Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth (Q5951618)

The authors of this paper investigate the existence of linking type solutions for the problem NEWLINE\[NEWLINE\begin{cases} -\Delta u(x)-\lambda u(x) = W(x)f(u) & \text{in }\Omega\subset \mathbb{R}^N,\\ u(x)|_{\partial \Omega}=0\end{cases}\tag{1}NEWLINE\]NEWLINE where \(w\in C(\overline\Omega)\) is a changing sign function, \(f\) has a superlinear growth and \(\lambda\) is a positive real parameter.NEWLINENEWLINENEWLINELet \(0<\lambda_1<\lambda_2\leq \cdots\leq \lambda_k\leq \lambda_{k+1}\leq \cdots\) be the sequence of eigenvalues of the operator \(-\Delta\) with respect to the zero boundary conditions on \(\Omega\). \(X_k\) denotes the \(k\)-dimensional subspace of the Sobolev space \(H^1_0(\Omega)\) spanned by the eigenfunctions related to \(\{\lambda_1,\dots,\lambda_k\}\). It is assumed that \(f\in C^0(\mathbb{R})\) and put \(F(t)=\int^t_0 f(\xi) d\xi\), \(t\in\mathbb{R}\) and, for \(N\geq 3\), \(2^*=\frac{2N}{N-2}\).NEWLINENEWLINENEWLINEOne of the main results states that the problem (1) admits a nontrivial solution \(u\) if the following conditions hold: NEWLINENEWLINENEWLINE(i) \(f(t)t\geq \beta F(t)\), \(t\in \mathbb{R}\); (ii) \(|f(t)|\leq C|t|^{\beta-1}\), \(t\in\mathbb{R}\), for some \(\beta \in (2,2^*)\); (iii) if \(k\) is the positive integer number such that \(\lambda\in [\lambda_k,\lambda_{k+1})\), then \(W\) verify \(W^-(f(t)t-\beta F(t))\leq \gamma(t)^2\) \(t\geq R>0\) sufficiently large for some \(\gamma\in (0,(\frac\beta 2-1)(\lambda_{k+1}-\lambda))\), where \(W^-=\max\{W^-(x):x\in\overline{\Omega}\}\), \(W^-(x)=-\min\{W(x),0\}\), \(x\in\Omega\); (iv) \(\text{meas}\{x\in\Omega: W(x)=0\}=0\); (v) \(W^+(x)=W(x)+W^-(x)\neq 0\); (vi) \(\int_\Omega W(x)F(v(x))dx\geq 0\), \(v\in X_k\); (vii) exists \(\overline v\in X^\perp_k\setminus\{0\}:\int_\Omega W(x)F(v(x)) dx\geq C_0\int_\Omega |v(x)|^\beta dx,\) \(v\in X_k\oplus\text{span}\{\overline v\}\), with \(\|v\|\geq R\).NEWLINENEWLINENEWLINEAnother result also states that the problem (1) admits a nontrivial solution \(u\) if the previous assumptions are satisfied with \(\beta=2^*\) and for \(N\geq 5\), \(F\) is convex, \(W\in C^3(\overline{\Omega})\), and \(\lim_{\varepsilon\to 0}(\varepsilon^{\frac{N+2}{2}})=|s|^{2^*-2}s\), uniformly w.r. to \(s\in\mathbb{R}\), NEWLINE\[NEWLINE\lim_{\mu\to 0}\mu^{\frac{N-2}{4}}\int^{\frac{1}{\sqrt{\mu}}}_0 \;\frac{\rho^2-1}{(1+\rho^2)^{N/2}}\left[f\left(\frac{\mu^{\frac{2-N}{4}}}{(1+p^2)^{\frac{N-2}{2}}}-\frac{\mu^{-\frac{N+ 2}{4}}}{(1+\rho^2)^{\frac{N+2}{2}}}\right)\right]\;\rho^{N-1} d\rho=0.NEWLINE\]NEWLINE In order to prove these results, it is shown that the functional \(I(u)=\frac 12\int_\Omega|\nabla u|^2 dx-\frac\lambda 2\int_\Omega u^2 dx-\int_\Omega W(x)F(u) dx\), \(u\in H^1_0(\Omega)\) satisfies the Palais-Smale condition, and also, the behaviour of Palais-Smale sequences is studied.