlocal problems. Under the Ambrosetti-Rabinowitz＇s 4-superlinearity condition, or no Ambrosetti- Rabinowitz＇s 4-superlinearity condition, we present two results of existence of infinitely many large energy solutions, respectively.     

Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents

In this paper, we study multiplicity of positive solutions for a class of Kirchhoff type of equations with the nonlinearity containing both singularity and critical exponents. We obtain two positive solutions via the variational and perturbation methods.     

Multiple positive solutions for Kirchhoff type problems with singularity and asymptotically linear nonlinearities

In this paper, we devote ourselves to investigating a nonlocal problem involving singularity and asymptotically linear nonlinearities. By using the variational and perturbation methods, we obtain the existence of two positive solutions which improve the existing result in the literature.     

On a class of Kirchhoff type problems

In this paper we consider the following Kirchhoff type problem: $$(\mathcal{K}) \quad \left(1 + \lambda \int\limits_{\mathbb{R}^3}\big(|\nabla u|^2 + V(y)u^2dy\big)\right)[-\Delta u + V(x)u] = |u|^{p-2}u, \quad {\rm in} \, \mathbb{R}^3,$$ where ${p\in (2, 6)}$ , λ > 0 is a parameter, and V(x) is a given potential. Some existence and nonexistence results are obtained by using variational methods. Also, the “energy doubling” property of nodal solutions of ${(\mathcal{K})}$ is discussed in this paper.     

Existence and concentration of solutions for a class of biharmonic Kirchhoff equations with discontinuous nonlinearity

This paper concerns the existence and concentration of positive solutions for a class of biharmonic Kirchhoff equations with discontinuous nonlinearity. The proof relies on variational method, truncated methods, and nonsmooth critical points theory. Some related results are improved.     

Nontrivial solutions of Kirchhoff type problems

In this work, we study Kirchhoff type problems on a bounded domain. We consider the cases where the nonlinearity is superlinear near zero but asymptotically 4-linear at infinity, and the nonlinearity is asymptotically linear near zero but 4-superlinear at infinity. By computing the relevant critical groups, we obtain nontrivial solutions via Morse theory.     

Uniqueness of non-negative solutions for a semipositone problem with concave nonlinearity

We study a semilinear Dirichlet problem where the nonlinearlity is monotone, negative at zero (Semipositone), and concave. We assume that the outer boundary of the region is convex and show that for large values of a parameter there can be only one non-negative solution. We show that such a solution is positive in the region.     

Multiple solutions to p-Laplacian problems with concave nonlinearities

In this paper we study the p-Laplacian type elliptic problems with concave nonlinearities. Using some asymptotic behavior of f at zero and infinity, three nontrivial solutions are established.     

Multiple solutions for elliptic problems with asymmetric nonlinearity

In this paper we establish the existence of multiple solutions for the semilinear elliptic problem

     

Existence and multiplicity of positive solutions for a class of singular Kirchhoff type problems with sign‐changing potential

In this paper, the existence and multiplicity of positive solutions are obtained for a class of Kirchhoff type problems with two singular terms and sign‐changing potential by the Nehari method.     

Finitely many solutions for a class of boundary value problems with superlinear convex nonlinearity

We consider the nonlinear Sturm-Liouville problem –u = f(u) + h in (0, 1), u(0) = u(1) = 0, where h L²(0,1) and f is a positive convex nonlinearity with superlinear growth at infinity. Our main result establishes that the above boundary value problem admits a finite number of solutions but it cannot have infinitely many solutions.Received: 8 July 2004     

Existence of positive solutions to Kirchhoff type problems with zero mass

The existence of positive solutions depending on a nonnegative parameter λ to Kirchhoff type problems with zero mass is proved by using variational method, and the new result does not require usual compactness conditions. A priori estimate and a Pohozaev type identity are used to obtain the bounded Palais–Smale sequences for constant coefficient nonlinearity, while a cut-off functional and Pohozaev type identity are utilized to obtain the bounded Palais–Smale sequences for the variable-coefficient case.